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

Percentage Accurate: 68.5% → 99.7%

Time: 17.2s

Alternatives: 24

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: 68.5% 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: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return (y * ((1.0 / (x + y)) * (x / (x + y)))) / (y - (-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 = (y * ((1.0d0 / (x + y)) * (x / (x + y)))) / (y - ((-1.0d0) - x))
end function

Java

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

Python

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

Julia

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

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = (y * ((1.0 / (x + y)) * (x / (x + y)))) / (y - (-1.0 - 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[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y - N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.3%

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

   1. associate-/l* 79.6%

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

   2. associate-+l+ 79.6%

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

3. Simplified 79.6%

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

   1. associate-*r/ 67.3%

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

   2. associate-+r+ 67.3%

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

   3. times-frac 86.7%

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

   4. associate-*r/ 86.7%

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

   5. pow2 86.7%

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

   6. +-commutative 86.7%

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

   7. associate-+r+ 86.7%

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

   8. +-commutative 86.7%

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

   9. associate-+l+ 86.7%

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

6. Applied egg-rr 86.7%

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

   1. *-un-lft-identity 86.7%

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

   2. unpow2 86.7%

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

   3. times-frac 99.7%

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

   4. +-commutative 99.7%

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

   5. +-commutative 99.7%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 81.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x + y}\\ \mathbf{if}\;x \leq -7 \cdot 10^{+17}:\\ \;\;\;\;\frac{t\_0}{x + y}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y}\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-46}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -5.1 \cdot 10^{-101}:\\ \;\;\;\;\frac{1}{\frac{y \cdot \left(y + 1\right)}{x}}\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-109}:\\ \;\;\;\;t\_0 \cdot \frac{-1}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y - \left(-1 - x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ x y))))
   (if (<= x -7e+17)
     (/ t_0 (+ x y))
     (if (<= x -5.2e-6)
       (/ (/ x (+ x y)) y)
       (if (<= x -2.6e-46)
         (/ y (* x (+ x 1.0)))
         (if (<= x -5.1e-101)
           (/ 1.0 (/ (* y (+ y 1.0)) x))
           (if (<= x -2.6e-109)
             (* t_0 (/ -1.0 (- -1.0 x)))
             (/ (/ x y) (- y (- -1.0 x))))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x + y);
	double tmp;
	if (x <= -7e+17) {
		tmp = t_0 / (x + y);
	} else if (x <= -5.2e-6) {
		tmp = (x / (x + y)) / y;
	} else if (x <= -2.6e-46) {
		tmp = y / (x * (x + 1.0));
	} else if (x <= -5.1e-101) {
		tmp = 1.0 / ((y * (y + 1.0)) / x);
	} else if (x <= -2.6e-109) {
		tmp = t_0 * (-1.0 / (-1.0 - x));
	} else {
		tmp = (x / y) / (y - (-1.0 - 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) :: t_0
    real(8) :: tmp
    t_0 = y / (x + y)
    if (x <= (-7d+17)) then
        tmp = t_0 / (x + y)
    else if (x <= (-5.2d-6)) then
        tmp = (x / (x + y)) / y
    else if (x <= (-2.6d-46)) then
        tmp = y / (x * (x + 1.0d0))
    else if (x <= (-5.1d-101)) then
        tmp = 1.0d0 / ((y * (y + 1.0d0)) / x)
    else if (x <= (-2.6d-109)) then
        tmp = t_0 * ((-1.0d0) / ((-1.0d0) - x))
    else
        tmp = (x / y) / (y - ((-1.0d0) - x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (x + y);
	double tmp;
	if (x <= -7e+17) {
		tmp = t_0 / (x + y);
	} else if (x <= -5.2e-6) {
		tmp = (x / (x + y)) / y;
	} else if (x <= -2.6e-46) {
		tmp = y / (x * (x + 1.0));
	} else if (x <= -5.1e-101) {
		tmp = 1.0 / ((y * (y + 1.0)) / x);
	} else if (x <= -2.6e-109) {
		tmp = t_0 * (-1.0 / (-1.0 - x));
	} else {
		tmp = (x / y) / (y - (-1.0 - x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (x + y)
	tmp = 0
	if x <= -7e+17:
		tmp = t_0 / (x + y)
	elif x <= -5.2e-6:
		tmp = (x / (x + y)) / y
	elif x <= -2.6e-46:
		tmp = y / (x * (x + 1.0))
	elif x <= -5.1e-101:
		tmp = 1.0 / ((y * (y + 1.0)) / x)
	elif x <= -2.6e-109:
		tmp = t_0 * (-1.0 / (-1.0 - x))
	else:
		tmp = (x / y) / (y - (-1.0 - x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(x + y))
	tmp = 0.0
	if (x <= -7e+17)
		tmp = Float64(t_0 / Float64(x + y));
	elseif (x <= -5.2e-6)
		tmp = Float64(Float64(x / Float64(x + y)) / y);
	elseif (x <= -2.6e-46)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (x <= -5.1e-101)
		tmp = Float64(1.0 / Float64(Float64(y * Float64(y + 1.0)) / x));
	elseif (x <= -2.6e-109)
		tmp = Float64(t_0 * Float64(-1.0 / Float64(-1.0 - x)));
	else
		tmp = Float64(Float64(x / y) / Float64(y - Float64(-1.0 - x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (x + y);
	tmp = 0.0;
	if (x <= -7e+17)
		tmp = t_0 / (x + y);
	elseif (x <= -5.2e-6)
		tmp = (x / (x + y)) / y;
	elseif (x <= -2.6e-46)
		tmp = y / (x * (x + 1.0));
	elseif (x <= -5.1e-101)
		tmp = 1.0 / ((y * (y + 1.0)) / x);
	elseif (x <= -2.6e-109)
		tmp = t_0 * (-1.0 / (-1.0 - x));
	else
		tmp = (x / y) / (y - (-1.0 - x));
	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[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7e+17], N[(t$95$0 / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.2e-6], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, -2.6e-46], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.1e-101], N[(1.0 / N[(N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.6e-109], N[(t$95$0 * N[(-1.0 / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y - N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -7e17

   1. Initial program 55.7%

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

   3. Taylor expanded in x around inf 55.7%

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

      1. *-commutative 55.7%

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

      2. associate-*l* 55.7%

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

      3. +-commutative 55.7%

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

      4. times-frac 77.7%

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

      5. +-commutative 77.7%

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

   5. Applied egg-rr 77.7%

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

      1. associate-*l/ 77.7%

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

      2. frac-2neg 77.7%

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

      3. *-commutative 77.7%

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

   7. Applied egg-rr 77.7%

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

      1. associate-/r* 88.6%

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

      2. *-inverses 88.6%

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

      3. associate-*r/ 88.6%

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

      4. *-rgt-identity 88.6%

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

      5. distribute-neg-in 88.6%

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

      6. unsub-neg 88.6%

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

   9. Simplified 88.6%

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

   if -7e17 < x < -5.20000000000000019e-6

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      3. times-frac 99.4%

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

      4. +-commutative 99.4%

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

      5. +-commutative 99.4%

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

      6. associate-+r+ 99.4%

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

      7. +-commutative 99.4%

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

      8. associate-+l+ 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. clear-num 99.4%

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

      2. associate-/r* 98.8%

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

      3. frac-times 99.1%

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

      4. *-un-lft-identity 99.1%

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

      5. +-commutative 99.1%

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

      6. +-commutative 99.1%

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

      7. +-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in y around inf 60.7%

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

   if -5.20000000000000019e-6 < x < -2.6000000000000002e-46

   1. Initial program 99.8%

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

      1. associate-/l* 99.6%

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

      2. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 86.0%

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

   if -2.6000000000000002e-46 < x < -5.1000000000000002e-101

   1. Initial program 87.6%

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

      1. associate-/l* 88.0%

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

      2. associate-+l+ 88.0%

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

   3. Simplified 88.0%

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

   5. Taylor expanded in x around 0 92.6%

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

      1. +-commutative 92.6%

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

   7. Simplified 92.6%

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

      1. *-un-lft-identity 92.6%

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

      2. times-frac 91.7%

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

   9. Applied egg-rr 91.7%

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

       1. frac-times 92.6%

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

       2. *-un-lft-identity 92.6%

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

       3. clear-num 92.6%

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

       4. +-commutative 92.6%

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

   11. Applied egg-rr 92.6%

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

   if -5.1000000000000002e-101 < x < -2.5999999999999998e-109

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.2%

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

      3. times-frac 99.2%

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

      4. +-commutative 99.2%

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

      5. +-commutative 99.2%

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

      6. associate-+r+ 99.2%

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

      7. +-commutative 99.2%

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

      8. associate-+l+ 99.2%

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in y around 0 37.2%

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

      1. +-commutative 37.2%

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

   7. Simplified 37.2%

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

   if -2.5999999999999998e-109 < x

   1. Initial program 67.8%

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

      1. associate-/l* 80.6%

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

      2. associate-+l+ 80.6%

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

   3. Simplified 80.6%

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

      1. associate-*r/ 67.8%

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

      2. associate-+r+ 67.8%

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

      3. times-frac 87.7%

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

      4. associate-*r/ 87.8%

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

      5. pow2 87.8%

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

      6. +-commutative 87.8%

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

      7. associate-+r+ 87.8%

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

      8. +-commutative 87.8%

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

      9. associate-+l+ 87.8%

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

   6. Applied egg-rr 87.8%

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

      1. *-un-lft-identity 87.8%

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

      2. unpow2 87.8%

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

      3. times-frac 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0 66.2%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y + \left(1 + x\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + y}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y}\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-46}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -5.1 \cdot 10^{-101}:\\ \;\;\;\;\frac{1}{\frac{y \cdot \left(y + 1\right)}{x}}\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{-1}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y - \left(-1 - x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x + y}\\ t_1 := y - \left(-1 - x\right)\\ \mathbf{if}\;x \leq -1.85 \cdot 10^{+16}:\\ \;\;\;\;\frac{t\_0}{x + y}\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{-1}{-1 - y}}{\frac{x + y}{x}}\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-46}:\\ \;\;\;\;\frac{y \cdot \frac{1}{x}}{t\_1}\\ \mathbf{elif}\;x \leq -5.6 \cdot 10^{-101}:\\ \;\;\;\;\frac{1}{\frac{y \cdot \left(y + 1\right)}{x}}\\ \mathbf{elif}\;x \leq -1.06 \cdot 10^{-109}:\\ \;\;\;\;t\_0 \cdot \frac{-1}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ x y))) (t_1 (- y (- -1.0 x))))
   (if (<= x -1.85e+16)
     (/ t_0 (+ x y))
     (if (<= x -1.75e-9)
       (/ (/ -1.0 (- -1.0 y)) (/ (+ x y) x))
       (if (<= x -1.9e-46)
         (/ (* y (/ 1.0 x)) t_1)
         (if (<= x -5.6e-101)
           (/ 1.0 (/ (* y (+ y 1.0)) x))
           (if (<= x -1.06e-109)
             (* t_0 (/ -1.0 (- -1.0 x)))
             (/ (/ x y) t_1))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x + y);
	double t_1 = y - (-1.0 - x);
	double tmp;
	if (x <= -1.85e+16) {
		tmp = t_0 / (x + y);
	} else if (x <= -1.75e-9) {
		tmp = (-1.0 / (-1.0 - y)) / ((x + y) / x);
	} else if (x <= -1.9e-46) {
		tmp = (y * (1.0 / x)) / t_1;
	} else if (x <= -5.6e-101) {
		tmp = 1.0 / ((y * (y + 1.0)) / x);
	} else if (x <= -1.06e-109) {
		tmp = t_0 * (-1.0 / (-1.0 - x));
	} else {
		tmp = (x / y) / t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = y / (x + y)
    t_1 = y - ((-1.0d0) - x)
    if (x <= (-1.85d+16)) then
        tmp = t_0 / (x + y)
    else if (x <= (-1.75d-9)) then
        tmp = ((-1.0d0) / ((-1.0d0) - y)) / ((x + y) / x)
    else if (x <= (-1.9d-46)) then
        tmp = (y * (1.0d0 / x)) / t_1
    else if (x <= (-5.6d-101)) then
        tmp = 1.0d0 / ((y * (y + 1.0d0)) / x)
    else if (x <= (-1.06d-109)) then
        tmp = t_0 * ((-1.0d0) / ((-1.0d0) - x))
    else
        tmp = (x / y) / t_1
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (x + y);
	double t_1 = y - (-1.0 - x);
	double tmp;
	if (x <= -1.85e+16) {
		tmp = t_0 / (x + y);
	} else if (x <= -1.75e-9) {
		tmp = (-1.0 / (-1.0 - y)) / ((x + y) / x);
	} else if (x <= -1.9e-46) {
		tmp = (y * (1.0 / x)) / t_1;
	} else if (x <= -5.6e-101) {
		tmp = 1.0 / ((y * (y + 1.0)) / x);
	} else if (x <= -1.06e-109) {
		tmp = t_0 * (-1.0 / (-1.0 - x));
	} else {
		tmp = (x / y) / t_1;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (x + y)
	t_1 = y - (-1.0 - x)
	tmp = 0
	if x <= -1.85e+16:
		tmp = t_0 / (x + y)
	elif x <= -1.75e-9:
		tmp = (-1.0 / (-1.0 - y)) / ((x + y) / x)
	elif x <= -1.9e-46:
		tmp = (y * (1.0 / x)) / t_1
	elif x <= -5.6e-101:
		tmp = 1.0 / ((y * (y + 1.0)) / x)
	elif x <= -1.06e-109:
		tmp = t_0 * (-1.0 / (-1.0 - x))
	else:
		tmp = (x / y) / t_1
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(x + y))
	t_1 = Float64(y - Float64(-1.0 - x))
	tmp = 0.0
	if (x <= -1.85e+16)
		tmp = Float64(t_0 / Float64(x + y));
	elseif (x <= -1.75e-9)
		tmp = Float64(Float64(-1.0 / Float64(-1.0 - y)) / Float64(Float64(x + y) / x));
	elseif (x <= -1.9e-46)
		tmp = Float64(Float64(y * Float64(1.0 / x)) / t_1);
	elseif (x <= -5.6e-101)
		tmp = Float64(1.0 / Float64(Float64(y * Float64(y + 1.0)) / x));
	elseif (x <= -1.06e-109)
		tmp = Float64(t_0 * Float64(-1.0 / Float64(-1.0 - x)));
	else
		tmp = Float64(Float64(x / y) / t_1);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (x + y);
	t_1 = y - (-1.0 - x);
	tmp = 0.0;
	if (x <= -1.85e+16)
		tmp = t_0 / (x + y);
	elseif (x <= -1.75e-9)
		tmp = (-1.0 / (-1.0 - y)) / ((x + y) / x);
	elseif (x <= -1.9e-46)
		tmp = (y * (1.0 / x)) / t_1;
	elseif (x <= -5.6e-101)
		tmp = 1.0 / ((y * (y + 1.0)) / x);
	elseif (x <= -1.06e-109)
		tmp = t_0 * (-1.0 / (-1.0 - x));
	else
		tmp = (x / y) / t_1;
	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[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y - N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.85e+16], N[(t$95$0 / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.75e-9], N[(N[(-1.0 / N[(-1.0 - y), $MachinePrecision]), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.9e-46], N[(N[(y * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[x, -5.6e-101], N[(1.0 / N[(N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.06e-109], N[(t$95$0 * N[(-1.0 / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -1.85e16

   1. Initial program 55.7%

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

   3. Taylor expanded in x around inf 55.7%

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

      1. *-commutative 55.7%

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

      2. associate-*l* 55.7%

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

      3. +-commutative 55.7%

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

      4. times-frac 77.7%

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

      5. +-commutative 77.7%

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

   5. Applied egg-rr 77.7%

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

      1. associate-*l/ 77.7%

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

      2. frac-2neg 77.7%

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

      3. *-commutative 77.7%

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

   7. Applied egg-rr 77.7%

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

      1. associate-/r* 88.6%

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

      2. *-inverses 88.6%

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

      3. associate-*r/ 88.6%

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

      4. *-rgt-identity 88.6%

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

      5. distribute-neg-in 88.6%

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

      6. unsub-neg 88.6%

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

   9. Simplified 88.6%

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

   if -1.85e16 < x < -1.75e-9

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      3. times-frac 99.4%

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

      4. +-commutative 99.4%

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

      5. +-commutative 99.4%

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

      6. associate-+r+ 99.4%

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

      7. +-commutative 99.4%

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

      8. associate-+l+ 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. frac-times 99.7%

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

      2. *-commutative 99.7%

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

      3. frac-times 99.1%

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

      4. *-commutative 99.1%

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

      5. clear-num 99.4%

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

      6. un-div-inv 99.4%

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

      7. +-commutative 99.4%

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

      8. +-commutative 99.4%

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

      9. +-commutative 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in x around 0 60.6%

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

      1. +-commutative 60.6%

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

   9. Simplified 60.6%

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

   if -1.75e-9 < x < -1.8999999999999998e-46

   1. Initial program 99.8%

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

      1. associate-/l* 99.6%

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

      2. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

      1. associate-*r/ 99.8%

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

      2. associate-+r+ 99.8%

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

      3. times-frac 99.6%

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

      4. associate-*r/ 99.6%

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

      5. pow2 99.6%

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

      6. +-commutative 99.6%

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

      7. associate-+r+ 99.6%

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

      8. +-commutative 99.6%

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

      9. associate-+l+ 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. unpow2 99.6%

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

      3. times-frac 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in x around inf 85.9%

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

   if -1.8999999999999998e-46 < x < -5.59999999999999978e-101

   1. Initial program 87.6%

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

      1. associate-/l* 88.0%

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

      2. associate-+l+ 88.0%

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

   3. Simplified 88.0%

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

   5. Taylor expanded in x around 0 92.6%

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

      1. +-commutative 92.6%

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

   7. Simplified 92.6%

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

      1. *-un-lft-identity 92.6%

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

      2. times-frac 91.7%

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

   9. Applied egg-rr 91.7%

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

       1. frac-times 92.6%

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

       2. *-un-lft-identity 92.6%

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

       3. clear-num 92.6%

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

       4. +-commutative 92.6%

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

   11. Applied egg-rr 92.6%

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

   if -5.59999999999999978e-101 < x < -1.06e-109

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.2%

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

      3. times-frac 99.2%

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

      4. +-commutative 99.2%

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

      5. +-commutative 99.2%

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

      6. associate-+r+ 99.2%

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

      7. +-commutative 99.2%

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

      8. associate-+l+ 99.2%

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in y around 0 37.2%

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

      1. +-commutative 37.2%

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

   7. Simplified 37.2%

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

   if -1.06e-109 < x

   1. Initial program 67.8%

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

      1. associate-/l* 80.6%

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

      2. associate-+l+ 80.6%

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

   3. Simplified 80.6%

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

      1. associate-*r/ 67.8%

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

      2. associate-+r+ 67.8%

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

      3. times-frac 87.7%

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

      4. associate-*r/ 87.8%

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

      5. pow2 87.8%

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

      6. +-commutative 87.8%

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

      7. associate-+r+ 87.8%

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

      8. +-commutative 87.8%

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

      9. associate-+l+ 87.8%

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

   6. Applied egg-rr 87.8%

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

      1. *-un-lft-identity 87.8%

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

      2. unpow2 87.8%

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

      3. times-frac 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0 66.2%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y + \left(1 + x\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + y}\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{-1}{-1 - y}}{\frac{x + y}{x}}\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-46}:\\ \;\;\;\;\frac{y \cdot \frac{1}{x}}{y - \left(-1 - x\right)}\\ \mathbf{elif}\;x \leq -5.6 \cdot 10^{-101}:\\ \;\;\;\;\frac{1}{\frac{y \cdot \left(y + 1\right)}{x}}\\ \mathbf{elif}\;x \leq -1.06 \cdot 10^{-109}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{-1}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y - \left(-1 - x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x + y}{y}}\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y}\\ \mathbf{elif}\;x \leq -2.25 \cdot 10^{-46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-101}:\\ \;\;\;\;\frac{1}{\frac{y \cdot \left(y + 1\right)}{x}}\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-109}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y - \left(-1 - x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* x (+ x 1.0)))))
   (if (<= x -1.8e+18)
     (/ (/ 1.0 x) (/ (+ x y) y))
     (if (<= x -1.25e-10)
       (/ (/ x (+ x y)) y)
       (if (<= x -2.25e-46)
         t_0
         (if (<= x -5.2e-101)
           (/ 1.0 (/ (* y (+ y 1.0)) x))
           (if (<= x -2.6e-109) t_0 (/ (/ x y) (- y (- -1.0 x))))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x * (x + 1.0));
	double tmp;
	if (x <= -1.8e+18) {
		tmp = (1.0 / x) / ((x + y) / y);
	} else if (x <= -1.25e-10) {
		tmp = (x / (x + y)) / y;
	} else if (x <= -2.25e-46) {
		tmp = t_0;
	} else if (x <= -5.2e-101) {
		tmp = 1.0 / ((y * (y + 1.0)) / x);
	} else if (x <= -2.6e-109) {
		tmp = t_0;
	} else {
		tmp = (x / y) / (y - (-1.0 - 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) :: t_0
    real(8) :: tmp
    t_0 = y / (x * (x + 1.0d0))
    if (x <= (-1.8d+18)) then
        tmp = (1.0d0 / x) / ((x + y) / y)
    else if (x <= (-1.25d-10)) then
        tmp = (x / (x + y)) / y
    else if (x <= (-2.25d-46)) then
        tmp = t_0
    else if (x <= (-5.2d-101)) then
        tmp = 1.0d0 / ((y * (y + 1.0d0)) / x)
    else if (x <= (-2.6d-109)) then
        tmp = t_0
    else
        tmp = (x / y) / (y - ((-1.0d0) - x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (x * (x + 1.0));
	double tmp;
	if (x <= -1.8e+18) {
		tmp = (1.0 / x) / ((x + y) / y);
	} else if (x <= -1.25e-10) {
		tmp = (x / (x + y)) / y;
	} else if (x <= -2.25e-46) {
		tmp = t_0;
	} else if (x <= -5.2e-101) {
		tmp = 1.0 / ((y * (y + 1.0)) / x);
	} else if (x <= -2.6e-109) {
		tmp = t_0;
	} else {
		tmp = (x / y) / (y - (-1.0 - x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (x * (x + 1.0))
	tmp = 0
	if x <= -1.8e+18:
		tmp = (1.0 / x) / ((x + y) / y)
	elif x <= -1.25e-10:
		tmp = (x / (x + y)) / y
	elif x <= -2.25e-46:
		tmp = t_0
	elif x <= -5.2e-101:
		tmp = 1.0 / ((y * (y + 1.0)) / x)
	elif x <= -2.6e-109:
		tmp = t_0
	else:
		tmp = (x / y) / (y - (-1.0 - x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(x * Float64(x + 1.0)))
	tmp = 0.0
	if (x <= -1.8e+18)
		tmp = Float64(Float64(1.0 / x) / Float64(Float64(x + y) / y));
	elseif (x <= -1.25e-10)
		tmp = Float64(Float64(x / Float64(x + y)) / y);
	elseif (x <= -2.25e-46)
		tmp = t_0;
	elseif (x <= -5.2e-101)
		tmp = Float64(1.0 / Float64(Float64(y * Float64(y + 1.0)) / x));
	elseif (x <= -2.6e-109)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / y) / Float64(y - Float64(-1.0 - x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (x * (x + 1.0));
	tmp = 0.0;
	if (x <= -1.8e+18)
		tmp = (1.0 / x) / ((x + y) / y);
	elseif (x <= -1.25e-10)
		tmp = (x / (x + y)) / y;
	elseif (x <= -2.25e-46)
		tmp = t_0;
	elseif (x <= -5.2e-101)
		tmp = 1.0 / ((y * (y + 1.0)) / x);
	elseif (x <= -2.6e-109)
		tmp = t_0;
	else
		tmp = (x / y) / (y - (-1.0 - x));
	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[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.8e+18], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.25e-10], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, -2.25e-46], t$95$0, If[LessEqual[x, -5.2e-101], N[(1.0 / N[(N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.6e-109], t$95$0, N[(N[(x / y), $MachinePrecision] / N[(y - N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.8e18

   1. Initial program 55.7%

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

      1. *-commutative 55.7%

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

      2. associate-*l* 55.7%

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

      3. times-frac 79.2%

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

      4. +-commutative 79.2%

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

      5. +-commutative 79.2%

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

      6. associate-+r+ 79.2%

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

      7. +-commutative 79.2%

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

      8. associate-+l+ 79.2%

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

   4. Applied egg-rr 79.2%

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

   5. Taylor expanded in x around inf 88.1%

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

      1. *-commutative 88.1%

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

      2. +-commutative 88.1%

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

      3. clear-num 88.1%

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

      4. un-div-inv 88.2%

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

   7. Applied egg-rr 88.2%

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

   if -1.8e18 < x < -1.25000000000000008e-10

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      3. times-frac 99.4%

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

      4. +-commutative 99.4%

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

      5. +-commutative 99.4%

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

      6. associate-+r+ 99.4%

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

      7. +-commutative 99.4%

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

      8. associate-+l+ 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. clear-num 99.4%

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

      2. associate-/r* 98.8%

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

      3. frac-times 99.1%

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

      4. *-un-lft-identity 99.1%

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

      5. +-commutative 99.1%

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

      6. +-commutative 99.1%

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

      7. +-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in y around inf 60.7%

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

   if -1.25000000000000008e-10 < x < -2.25e-46 or -5.2000000000000002e-101 < x < -2.5999999999999998e-109

   1. Initial program 99.7%

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

      1. associate-/l* 99.5%

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

      2. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 74.8%

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

   if -2.25e-46 < x < -5.2000000000000002e-101

   1. Initial program 87.6%

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

      1. associate-/l* 88.0%

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

      2. associate-+l+ 88.0%

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

   3. Simplified 88.0%

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

   5. Taylor expanded in x around 0 92.6%

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

      1. +-commutative 92.6%

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

   7. Simplified 92.6%

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

      1. *-un-lft-identity 92.6%

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

      2. times-frac 91.7%

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

   9. Applied egg-rr 91.7%

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

       1. frac-times 92.6%

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

       2. *-un-lft-identity 92.6%

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

       3. clear-num 92.6%

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

       4. +-commutative 92.6%

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

   11. Applied egg-rr 92.6%

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

   if -2.5999999999999998e-109 < x

   1. Initial program 67.8%

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

      1. associate-/l* 80.6%

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

      2. associate-+l+ 80.6%

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

   3. Simplified 80.6%

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

      1. associate-*r/ 67.8%

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

      2. associate-+r+ 67.8%

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

      3. times-frac 87.7%

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

      4. associate-*r/ 87.8%

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

      5. pow2 87.8%

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

      6. +-commutative 87.8%

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

      7. associate-+r+ 87.8%

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

      8. +-commutative 87.8%

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

      9. associate-+l+ 87.8%

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

   6. Applied egg-rr 87.8%

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

      1. *-un-lft-identity 87.8%

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

      2. unpow2 87.8%

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

      3. times-frac 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0 66.2%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y + \left(1 + x\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x + y}{y}}\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y}\\ \mathbf{elif}\;x \leq -2.25 \cdot 10^{-46}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-101}:\\ \;\;\;\;\frac{1}{\frac{y \cdot \left(y + 1\right)}{x}}\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y - \left(-1 - x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{+129}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -5.3 \cdot 10^{+14}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + y\right)}\\ \mathbf{elif}\;x \leq -1.36 \cdot 10^{-46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -5.1 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -1.06 \cdot 10^{-109}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* x (+ x 1.0)))))
   (if (<= x -9.5e+129)
     (* (/ 1.0 x) (/ y x))
     (if (<= x -5.3e+14)
       (/ y (* x (+ x y)))
       (if (<= x -1.36e-46)
         t_0
         (if (<= x -5.1e-101)
           (/ x (* y (+ y 1.0)))
           (if (<= x -1.06e-109) t_0 (/ (/ x (+ y 1.0)) y))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x * (x + 1.0));
	double tmp;
	if (x <= -9.5e+129) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -5.3e+14) {
		tmp = y / (x * (x + y));
	} else if (x <= -1.36e-46) {
		tmp = t_0;
	} else if (x <= -5.1e-101) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -1.06e-109) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = y / (x * (x + 1.0d0))
    if (x <= (-9.5d+129)) then
        tmp = (1.0d0 / x) * (y / x)
    else if (x <= (-5.3d+14)) then
        tmp = y / (x * (x + y))
    else if (x <= (-1.36d-46)) then
        tmp = t_0
    else if (x <= (-5.1d-101)) then
        tmp = x / (y * (y + 1.0d0))
    else if (x <= (-1.06d-109)) then
        tmp = t_0
    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 t_0 = y / (x * (x + 1.0));
	double tmp;
	if (x <= -9.5e+129) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -5.3e+14) {
		tmp = y / (x * (x + y));
	} else if (x <= -1.36e-46) {
		tmp = t_0;
	} else if (x <= -5.1e-101) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -1.06e-109) {
		tmp = t_0;
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (x * (x + 1.0))
	tmp = 0
	if x <= -9.5e+129:
		tmp = (1.0 / x) * (y / x)
	elif x <= -5.3e+14:
		tmp = y / (x * (x + y))
	elif x <= -1.36e-46:
		tmp = t_0
	elif x <= -5.1e-101:
		tmp = x / (y * (y + 1.0))
	elif x <= -1.06e-109:
		tmp = t_0
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(x * Float64(x + 1.0)))
	tmp = 0.0
	if (x <= -9.5e+129)
		tmp = Float64(Float64(1.0 / x) * Float64(y / x));
	elseif (x <= -5.3e+14)
		tmp = Float64(y / Float64(x * Float64(x + y)));
	elseif (x <= -1.36e-46)
		tmp = t_0;
	elseif (x <= -5.1e-101)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	elseif (x <= -1.06e-109)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (x * (x + 1.0));
	tmp = 0.0;
	if (x <= -9.5e+129)
		tmp = (1.0 / x) * (y / x);
	elseif (x <= -5.3e+14)
		tmp = y / (x * (x + y));
	elseif (x <= -1.36e-46)
		tmp = t_0;
	elseif (x <= -5.1e-101)
		tmp = x / (y * (y + 1.0));
	elseif (x <= -1.06e-109)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9.5e+129], N[(N[(1.0 / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.3e+14], N[(y / N[(x * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.36e-46], t$95$0, If[LessEqual[x, -5.1e-101], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.06e-109], t$95$0, N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -9.5000000000000004e129

   1. Initial program 50.0%

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

      1. *-commutative 50.0%

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

      2. associate-*l* 50.0%

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

      3. times-frac 73.7%

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

      4. +-commutative 73.7%

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

      5. +-commutative 73.7%

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

      6. associate-+r+ 73.7%

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

      7. +-commutative 73.7%

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

      8. associate-+l+ 73.7%

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

   4. Applied egg-rr 73.7%

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

   5. Taylor expanded in x around inf 93.1%

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

   6. Taylor expanded in y around 0 93.0%

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

   if -9.5000000000000004e129 < x < -5.3e14

   1. Initial program 78.3%

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

      1. *-commutative 78.3%

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

      2. associate-*l* 78.3%

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

      3. times-frac 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r+ 99.6%

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

      7. +-commutative 99.6%

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

      8. associate-+l+ 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around inf 65.2%

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

      1. frac-times 85.9%

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

      2. *-rgt-identity 85.9%

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

      3. +-commutative 85.9%

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

   7. Applied egg-rr 85.9%

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

   if -5.3e14 < x < -1.3600000000000001e-46 or -5.1000000000000002e-101 < x < -1.06e-109

   1. Initial program 99.6%

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

      1. associate-/l* 99.3%

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

      2. associate-+l+ 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around 0 67.6%

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

   if -1.3600000000000001e-46 < x < -5.1000000000000002e-101

   1. Initial program 87.6%

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

      1. associate-/l* 88.0%

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

      2. associate-+l+ 88.0%

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

   3. Simplified 88.0%

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

   5. Taylor expanded in x around 0 92.6%

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

      1. +-commutative 92.6%

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

   7. Simplified 92.6%

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

   if -1.06e-109 < x

   1. Initial program 67.8%

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

      1. associate-/l* 80.6%

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

      2. associate-+l+ 80.6%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in x around 0 64.7%

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

      1. +-commutative 64.7%

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

   7. Simplified 64.7%

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

      1. *-un-lft-identity 64.7%

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

      2. times-frac 66.0%

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

   9. Applied egg-rr 66.0%

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

       1. associate-*l/ 66.0%

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

       2. *-un-lft-identity 66.0%

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

       3. +-commutative 66.0%

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

   11. Applied egg-rr 66.0%

       \[\leadsto \color{blue}{\frac{\frac{x}{1 + y}}{y}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+129}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -5.3 \cdot 10^{+14}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + y\right)}\\ \mathbf{elif}\;x \leq -1.36 \cdot 10^{-46}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -5.1 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -1.06 \cdot 10^{-109}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -0.000106:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y}\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -5.1 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -1.06 \cdot 10^{-109}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* x (+ x 1.0)))))
   (if (<= x -1.8e+18)
     (/ (/ y x) (+ x y))
     (if (<= x -0.000106)
       (/ (/ x (+ x y)) y)
       (if (<= x -2.6e-46)
         t_0
         (if (<= x -5.1e-101)
           (/ x (* y (+ y 1.0)))
           (if (<= x -1.06e-109) t_0 (/ (/ x (+ y 1.0)) y))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x * (x + 1.0));
	double tmp;
	if (x <= -1.8e+18) {
		tmp = (y / x) / (x + y);
	} else if (x <= -0.000106) {
		tmp = (x / (x + y)) / y;
	} else if (x <= -2.6e-46) {
		tmp = t_0;
	} else if (x <= -5.1e-101) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -1.06e-109) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = y / (x * (x + 1.0d0))
    if (x <= (-1.8d+18)) then
        tmp = (y / x) / (x + y)
    else if (x <= (-0.000106d0)) then
        tmp = (x / (x + y)) / y
    else if (x <= (-2.6d-46)) then
        tmp = t_0
    else if (x <= (-5.1d-101)) then
        tmp = x / (y * (y + 1.0d0))
    else if (x <= (-1.06d-109)) then
        tmp = t_0
    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 t_0 = y / (x * (x + 1.0));
	double tmp;
	if (x <= -1.8e+18) {
		tmp = (y / x) / (x + y);
	} else if (x <= -0.000106) {
		tmp = (x / (x + y)) / y;
	} else if (x <= -2.6e-46) {
		tmp = t_0;
	} else if (x <= -5.1e-101) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -1.06e-109) {
		tmp = t_0;
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (x * (x + 1.0))
	tmp = 0
	if x <= -1.8e+18:
		tmp = (y / x) / (x + y)
	elif x <= -0.000106:
		tmp = (x / (x + y)) / y
	elif x <= -2.6e-46:
		tmp = t_0
	elif x <= -5.1e-101:
		tmp = x / (y * (y + 1.0))
	elif x <= -1.06e-109:
		tmp = t_0
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(x * Float64(x + 1.0)))
	tmp = 0.0
	if (x <= -1.8e+18)
		tmp = Float64(Float64(y / x) / Float64(x + y));
	elseif (x <= -0.000106)
		tmp = Float64(Float64(x / Float64(x + y)) / y);
	elseif (x <= -2.6e-46)
		tmp = t_0;
	elseif (x <= -5.1e-101)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	elseif (x <= -1.06e-109)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (x * (x + 1.0));
	tmp = 0.0;
	if (x <= -1.8e+18)
		tmp = (y / x) / (x + y);
	elseif (x <= -0.000106)
		tmp = (x / (x + y)) / y;
	elseif (x <= -2.6e-46)
		tmp = t_0;
	elseif (x <= -5.1e-101)
		tmp = x / (y * (y + 1.0));
	elseif (x <= -1.06e-109)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.8e+18], N[(N[(y / x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -0.000106], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, -2.6e-46], t$95$0, If[LessEqual[x, -5.1e-101], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.06e-109], t$95$0, N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.8e18

   1. Initial program 55.7%

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

      1. *-commutative 55.7%

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

      2. associate-*l* 55.7%

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

      3. times-frac 79.2%

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

      4. +-commutative 79.2%

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

      5. +-commutative 79.2%

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

      6. associate-+r+ 79.2%

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

      7. +-commutative 79.2%

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

      8. associate-+l+ 79.2%

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

   4. Applied egg-rr 79.2%

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

   5. Taylor expanded in x around inf 88.1%

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

      1. frac-times 77.7%

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

      2. *-rgt-identity 77.7%

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

      3. +-commutative 77.7%

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

   7. Applied egg-rr 77.7%

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

      1. associate-/l/ 88.1%

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

   9. Simplified 88.1%

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

   if -1.8e18 < x < -1.06e-4

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      3. times-frac 99.4%

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

      4. +-commutative 99.4%

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

      5. +-commutative 99.4%

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

      6. associate-+r+ 99.4%

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

      7. +-commutative 99.4%

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

      8. associate-+l+ 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. clear-num 99.4%

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

      2. associate-/r* 98.8%

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

      3. frac-times 99.1%

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

      4. *-un-lft-identity 99.1%

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

      5. +-commutative 99.1%

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

      6. +-commutative 99.1%

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

      7. +-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in y around inf 60.7%

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

   if -1.06e-4 < x < -2.6000000000000002e-46 or -5.1000000000000002e-101 < x < -1.06e-109

   1. Initial program 99.7%

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

      1. associate-/l* 99.5%

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

      2. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 74.8%

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

   if -2.6000000000000002e-46 < x < -5.1000000000000002e-101

   1. Initial program 87.6%

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

      1. associate-/l* 88.0%

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

      2. associate-+l+ 88.0%

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

   3. Simplified 88.0%

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

   5. Taylor expanded in x around 0 92.6%

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

      1. +-commutative 92.6%

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

   7. Simplified 92.6%

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

   if -1.06e-109 < x

   1. Initial program 67.8%

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

      1. associate-/l* 80.6%

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

      2. associate-+l+ 80.6%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in x around 0 64.7%

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

      1. +-commutative 64.7%

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

   7. Simplified 64.7%

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

      1. *-un-lft-identity 64.7%

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

      2. times-frac 66.0%

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

   9. Applied egg-rr 66.0%

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

       1. associate-*l/ 66.0%

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

       2. *-un-lft-identity 66.0%

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

       3. +-commutative 66.0%

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

   11. Applied egg-rr 66.0%

       \[\leadsto \color{blue}{\frac{\frac{x}{1 + y}}{y}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -0.000106:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y}\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-46}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -5.1 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -1.06 \cdot 10^{-109}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{+18}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y}\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-47}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -5.1 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -1.06 \cdot 10^{-109}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* x (+ x 1.0)))))
   (if (<= x -1.25e+18)
     (* (/ y (+ x y)) (/ 1.0 x))
     (if (<= x -3e-10)
       (/ (/ x (+ x y)) y)
       (if (<= x -6.8e-47)
         t_0
         (if (<= x -5.1e-101)
           (/ x (* y (+ y 1.0)))
           (if (<= x -1.06e-109) t_0 (/ (/ x (+ y 1.0)) y))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x * (x + 1.0));
	double tmp;
	if (x <= -1.25e+18) {
		tmp = (y / (x + y)) * (1.0 / x);
	} else if (x <= -3e-10) {
		tmp = (x / (x + y)) / y;
	} else if (x <= -6.8e-47) {
		tmp = t_0;
	} else if (x <= -5.1e-101) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -1.06e-109) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = y / (x * (x + 1.0d0))
    if (x <= (-1.25d+18)) then
        tmp = (y / (x + y)) * (1.0d0 / x)
    else if (x <= (-3d-10)) then
        tmp = (x / (x + y)) / y
    else if (x <= (-6.8d-47)) then
        tmp = t_0
    else if (x <= (-5.1d-101)) then
        tmp = x / (y * (y + 1.0d0))
    else if (x <= (-1.06d-109)) then
        tmp = t_0
    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 t_0 = y / (x * (x + 1.0));
	double tmp;
	if (x <= -1.25e+18) {
		tmp = (y / (x + y)) * (1.0 / x);
	} else if (x <= -3e-10) {
		tmp = (x / (x + y)) / y;
	} else if (x <= -6.8e-47) {
		tmp = t_0;
	} else if (x <= -5.1e-101) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -1.06e-109) {
		tmp = t_0;
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (x * (x + 1.0))
	tmp = 0
	if x <= -1.25e+18:
		tmp = (y / (x + y)) * (1.0 / x)
	elif x <= -3e-10:
		tmp = (x / (x + y)) / y
	elif x <= -6.8e-47:
		tmp = t_0
	elif x <= -5.1e-101:
		tmp = x / (y * (y + 1.0))
	elif x <= -1.06e-109:
		tmp = t_0
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(x * Float64(x + 1.0)))
	tmp = 0.0
	if (x <= -1.25e+18)
		tmp = Float64(Float64(y / Float64(x + y)) * Float64(1.0 / x));
	elseif (x <= -3e-10)
		tmp = Float64(Float64(x / Float64(x + y)) / y);
	elseif (x <= -6.8e-47)
		tmp = t_0;
	elseif (x <= -5.1e-101)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	elseif (x <= -1.06e-109)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (x * (x + 1.0));
	tmp = 0.0;
	if (x <= -1.25e+18)
		tmp = (y / (x + y)) * (1.0 / x);
	elseif (x <= -3e-10)
		tmp = (x / (x + y)) / y;
	elseif (x <= -6.8e-47)
		tmp = t_0;
	elseif (x <= -5.1e-101)
		tmp = x / (y * (y + 1.0));
	elseif (x <= -1.06e-109)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.25e+18], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3e-10], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, -6.8e-47], t$95$0, If[LessEqual[x, -5.1e-101], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.06e-109], t$95$0, N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.25e18

   1. Initial program 55.7%

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

      1. *-commutative 55.7%

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

      2. associate-*l* 55.7%

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

      3. times-frac 79.2%

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

      4. +-commutative 79.2%

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

      5. +-commutative 79.2%

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

      6. associate-+r+ 79.2%

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

      7. +-commutative 79.2%

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

      8. associate-+l+ 79.2%

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

   4. Applied egg-rr 79.2%

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

   5. Taylor expanded in x around inf 88.1%

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

   if -1.25e18 < x < -3e-10

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      3. times-frac 99.4%

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

      4. +-commutative 99.4%

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

      5. +-commutative 99.4%

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

      6. associate-+r+ 99.4%

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

      7. +-commutative 99.4%

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

      8. associate-+l+ 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. clear-num 99.4%

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

      2. associate-/r* 98.8%

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

      3. frac-times 99.1%

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

      4. *-un-lft-identity 99.1%

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

      5. +-commutative 99.1%

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

      6. +-commutative 99.1%

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

      7. +-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in y around inf 60.7%

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

   if -3e-10 < x < -6.8000000000000003e-47 or -5.1000000000000002e-101 < x < -1.06e-109

   1. Initial program 99.7%

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

      1. associate-/l* 99.5%

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

      2. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 74.8%

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

   if -6.8000000000000003e-47 < x < -5.1000000000000002e-101

   1. Initial program 87.6%

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

      1. associate-/l* 88.0%

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

      2. associate-+l+ 88.0%

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

   3. Simplified 88.0%

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

   5. Taylor expanded in x around 0 92.6%

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

      1. +-commutative 92.6%

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

   7. Simplified 92.6%

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

   if -1.06e-109 < x

   1. Initial program 67.8%

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

      1. associate-/l* 80.6%

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

      2. associate-+l+ 80.6%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in x around 0 64.7%

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

      1. +-commutative 64.7%

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

   7. Simplified 64.7%

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

      1. *-un-lft-identity 64.7%

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

      2. times-frac 66.0%

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

   9. Applied egg-rr 66.0%

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

       1. associate-*l/ 66.0%

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

       2. *-un-lft-identity 66.0%

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

       3. +-commutative 66.0%

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

   11. Applied egg-rr 66.0%

       \[\leadsto \color{blue}{\frac{\frac{x}{1 + y}}{y}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+18}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y}\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -5.1 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -1.06 \cdot 10^{-109}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{if}\;x \leq -9.2 \cdot 10^{+17}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y}\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -5.1 \cdot 10^{-101}:\\ \;\;\;\;\frac{1}{\frac{y \cdot \left(y + 1\right)}{x}}\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-109}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* x (+ x 1.0)))))
   (if (<= x -9.2e+17)
     (* (/ y (+ x y)) (/ 1.0 x))
     (if (<= x -1.7e-6)
       (/ (/ x (+ x y)) y)
       (if (<= x -1.9e-46)
         t_0
         (if (<= x -5.1e-101)
           (/ 1.0 (/ (* y (+ y 1.0)) x))
           (if (<= x -2.6e-109) t_0 (/ (/ x (+ y 1.0)) y))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x * (x + 1.0));
	double tmp;
	if (x <= -9.2e+17) {
		tmp = (y / (x + y)) * (1.0 / x);
	} else if (x <= -1.7e-6) {
		tmp = (x / (x + y)) / y;
	} else if (x <= -1.9e-46) {
		tmp = t_0;
	} else if (x <= -5.1e-101) {
		tmp = 1.0 / ((y * (y + 1.0)) / x);
	} else if (x <= -2.6e-109) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = y / (x * (x + 1.0d0))
    if (x <= (-9.2d+17)) then
        tmp = (y / (x + y)) * (1.0d0 / x)
    else if (x <= (-1.7d-6)) then
        tmp = (x / (x + y)) / y
    else if (x <= (-1.9d-46)) then
        tmp = t_0
    else if (x <= (-5.1d-101)) then
        tmp = 1.0d0 / ((y * (y + 1.0d0)) / x)
    else if (x <= (-2.6d-109)) then
        tmp = t_0
    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 t_0 = y / (x * (x + 1.0));
	double tmp;
	if (x <= -9.2e+17) {
		tmp = (y / (x + y)) * (1.0 / x);
	} else if (x <= -1.7e-6) {
		tmp = (x / (x + y)) / y;
	} else if (x <= -1.9e-46) {
		tmp = t_0;
	} else if (x <= -5.1e-101) {
		tmp = 1.0 / ((y * (y + 1.0)) / x);
	} else if (x <= -2.6e-109) {
		tmp = t_0;
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (x * (x + 1.0))
	tmp = 0
	if x <= -9.2e+17:
		tmp = (y / (x + y)) * (1.0 / x)
	elif x <= -1.7e-6:
		tmp = (x / (x + y)) / y
	elif x <= -1.9e-46:
		tmp = t_0
	elif x <= -5.1e-101:
		tmp = 1.0 / ((y * (y + 1.0)) / x)
	elif x <= -2.6e-109:
		tmp = t_0
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(x * Float64(x + 1.0)))
	tmp = 0.0
	if (x <= -9.2e+17)
		tmp = Float64(Float64(y / Float64(x + y)) * Float64(1.0 / x));
	elseif (x <= -1.7e-6)
		tmp = Float64(Float64(x / Float64(x + y)) / y);
	elseif (x <= -1.9e-46)
		tmp = t_0;
	elseif (x <= -5.1e-101)
		tmp = Float64(1.0 / Float64(Float64(y * Float64(y + 1.0)) / x));
	elseif (x <= -2.6e-109)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (x * (x + 1.0));
	tmp = 0.0;
	if (x <= -9.2e+17)
		tmp = (y / (x + y)) * (1.0 / x);
	elseif (x <= -1.7e-6)
		tmp = (x / (x + y)) / y;
	elseif (x <= -1.9e-46)
		tmp = t_0;
	elseif (x <= -5.1e-101)
		tmp = 1.0 / ((y * (y + 1.0)) / x);
	elseif (x <= -2.6e-109)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9.2e+17], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.7e-6], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, -1.9e-46], t$95$0, If[LessEqual[x, -5.1e-101], N[(1.0 / N[(N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.6e-109], t$95$0, N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -9.2e17

   1. Initial program 55.7%

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

      1. *-commutative 55.7%

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

      2. associate-*l* 55.7%

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

      3. times-frac 79.2%

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

      4. +-commutative 79.2%

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

      5. +-commutative 79.2%

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

      6. associate-+r+ 79.2%

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

      7. +-commutative 79.2%

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

      8. associate-+l+ 79.2%

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

   4. Applied egg-rr 79.2%

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

   5. Taylor expanded in x around inf 88.1%

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

   if -9.2e17 < x < -1.70000000000000003e-6

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      3. times-frac 99.4%

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

      4. +-commutative 99.4%

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

      5. +-commutative 99.4%

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

      6. associate-+r+ 99.4%

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

      7. +-commutative 99.4%

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

      8. associate-+l+ 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. clear-num 99.4%

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

      2. associate-/r* 98.8%

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

      3. frac-times 99.1%

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

      4. *-un-lft-identity 99.1%

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

      5. +-commutative 99.1%

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

      6. +-commutative 99.1%

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

      7. +-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in y around inf 60.7%

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

   if -1.70000000000000003e-6 < x < -1.8999999999999998e-46 or -5.1000000000000002e-101 < x < -2.5999999999999998e-109

   1. Initial program 99.7%

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

      1. associate-/l* 99.5%

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

      2. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 74.8%

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

   if -1.8999999999999998e-46 < x < -5.1000000000000002e-101

   1. Initial program 87.6%

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

      1. associate-/l* 88.0%

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

      2. associate-+l+ 88.0%

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

   3. Simplified 88.0%

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

   5. Taylor expanded in x around 0 92.6%

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

      1. +-commutative 92.6%

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

   7. Simplified 92.6%

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

      1. *-un-lft-identity 92.6%

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

      2. times-frac 91.7%

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

   9. Applied egg-rr 91.7%

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

       1. frac-times 92.6%

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

       2. *-un-lft-identity 92.6%

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

       3. clear-num 92.6%

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

       4. +-commutative 92.6%

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

   11. Applied egg-rr 92.6%

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

   if -2.5999999999999998e-109 < x

   1. Initial program 67.8%

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

      1. associate-/l* 80.6%

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

      2. associate-+l+ 80.6%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in x around 0 64.7%

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

      1. +-commutative 64.7%

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

   7. Simplified 64.7%

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

      1. *-un-lft-identity 64.7%

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

      2. times-frac 66.0%

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

   9. Applied egg-rr 66.0%

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

       1. associate-*l/ 66.0%

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

       2. *-un-lft-identity 66.0%

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

       3. +-commutative 66.0%

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

   11. Applied egg-rr 66.0%

       \[\leadsto \color{blue}{\frac{\frac{x}{1 + y}}{y}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+17}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y}\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-46}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -5.1 \cdot 10^{-101}:\\ \;\;\;\;\frac{1}{\frac{y \cdot \left(y + 1\right)}{x}}\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{if}\;x \leq -2.8 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x + y}{y}}\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y}\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-100}:\\ \;\;\;\;\frac{1}{\frac{y \cdot \left(y + 1\right)}{x}}\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-109}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* x (+ x 1.0)))))
   (if (<= x -2.8e+17)
     (/ (/ 1.0 x) (/ (+ x y) y))
     (if (<= x -1.85e-10)
       (/ (/ x (+ x y)) y)
       (if (<= x -2.5e-46)
         t_0
         (if (<= x -3.8e-100)
           (/ 1.0 (/ (* y (+ y 1.0)) x))
           (if (<= x -2.6e-109) t_0 (/ (/ x (+ y 1.0)) y))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x * (x + 1.0));
	double tmp;
	if (x <= -2.8e+17) {
		tmp = (1.0 / x) / ((x + y) / y);
	} else if (x <= -1.85e-10) {
		tmp = (x / (x + y)) / y;
	} else if (x <= -2.5e-46) {
		tmp = t_0;
	} else if (x <= -3.8e-100) {
		tmp = 1.0 / ((y * (y + 1.0)) / x);
	} else if (x <= -2.6e-109) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = y / (x * (x + 1.0d0))
    if (x <= (-2.8d+17)) then
        tmp = (1.0d0 / x) / ((x + y) / y)
    else if (x <= (-1.85d-10)) then
        tmp = (x / (x + y)) / y
    else if (x <= (-2.5d-46)) then
        tmp = t_0
    else if (x <= (-3.8d-100)) then
        tmp = 1.0d0 / ((y * (y + 1.0d0)) / x)
    else if (x <= (-2.6d-109)) then
        tmp = t_0
    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 t_0 = y / (x * (x + 1.0));
	double tmp;
	if (x <= -2.8e+17) {
		tmp = (1.0 / x) / ((x + y) / y);
	} else if (x <= -1.85e-10) {
		tmp = (x / (x + y)) / y;
	} else if (x <= -2.5e-46) {
		tmp = t_0;
	} else if (x <= -3.8e-100) {
		tmp = 1.0 / ((y * (y + 1.0)) / x);
	} else if (x <= -2.6e-109) {
		tmp = t_0;
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (x * (x + 1.0))
	tmp = 0
	if x <= -2.8e+17:
		tmp = (1.0 / x) / ((x + y) / y)
	elif x <= -1.85e-10:
		tmp = (x / (x + y)) / y
	elif x <= -2.5e-46:
		tmp = t_0
	elif x <= -3.8e-100:
		tmp = 1.0 / ((y * (y + 1.0)) / x)
	elif x <= -2.6e-109:
		tmp = t_0
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(x * Float64(x + 1.0)))
	tmp = 0.0
	if (x <= -2.8e+17)
		tmp = Float64(Float64(1.0 / x) / Float64(Float64(x + y) / y));
	elseif (x <= -1.85e-10)
		tmp = Float64(Float64(x / Float64(x + y)) / y);
	elseif (x <= -2.5e-46)
		tmp = t_0;
	elseif (x <= -3.8e-100)
		tmp = Float64(1.0 / Float64(Float64(y * Float64(y + 1.0)) / x));
	elseif (x <= -2.6e-109)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (x * (x + 1.0));
	tmp = 0.0;
	if (x <= -2.8e+17)
		tmp = (1.0 / x) / ((x + y) / y);
	elseif (x <= -1.85e-10)
		tmp = (x / (x + y)) / y;
	elseif (x <= -2.5e-46)
		tmp = t_0;
	elseif (x <= -3.8e-100)
		tmp = 1.0 / ((y * (y + 1.0)) / x);
	elseif (x <= -2.6e-109)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.8e+17], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.85e-10], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, -2.5e-46], t$95$0, If[LessEqual[x, -3.8e-100], N[(1.0 / N[(N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.6e-109], t$95$0, N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -2.8e17

   1. Initial program 55.7%

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

      1. *-commutative 55.7%

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

      2. associate-*l* 55.7%

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

      3. times-frac 79.2%

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

      4. +-commutative 79.2%

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

      5. +-commutative 79.2%

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

      6. associate-+r+ 79.2%

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

      7. +-commutative 79.2%

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

      8. associate-+l+ 79.2%

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

   4. Applied egg-rr 79.2%

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

   5. Taylor expanded in x around inf 88.1%

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

      1. *-commutative 88.1%

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

      2. +-commutative 88.1%

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

      3. clear-num 88.1%

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

      4. un-div-inv 88.2%

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

   7. Applied egg-rr 88.2%

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

   if -2.8e17 < x < -1.85000000000000007e-10

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      3. times-frac 99.4%

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

      4. +-commutative 99.4%

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

      5. +-commutative 99.4%

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

      6. associate-+r+ 99.4%

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

      7. +-commutative 99.4%

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

      8. associate-+l+ 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. clear-num 99.4%

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

      2. associate-/r* 98.8%

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

      3. frac-times 99.1%

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

      4. *-un-lft-identity 99.1%

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

      5. +-commutative 99.1%

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

      6. +-commutative 99.1%

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

      7. +-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in y around inf 60.7%

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

   if -1.85000000000000007e-10 < x < -2.49999999999999996e-46 or -3.79999999999999997e-100 < x < -2.5999999999999998e-109

   1. Initial program 99.7%

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

      1. associate-/l* 99.5%

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

      2. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 74.8%

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

   if -2.49999999999999996e-46 < x < -3.79999999999999997e-100

   1. Initial program 87.6%

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

      1. associate-/l* 88.0%

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

      2. associate-+l+ 88.0%

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

   3. Simplified 88.0%

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

   5. Taylor expanded in x around 0 92.6%

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

      1. +-commutative 92.6%

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

   7. Simplified 92.6%

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

      1. *-un-lft-identity 92.6%

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

      2. times-frac 91.7%

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

   9. Applied egg-rr 91.7%

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

       1. frac-times 92.6%

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

       2. *-un-lft-identity 92.6%

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

       3. clear-num 92.6%

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

       4. +-commutative 92.6%

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

   11. Applied egg-rr 92.6%

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

   if -2.5999999999999998e-109 < x

   1. Initial program 67.8%

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

      1. associate-/l* 80.6%

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

      2. associate-+l+ 80.6%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in x around 0 64.7%

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

      1. +-commutative 64.7%

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

   7. Simplified 64.7%

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

      1. *-un-lft-identity 64.7%

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

      2. times-frac 66.0%

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

   9. Applied egg-rr 66.0%

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

       1. associate-*l/ 66.0%

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

       2. *-un-lft-identity 66.0%

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

       3. +-commutative 66.0%

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

   11. Applied egg-rr 66.0%

       \[\leadsto \color{blue}{\frac{\frac{x}{1 + y}}{y}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x + y}{y}}\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y}\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-46}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-100}:\\ \;\;\;\;\frac{1}{\frac{y \cdot \left(y + 1\right)}{x}}\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+129}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-46} \lor \neg \left(x \leq -1.4 \cdot 10^{-100}\right) \land x \leq -2.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5e+129)
   (* (/ 1.0 x) (/ y x))
   (if (or (<= x -2.6e-46) (and (not (<= x -1.4e-100)) (<= x -2.6e-109)))
     (/ y (* x (+ x 1.0)))
     (/ x (* y (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5e+129) {
		tmp = (1.0 / x) * (y / x);
	} else if ((x <= -2.6e-46) || (!(x <= -1.4e-100) && (x <= -2.6e-109))) {
		tmp = y / (x * (x + 1.0));
	} 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 <= (-5d+129)) then
        tmp = (1.0d0 / x) * (y / x)
    else if ((x <= (-2.6d-46)) .or. (.not. (x <= (-1.4d-100))) .and. (x <= (-2.6d-109))) then
        tmp = y / (x * (x + 1.0d0))
    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 <= -5e+129) {
		tmp = (1.0 / x) * (y / x);
	} else if ((x <= -2.6e-46) || (!(x <= -1.4e-100) && (x <= -2.6e-109))) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -5e+129:
		tmp = (1.0 / x) * (y / x)
	elif (x <= -2.6e-46) or (not (x <= -1.4e-100) and (x <= -2.6e-109)):
		tmp = y / (x * (x + 1.0))
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5e+129)
		tmp = Float64(Float64(1.0 / x) * Float64(y / x));
	elseif ((x <= -2.6e-46) || (!(x <= -1.4e-100) && (x <= -2.6e-109)))
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	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 <= -5e+129)
		tmp = (1.0 / x) * (y / x);
	elseif ((x <= -2.6e-46) || (~((x <= -1.4e-100)) && (x <= -2.6e-109)))
		tmp = y / (x * (x + 1.0));
	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, -5e+129], N[(N[(1.0 / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -2.6e-46], And[N[Not[LessEqual[x, -1.4e-100]], $MachinePrecision], LessEqual[x, -2.6e-109]]], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.0000000000000003e129

   1. Initial program 50.0%

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

      1. *-commutative 50.0%

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

      2. associate-*l* 50.0%

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

      3. times-frac 73.7%

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

      4. +-commutative 73.7%

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

      5. +-commutative 73.7%

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

      6. associate-+r+ 73.7%

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

      7. +-commutative 73.7%

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

      8. associate-+l+ 73.7%

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

   4. Applied egg-rr 73.7%

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

   5. Taylor expanded in x around inf 93.1%

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

   6. Taylor expanded in y around 0 93.0%

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

   if -5.0000000000000003e129 < x < -2.6000000000000002e-46 or -1.39999999999999998e-100 < x < -2.5999999999999998e-109

   1. Initial program 88.6%

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

      1. associate-/l* 95.0%

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

      2. associate-+l+ 95.0%

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

   3. Simplified 95.0%

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

   5. Taylor expanded in y around 0 66.1%

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

   if -2.6000000000000002e-46 < x < -1.39999999999999998e-100 or -2.5999999999999998e-109 < x

   1. Initial program 68.7%

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

      1. associate-/l* 80.9%

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

      2. associate-+l+ 80.9%

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

   3. Simplified 80.9%

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

   5. Taylor expanded in x around 0 66.0%

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

      1. +-commutative 66.0%

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

   7. Simplified 66.0%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+129}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-46} \lor \neg \left(x \leq -1.4 \cdot 10^{-100}\right) \land x \leq -2.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{1}{x + y}}{\frac{x + y}{y}}\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-177}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.4e+103)
   (/ (/ 1.0 (+ x y)) (/ (+ x y) y))
   (if (<= x -2.3e-177)
     (* x (/ y (* (* (+ x y) (+ x y)) (+ x (+ y 1.0)))))
     (/ (/ x (+ x y)) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.4e+103) {
		tmp = (1.0 / (x + y)) / ((x + y) / y);
	} else if (x <= -2.3e-177) {
		tmp = x * (y / (((x + y) * (x + y)) * (x + (y + 1.0))));
	} else {
		tmp = (x / (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 <= (-3.4d+103)) then
        tmp = (1.0d0 / (x + y)) / ((x + y) / y)
    else if (x <= (-2.3d-177)) then
        tmp = x * (y / (((x + y) * (x + y)) * (x + (y + 1.0d0))))
    else
        tmp = (x / (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 <= -3.4e+103) {
		tmp = (1.0 / (x + y)) / ((x + y) / y);
	} else if (x <= -2.3e-177) {
		tmp = x * (y / (((x + y) * (x + y)) * (x + (y + 1.0))));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.4e+103:
		tmp = (1.0 / (x + y)) / ((x + y) / y)
	elif x <= -2.3e-177:
		tmp = x * (y / (((x + y) * (x + y)) * (x + (y + 1.0))))
	else:
		tmp = (x / (x + y)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.4e+103)
		tmp = Float64(Float64(1.0 / Float64(x + y)) / Float64(Float64(x + y) / y));
	elseif (x <= -2.3e-177)
		tmp = Float64(x * Float64(y / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(x + Float64(y + 1.0)))));
	else
		tmp = Float64(Float64(x / 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 <= -3.4e+103)
		tmp = (1.0 / (x + y)) / ((x + y) / y);
	elseif (x <= -2.3e-177)
		tmp = x * (y / (((x + y) * (x + y)) * (x + (y + 1.0))));
	else
		tmp = (x / (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, -3.4e+103], N[(N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.3e-177], N[(x * N[(y / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.3999999999999998e103

   1. Initial program 50.0%

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

   3. Taylor expanded in x around inf 50.0%

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

      1. *-commutative 50.0%

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

      2. associate-*l* 50.0%

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

      3. +-commutative 50.0%

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

      4. times-frac 73.7%

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

      5. +-commutative 73.7%

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

   5. Applied egg-rr 73.7%

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

      1. clear-num 73.7%

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

      2. associate-/r* 93.2%

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

      3. frac-times 88.1%

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

      4. *-un-lft-identity 88.1%

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

   7. Applied egg-rr 88.1%

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

      1. div-inv 88.1%

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

      2. *-commutative 88.1%

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

      3. times-frac 93.2%

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

   9. Applied egg-rr 93.2%

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

       1. *-inverses 93.2%

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

       2. associate-*r/ 93.2%

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

   11. Simplified 93.2%

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

   if -3.3999999999999998e103 < x < -2.30000000000000022e-177

   1. Initial program 90.5%

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

      1. associate-/l* 94.5%

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

      2. associate-+l+ 94.5%

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

   3. Simplified 94.5%

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

   if -2.30000000000000022e-177 < x

   1. Initial program 66.3%

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

      1. *-commutative 66.3%

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

      2. associate-*l* 66.3%

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

      3. times-frac 96.0%

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

      4. +-commutative 96.0%

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

      5. +-commutative 96.0%

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

      6. associate-+r+ 96.0%

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

      7. +-commutative 96.0%

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

      8. associate-+l+ 96.0%

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

   4. Applied egg-rr 96.0%

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

      1. clear-num 96.0%

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

      2. associate-/r* 99.8%

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

      3. frac-times 99.3%

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

      4. *-un-lft-identity 99.3%

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

      5. +-commutative 99.3%

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

      6. +-commutative 99.3%

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

      7. +-commutative 99.3%

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

   6. Applied egg-rr 99.3%

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

   7. Taylor expanded in x around 0 65.3%

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

      1. +-commutative 65.3%

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

   9. Simplified 65.3%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{1}{x + y}}{\frac{x + y}{y}}\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-177}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 82.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x + y} \cdot \frac{-1}{-1 - x}\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{-46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -5.1 \cdot 10^{-101}:\\ \;\;\;\;\frac{1}{\frac{y \cdot \left(y + 1\right)}{x}}\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-109}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y - \left(-1 - x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ y (+ x y)) (/ -1.0 (- -1.0 x)))))
   (if (<= x -1.65e-46)
     t_0
     (if (<= x -5.1e-101)
       (/ 1.0 (/ (* y (+ y 1.0)) x))
       (if (<= x -2.3e-109) t_0 (/ (/ x y) (- y (- -1.0 x))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y / (x + y)) * (-1.0 / (-1.0 - x));
	double tmp;
	if (x <= -1.65e-46) {
		tmp = t_0;
	} else if (x <= -5.1e-101) {
		tmp = 1.0 / ((y * (y + 1.0)) / x);
	} else if (x <= -2.3e-109) {
		tmp = t_0;
	} else {
		tmp = (x / y) / (y - (-1.0 - 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) :: t_0
    real(8) :: tmp
    t_0 = (y / (x + y)) * ((-1.0d0) / ((-1.0d0) - x))
    if (x <= (-1.65d-46)) then
        tmp = t_0
    else if (x <= (-5.1d-101)) then
        tmp = 1.0d0 / ((y * (y + 1.0d0)) / x)
    else if (x <= (-2.3d-109)) then
        tmp = t_0
    else
        tmp = (x / y) / (y - ((-1.0d0) - x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y / (x + y)) * (-1.0 / (-1.0 - x));
	double tmp;
	if (x <= -1.65e-46) {
		tmp = t_0;
	} else if (x <= -5.1e-101) {
		tmp = 1.0 / ((y * (y + 1.0)) / x);
	} else if (x <= -2.3e-109) {
		tmp = t_0;
	} else {
		tmp = (x / y) / (y - (-1.0 - x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y / (x + y)) * (-1.0 / (-1.0 - x))
	tmp = 0
	if x <= -1.65e-46:
		tmp = t_0
	elif x <= -5.1e-101:
		tmp = 1.0 / ((y * (y + 1.0)) / x)
	elif x <= -2.3e-109:
		tmp = t_0
	else:
		tmp = (x / y) / (y - (-1.0 - x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y / Float64(x + y)) * Float64(-1.0 / Float64(-1.0 - x)))
	tmp = 0.0
	if (x <= -1.65e-46)
		tmp = t_0;
	elseif (x <= -5.1e-101)
		tmp = Float64(1.0 / Float64(Float64(y * Float64(y + 1.0)) / x));
	elseif (x <= -2.3e-109)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / y) / Float64(y - Float64(-1.0 - x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (y / (x + y)) * (-1.0 / (-1.0 - x));
	tmp = 0.0;
	if (x <= -1.65e-46)
		tmp = t_0;
	elseif (x <= -5.1e-101)
		tmp = 1.0 / ((y * (y + 1.0)) / x);
	elseif (x <= -2.3e-109)
		tmp = t_0;
	else
		tmp = (x / y) / (y - (-1.0 - x));
	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[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.65e-46], t$95$0, If[LessEqual[x, -5.1e-101], N[(1.0 / N[(N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.3e-109], t$95$0, N[(N[(x / y), $MachinePrecision] / N[(y - N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.65000000000000007e-46 or -5.1000000000000002e-101 < x < -2.3000000000000001e-109

   1. Initial program 63.9%

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

      1. *-commutative 63.9%

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

      2. associate-*l* 63.9%

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

      3. times-frac 83.0%

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

      4. +-commutative 83.0%

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

      5. +-commutative 83.0%

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

      6. associate-+r+ 83.0%

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

      7. +-commutative 83.0%

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

      8. associate-+l+ 83.0%

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

   4. Applied egg-rr 83.0%

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

   5. Taylor expanded in y around 0 83.6%

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

      1. +-commutative 83.6%

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

   7. Simplified 83.6%

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

   if -1.65000000000000007e-46 < x < -5.1000000000000002e-101

   1. Initial program 87.6%

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

      1. associate-/l* 88.0%

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

      2. associate-+l+ 88.0%

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

   3. Simplified 88.0%

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

   5. Taylor expanded in x around 0 92.6%

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

      1. +-commutative 92.6%

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

   7. Simplified 92.6%

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

      1. *-un-lft-identity 92.6%

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

      2. times-frac 91.7%

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

   9. Applied egg-rr 91.7%

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

       1. frac-times 92.6%

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

       2. *-un-lft-identity 92.6%

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

       3. clear-num 92.6%

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

       4. +-commutative 92.6%

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

   11. Applied egg-rr 92.6%

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

   if -2.3000000000000001e-109 < x

   1. Initial program 67.8%

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

      1. associate-/l* 80.6%

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

      2. associate-+l+ 80.6%

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

   3. Simplified 80.6%

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

      1. associate-*r/ 67.8%

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

      2. associate-+r+ 67.8%

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

      3. times-frac 87.7%

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

      4. associate-*r/ 87.8%

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

      5. pow2 87.8%

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

      6. +-commutative 87.8%

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

      7. associate-+r+ 87.8%

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

      8. +-commutative 87.8%

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

      9. associate-+l+ 87.8%

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

   6. Applied egg-rr 87.8%

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

      1. *-un-lft-identity 87.8%

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

      2. unpow2 87.8%

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

      3. times-frac 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0 66.2%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-46}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{-1}{-1 - x}\\ \mathbf{elif}\;x \leq -5.1 \cdot 10^{-101}:\\ \;\;\;\;\frac{1}{\frac{y \cdot \left(y + 1\right)}{x}}\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-109}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{-1}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y - \left(-1 - x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 82.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y - \left(-1 - x\right)\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{-46}:\\ \;\;\;\;\frac{y \cdot \frac{1}{x}}{t\_0}\\ \mathbf{elif}\;x \leq -5.1 \cdot 10^{-101}:\\ \;\;\;\;\frac{1}{\frac{y \cdot \left(y + 1\right)}{x}}\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{-1}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- y (- -1.0 x))))
   (if (<= x -2.6e-46)
     (/ (* y (/ 1.0 x)) t_0)
     (if (<= x -5.1e-101)
       (/ 1.0 (/ (* y (+ y 1.0)) x))
       (if (<= x -2.6e-109)
         (* (/ y (+ x y)) (/ -1.0 (- -1.0 x)))
         (/ (/ x y) t_0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y - (-1.0 - x);
	double tmp;
	if (x <= -2.6e-46) {
		tmp = (y * (1.0 / x)) / t_0;
	} else if (x <= -5.1e-101) {
		tmp = 1.0 / ((y * (y + 1.0)) / x);
	} else if (x <= -2.6e-109) {
		tmp = (y / (x + y)) * (-1.0 / (-1.0 - x));
	} else {
		tmp = (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 = y - ((-1.0d0) - x)
    if (x <= (-2.6d-46)) then
        tmp = (y * (1.0d0 / x)) / t_0
    else if (x <= (-5.1d-101)) then
        tmp = 1.0d0 / ((y * (y + 1.0d0)) / x)
    else if (x <= (-2.6d-109)) then
        tmp = (y / (x + y)) * ((-1.0d0) / ((-1.0d0) - x))
    else
        tmp = (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 = y - (-1.0 - x);
	double tmp;
	if (x <= -2.6e-46) {
		tmp = (y * (1.0 / x)) / t_0;
	} else if (x <= -5.1e-101) {
		tmp = 1.0 / ((y * (y + 1.0)) / x);
	} else if (x <= -2.6e-109) {
		tmp = (y / (x + y)) * (-1.0 / (-1.0 - x));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y - (-1.0 - x)
	tmp = 0
	if x <= -2.6e-46:
		tmp = (y * (1.0 / x)) / t_0
	elif x <= -5.1e-101:
		tmp = 1.0 / ((y * (y + 1.0)) / x)
	elif x <= -2.6e-109:
		tmp = (y / (x + y)) * (-1.0 / (-1.0 - x))
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y - Float64(-1.0 - x))
	tmp = 0.0
	if (x <= -2.6e-46)
		tmp = Float64(Float64(y * Float64(1.0 / x)) / t_0);
	elseif (x <= -5.1e-101)
		tmp = Float64(1.0 / Float64(Float64(y * Float64(y + 1.0)) / x));
	elseif (x <= -2.6e-109)
		tmp = Float64(Float64(y / Float64(x + y)) * Float64(-1.0 / Float64(-1.0 - x)));
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y - (-1.0 - x);
	tmp = 0.0;
	if (x <= -2.6e-46)
		tmp = (y * (1.0 / x)) / t_0;
	elseif (x <= -5.1e-101)
		tmp = 1.0 / ((y * (y + 1.0)) / x);
	elseif (x <= -2.6e-109)
		tmp = (y / (x + y)) * (-1.0 / (-1.0 - x));
	else
		tmp = (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[(y - N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.6e-46], N[(N[(y * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, -5.1e-101], N[(1.0 / N[(N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.6e-109], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.6000000000000002e-46

   1. Initial program 62.9%

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

      1. associate-/l* 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. Step-by-step derivation

      1. associate-*r/ 62.9%

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

      2. associate-+r+ 62.9%

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

      3. times-frac 82.5%

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

      4. associate-*r/ 82.5%

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

      5. pow2 82.5%

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

      6. +-commutative 82.5%

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

      7. associate-+r+ 82.5%

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

      8. +-commutative 82.5%

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

      9. associate-+l+ 82.5%

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

   6. Applied egg-rr 82.5%

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

      1. *-un-lft-identity 82.5%

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

      2. unpow2 82.5%

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

      3. times-frac 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in x around inf 84.8%

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

   if -2.6000000000000002e-46 < x < -5.1000000000000002e-101

   1. Initial program 87.6%

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

      1. associate-/l* 88.0%

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

      2. associate-+l+ 88.0%

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

   3. Simplified 88.0%

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

   5. Taylor expanded in x around 0 92.6%

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

      1. +-commutative 92.6%

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

   7. Simplified 92.6%

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

      1. *-un-lft-identity 92.6%

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

      2. times-frac 91.7%

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

   9. Applied egg-rr 91.7%

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

       1. frac-times 92.6%

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

       2. *-un-lft-identity 92.6%

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

       3. clear-num 92.6%

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

       4. +-commutative 92.6%

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

   11. Applied egg-rr 92.6%

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

   if -5.1000000000000002e-101 < x < -2.5999999999999998e-109

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.2%

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

      3. times-frac 99.2%

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

      4. +-commutative 99.2%

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

      5. +-commutative 99.2%

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

      6. associate-+r+ 99.2%

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

      7. +-commutative 99.2%

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

      8. associate-+l+ 99.2%

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in y around 0 37.2%

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

      1. +-commutative 37.2%

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

   7. Simplified 37.2%

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

   if -2.5999999999999998e-109 < x

   1. Initial program 67.8%

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

      1. associate-/l* 80.6%

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

      2. associate-+l+ 80.6%

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

   3. Simplified 80.6%

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

      1. associate-*r/ 67.8%

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

      2. associate-+r+ 67.8%

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

      3. times-frac 87.7%

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

      4. associate-*r/ 87.8%

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

      5. pow2 87.8%

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

      6. +-commutative 87.8%

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

      7. associate-+r+ 87.8%

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

      8. +-commutative 87.8%

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

      9. associate-+l+ 87.8%

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

   6. Applied egg-rr 87.8%

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

      1. *-un-lft-identity 87.8%

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

      2. unpow2 87.8%

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

      3. times-frac 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0 66.2%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-46}:\\ \;\;\;\;\frac{y \cdot \frac{1}{x}}{y - \left(-1 - x\right)}\\ \mathbf{elif}\;x \leq -5.1 \cdot 10^{-101}:\\ \;\;\;\;\frac{1}{\frac{y \cdot \left(y + 1\right)}{x}}\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{-1}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y - \left(-1 - x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 85.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+93}:\\ \;\;\;\;\frac{\frac{1}{x + y}}{\frac{x + y}{y}}\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-120}:\\ \;\;\;\;\frac{x \cdot y}{\left(x + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.02e+93)
   (/ (/ 1.0 (+ x y)) (/ (+ x y) y))
   (if (<= x -2.7e-120)
     (/ (* x y) (* (+ x 1.0) (* (+ x y) (+ x y))))
     (/ (/ x (+ x y)) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.02e+93) {
		tmp = (1.0 / (x + y)) / ((x + y) / y);
	} else if (x <= -2.7e-120) {
		tmp = (x * y) / ((x + 1.0) * ((x + y) * (x + y)));
	} else {
		tmp = (x / (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 <= (-1.02d+93)) then
        tmp = (1.0d0 / (x + y)) / ((x + y) / y)
    else if (x <= (-2.7d-120)) then
        tmp = (x * y) / ((x + 1.0d0) * ((x + y) * (x + y)))
    else
        tmp = (x / (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 <= -1.02e+93) {
		tmp = (1.0 / (x + y)) / ((x + y) / y);
	} else if (x <= -2.7e-120) {
		tmp = (x * y) / ((x + 1.0) * ((x + y) * (x + y)));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.02e+93:
		tmp = (1.0 / (x + y)) / ((x + y) / y)
	elif x <= -2.7e-120:
		tmp = (x * y) / ((x + 1.0) * ((x + y) * (x + y)))
	else:
		tmp = (x / (x + y)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.02e+93)
		tmp = Float64(Float64(1.0 / Float64(x + y)) / Float64(Float64(x + y) / y));
	elseif (x <= -2.7e-120)
		tmp = Float64(Float64(x * y) / Float64(Float64(x + 1.0) * Float64(Float64(x + y) * Float64(x + y))));
	else
		tmp = Float64(Float64(x / 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 <= -1.02e+93)
		tmp = (1.0 / (x + y)) / ((x + y) / y);
	elseif (x <= -2.7e-120)
		tmp = (x * y) / ((x + 1.0) * ((x + y) * (x + y)));
	else
		tmp = (x / (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, -1.02e+93], N[(N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.7e-120], N[(N[(x * y), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.0200000000000001e93

   1. Initial program 51.8%

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

   3. Taylor expanded in x around inf 51.8%

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

      1. *-commutative 51.8%

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

      2. associate-*l* 51.8%

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

      3. +-commutative 51.8%

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

      4. times-frac 76.6%

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

      5. +-commutative 76.6%

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

   5. Applied egg-rr 76.6%

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

      1. clear-num 76.6%

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

      2. associate-/r* 90.5%

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

      3. frac-times 85.9%

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

      4. *-un-lft-identity 85.9%

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

   7. Applied egg-rr 85.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{x + y}{y} \cdot x}} \]
   8. Step-by-step derivation

      1. div-inv 86.0%

         \[\leadsto \frac{\color{blue}{x \cdot \frac{1}{x + y}}}{\frac{x + y}{y} \cdot x} \]

      2. *-commutative 86.0%

         \[\leadsto \frac{x \cdot \frac{1}{x + y}}{\color{blue}{x \cdot \frac{x + y}{y}}} \]

      3. times-frac 90.5%

         \[\leadsto \color{blue}{\frac{x}{x} \cdot \frac{\frac{1}{x + y}}{\frac{x + y}{y}}} \]

   9. Applied egg-rr 90.5%

      \[\leadsto \color{blue}{\frac{x}{x} \cdot \frac{\frac{1}{x + y}}{\frac{x + y}{y}}} \]
   10. Step-by-step derivation

       1. *-inverses 90.5%

          \[\leadsto \color{blue}{1} \cdot \frac{\frac{1}{x + y}}{\frac{x + y}{y}} \]

       2. associate-*r/ 90.5%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{1}{x + y}}{\frac{x + y}{y}}} \]

   11. Simplified 90.5%

       \[\leadsto \color{blue}{\frac{1 \cdot \frac{1}{x + y}}{\frac{x + y}{y}}} \]

   if -1.0200000000000001e93 < x < -2.6999999999999999e-120

   1. Initial program 93.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 76.4%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. +-commutative 76.4%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + 1\right)}} \]

   5. Simplified 76.4%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + 1\right)}} \]

   if -2.6999999999999999e-120 < x

   1. Initial program 67.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 67.3%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. associate-*l* 67.3%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      3. times-frac 96.1%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. +-commutative 96.1%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. +-commutative 96.1%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-+r+ 96.1%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      7. +-commutative 96.1%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      8. associate-+l+ 96.1%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

   4. Applied egg-rr 96.1%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]
   5. Step-by-step derivation

      1. clear-num 96.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{y + x}{y}}} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)} \]

      2. associate-/r* 99.8%

         \[\leadsto \frac{1}{\frac{y + x}{y}} \cdot \color{blue}{\frac{\frac{x}{y + x}}{y + \left(1 + x\right)}} \]

      3. frac-times 99.3%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y + x}}{\frac{y + x}{y} \cdot \left(y + \left(1 + x\right)\right)}} \]

      4. *-un-lft-identity 99.3%

         \[\leadsto \frac{\color{blue}{\frac{x}{y + x}}}{\frac{y + x}{y} \cdot \left(y + \left(1 + x\right)\right)} \]

      5. +-commutative 99.3%

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{\frac{y + x}{y} \cdot \left(y + \left(1 + x\right)\right)} \]

      6. +-commutative 99.3%

         \[\leadsto \frac{\frac{x}{x + y}}{\frac{\color{blue}{x + y}}{y} \cdot \left(y + \left(1 + x\right)\right)} \]

      7. +-commutative 99.3%

         \[\leadsto \frac{\frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(y + \color{blue}{\left(x + 1\right)}\right)} \]

   6. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(y + \left(x + 1\right)\right)}} \]

   7. Taylor expanded in x around 0 66.3%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{1 + y}} \]
   8. Step-by-step derivation

      1. +-commutative 66.3%

         \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{y + 1}} \]

   9. Simplified 66.3%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{y + 1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+93}:\\ \;\;\;\;\frac{\frac{1}{x + y}}{\frac{x + y}{y}}\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-120}:\\ \;\;\;\;\frac{x \cdot y}{\left(x + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 86.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -60000000:\\ \;\;\;\;\frac{y \cdot \frac{1}{x}}{y - \left(-1 - x\right)}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-159}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -60000000.0)
   (/ (* y (/ 1.0 x)) (- y (- -1.0 x)))
   (if (<= x -5e-159)
     (/ (* x y) (* (* (+ x y) (+ x y)) (+ y 1.0)))
     (/ (/ x (+ x y)) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -60000000.0) {
		tmp = (y * (1.0 / x)) / (y - (-1.0 - x));
	} else if (x <= -5e-159) {
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	} else {
		tmp = (x / (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 <= (-60000000.0d0)) then
        tmp = (y * (1.0d0 / x)) / (y - ((-1.0d0) - x))
    else if (x <= (-5d-159)) then
        tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0d0))
    else
        tmp = (x / (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 <= -60000000.0) {
		tmp = (y * (1.0 / x)) / (y - (-1.0 - x));
	} else if (x <= -5e-159) {
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -60000000.0:
		tmp = (y * (1.0 / x)) / (y - (-1.0 - x))
	elif x <= -5e-159:
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0))
	else:
		tmp = (x / (x + y)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -60000000.0)
		tmp = Float64(Float64(y * Float64(1.0 / x)) / Float64(y - Float64(-1.0 - x)));
	elseif (x <= -5e-159)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / 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 <= -60000000.0)
		tmp = (y * (1.0 / x)) / (y - (-1.0 - x));
	elseif (x <= -5e-159)
		tmp = (x * y) / (((x + y) * (x + y)) * (y + 1.0));
	else
		tmp = (x / (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, -60000000.0], N[(N[(y * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(y - N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5e-159], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6e7

   1. Initial program 57.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 72.4%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 72.4%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 72.4%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 57.7%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      2. associate-+r+ 57.7%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      3. times-frac 80.1%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      4. associate-*r/ 80.2%

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1}} \]

      5. pow2 80.2%

         \[\leadsto \frac{\frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot y}{\left(x + y\right) + 1} \]

      6. +-commutative 80.2%

         \[\leadsto \frac{\frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot y}{\left(x + y\right) + 1} \]

      7. associate-+r+ 80.2%

         \[\leadsto \frac{\frac{x}{{\left(y + x\right)}^{2}} \cdot y}{\color{blue}{x + \left(y + 1\right)}} \]

      8. +-commutative 80.2%

         \[\leadsto \frac{\frac{x}{{\left(y + x\right)}^{2}} \cdot y}{\color{blue}{\left(y + 1\right) + x}} \]

      9. associate-+l+ 80.2%

         \[\leadsto \frac{\frac{x}{{\left(y + x\right)}^{2}} \cdot y}{\color{blue}{y + \left(1 + x\right)}} \]

   6. Applied egg-rr 80.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{{\left(y + x\right)}^{2}} \cdot y}{y + \left(1 + x\right)}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 80.2%

         \[\leadsto \frac{\frac{\color{blue}{1 \cdot x}}{{\left(y + x\right)}^{2}} \cdot y}{y + \left(1 + x\right)} \]

      2. unpow2 80.2%

         \[\leadsto \frac{\frac{1 \cdot x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot y}{y + \left(1 + x\right)} \]

      3. times-frac 99.8%

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{y + x} \cdot \frac{x}{y + x}\right)} \cdot y}{y + \left(1 + x\right)} \]

      4. +-commutative 99.8%

         \[\leadsto \frac{\left(\frac{1}{\color{blue}{x + y}} \cdot \frac{x}{y + x}\right) \cdot y}{y + \left(1 + x\right)} \]

      5. +-commutative 99.8%

         \[\leadsto \frac{\left(\frac{1}{x + y} \cdot \frac{x}{\color{blue}{x + y}}\right) \cdot y}{y + \left(1 + x\right)} \]

   8. Applied egg-rr 99.8%

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{x + y} \cdot \frac{x}{x + y}\right)} \cdot y}{y + \left(1 + x\right)} \]

   9. Taylor expanded in x around inf 87.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}} \cdot y}{y + \left(1 + x\right)} \]

   if -6e7 < x < -5.00000000000000032e-159

   1. Initial program 96.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 95.4%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. +-commutative 95.4%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + 1\right)}} \]

   5. Simplified 95.4%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + 1\right)}} \]

   if -5.00000000000000032e-159 < x

   1. Initial program 66.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 66.5%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. associate-*l* 66.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      3. times-frac 96.0%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. +-commutative 96.0%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. +-commutative 96.0%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-+r+ 96.0%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      7. +-commutative 96.0%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      8. associate-+l+ 96.0%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

   4. Applied egg-rr 96.0%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]
   5. Step-by-step derivation

      1. clear-num 96.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{y + x}{y}}} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)} \]

      2. associate-/r* 99.8%

         \[\leadsto \frac{1}{\frac{y + x}{y}} \cdot \color{blue}{\frac{\frac{x}{y + x}}{y + \left(1 + x\right)}} \]

      3. frac-times 99.3%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y + x}}{\frac{y + x}{y} \cdot \left(y + \left(1 + x\right)\right)}} \]

      4. *-un-lft-identity 99.3%

         \[\leadsto \frac{\color{blue}{\frac{x}{y + x}}}{\frac{y + x}{y} \cdot \left(y + \left(1 + x\right)\right)} \]

      5. +-commutative 99.3%

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{\frac{y + x}{y} \cdot \left(y + \left(1 + x\right)\right)} \]

      6. +-commutative 99.3%

         \[\leadsto \frac{\frac{x}{x + y}}{\frac{\color{blue}{x + y}}{y} \cdot \left(y + \left(1 + x\right)\right)} \]

      7. +-commutative 99.3%

         \[\leadsto \frac{\frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(y + \color{blue}{\left(x + 1\right)}\right)} \]

   6. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(y + \left(x + 1\right)\right)}} \]

   7. Taylor expanded in x around 0 65.5%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{1 + y}} \]
   8. Step-by-step derivation

      1. +-commutative 65.5%

         \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{y + 1}} \]

   9. Simplified 65.5%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{y + 1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -60000000:\\ \;\;\;\;\frac{y \cdot \frac{1}{x}}{y - \left(-1 - x\right)}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-159}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 95.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{+156}:\\ \;\;\;\;\frac{\frac{1}{x + y}}{\frac{x + y}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(y - \left(-1 - x\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -4.3e+156)
   (/ (/ 1.0 (+ x y)) (/ (+ x y) y))
   (* (/ x (+ x y)) (/ y (* (+ x y) (- y (- -1.0 x)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4.3e+156) {
		tmp = (1.0 / (x + y)) / ((x + y) / y);
	} else {
		tmp = (x / (x + y)) * (y / ((x + y) * (y - (-1.0 - 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 (x <= (-4.3d+156)) then
        tmp = (1.0d0 / (x + y)) / ((x + y) / y)
    else
        tmp = (x / (x + y)) * (y / ((x + y) * (y - ((-1.0d0) - x))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -4.3e+156) {
		tmp = (1.0 / (x + y)) / ((x + y) / y);
	} else {
		tmp = (x / (x + y)) * (y / ((x + y) * (y - (-1.0 - x))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -4.3e+156:
		tmp = (1.0 / (x + y)) / ((x + y) / y)
	else:
		tmp = (x / (x + y)) * (y / ((x + y) * (y - (-1.0 - x))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -4.3e+156)
		tmp = Float64(Float64(1.0 / Float64(x + y)) / Float64(Float64(x + y) / y));
	else
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(y / Float64(Float64(x + y) * Float64(y - Float64(-1.0 - x)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.3e+156)
		tmp = (1.0 / (x + y)) / ((x + y) / y);
	else
		tmp = (x / (x + y)) * (y / ((x + y) * (y - (-1.0 - 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[x, -4.3e+156], N[(N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(x + y), $MachinePrecision] * N[(y - N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.29999999999999985e156

   1. Initial program 49.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 49.4%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{x}} \]
   4. Step-by-step derivation

      1. *-commutative 49.4%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot x} \]

      2. associate-*l* 49.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot x\right)}} \]

      3. +-commutative 49.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot x\right)} \]

      4. times-frac 68.6%

         \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(x + y\right) \cdot x}} \]

      5. +-commutative 68.6%

         \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot x} \]

   5. Applied egg-rr 68.6%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot x}} \]
   6. Step-by-step derivation

      1. clear-num 68.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + y}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot x} \]

      2. associate-/r* 91.8%

         \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x}} \]

      3. frac-times 85.8%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot x}} \]

      4. *-un-lft-identity 85.8%

         \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{\frac{x + y}{y} \cdot x} \]

   7. Applied egg-rr 85.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{x + y}{y} \cdot x}} \]
   8. Step-by-step derivation

      1. div-inv 85.8%

         \[\leadsto \frac{\color{blue}{x \cdot \frac{1}{x + y}}}{\frac{x + y}{y} \cdot x} \]

      2. *-commutative 85.8%

         \[\leadsto \frac{x \cdot \frac{1}{x + y}}{\color{blue}{x \cdot \frac{x + y}{y}}} \]

      3. times-frac 91.9%

         \[\leadsto \color{blue}{\frac{x}{x} \cdot \frac{\frac{1}{x + y}}{\frac{x + y}{y}}} \]

   9. Applied egg-rr 91.9%

      \[\leadsto \color{blue}{\frac{x}{x} \cdot \frac{\frac{1}{x + y}}{\frac{x + y}{y}}} \]
   10. Step-by-step derivation

       1. *-inverses 91.9%

          \[\leadsto \color{blue}{1} \cdot \frac{\frac{1}{x + y}}{\frac{x + y}{y}} \]

       2. associate-*r/ 91.9%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{1}{x + y}}{\frac{x + y}{y}}} \]

   11. Simplified 91.9%

       \[\leadsto \color{blue}{\frac{1 \cdot \frac{1}{x + y}}{\frac{x + y}{y}}} \]

   if -4.29999999999999985e156 < x

   1. Initial program 70.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* 70.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. times-frac 96.8%

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. +-commutative 96.8%

         \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. +-commutative 96.8%

         \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      5. associate-+r+ 96.8%

         \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      6. +-commutative 96.8%

         \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      7. associate-+l+ 96.8%

         \[\leadsto \frac{x}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

   4. Applied egg-rr 96.8%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{+156}:\\ \;\;\;\;\frac{\frac{1}{x + y}}{\frac{x + y}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(y - \left(-1 - x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 95.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{1}{x + y}}{\frac{x + y}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(y - \left(-1 - x\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.65e+155)
   (/ (/ 1.0 (+ x y)) (/ (+ x y) y))
   (* (/ y (+ x y)) (/ x (* (+ x y) (- y (- -1.0 x)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.65e+155) {
		tmp = (1.0 / (x + y)) / ((x + y) / y);
	} else {
		tmp = (y / (x + y)) * (x / ((x + y) * (y - (-1.0 - 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 (x <= (-2.65d+155)) then
        tmp = (1.0d0 / (x + y)) / ((x + y) / y)
    else
        tmp = (y / (x + y)) * (x / ((x + y) * (y - ((-1.0d0) - x))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.65e+155) {
		tmp = (1.0 / (x + y)) / ((x + y) / y);
	} else {
		tmp = (y / (x + y)) * (x / ((x + y) * (y - (-1.0 - x))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.65e+155:
		tmp = (1.0 / (x + y)) / ((x + y) / y)
	else:
		tmp = (y / (x + y)) * (x / ((x + y) * (y - (-1.0 - x))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.65e+155)
		tmp = Float64(Float64(1.0 / Float64(x + y)) / Float64(Float64(x + y) / y));
	else
		tmp = Float64(Float64(y / Float64(x + y)) * Float64(x / Float64(Float64(x + y) * Float64(y - Float64(-1.0 - x)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.65e+155)
		tmp = (1.0 / (x + y)) / ((x + y) / y);
	else
		tmp = (y / (x + y)) * (x / ((x + y) * (y - (-1.0 - 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[x, -2.65e+155], N[(N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] * N[(y - N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.64999999999999982e155

   1. Initial program 49.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 49.4%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{x}} \]
   4. Step-by-step derivation

      1. *-commutative 49.4%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot x} \]

      2. associate-*l* 49.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot x\right)}} \]

      3. +-commutative 49.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot x\right)} \]

      4. times-frac 68.6%

         \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(x + y\right) \cdot x}} \]

      5. +-commutative 68.6%

         \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{x}{\left(x + y\right) \cdot x} \]

   5. Applied egg-rr 68.6%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot x}} \]
   6. Step-by-step derivation

      1. clear-num 68.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + y}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot x} \]

      2. associate-/r* 91.8%

         \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x}} \]

      3. frac-times 85.8%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot x}} \]

      4. *-un-lft-identity 85.8%

         \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{\frac{x + y}{y} \cdot x} \]

   7. Applied egg-rr 85.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{x + y}{y} \cdot x}} \]
   8. Step-by-step derivation

      1. div-inv 85.8%

         \[\leadsto \frac{\color{blue}{x \cdot \frac{1}{x + y}}}{\frac{x + y}{y} \cdot x} \]

      2. *-commutative 85.8%

         \[\leadsto \frac{x \cdot \frac{1}{x + y}}{\color{blue}{x \cdot \frac{x + y}{y}}} \]

      3. times-frac 91.9%

         \[\leadsto \color{blue}{\frac{x}{x} \cdot \frac{\frac{1}{x + y}}{\frac{x + y}{y}}} \]

   9. Applied egg-rr 91.9%

      \[\leadsto \color{blue}{\frac{x}{x} \cdot \frac{\frac{1}{x + y}}{\frac{x + y}{y}}} \]
   10. Step-by-step derivation

       1. *-inverses 91.9%

          \[\leadsto \color{blue}{1} \cdot \frac{\frac{1}{x + y}}{\frac{x + y}{y}} \]

       2. associate-*r/ 91.9%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{1}{x + y}}{\frac{x + y}{y}}} \]

   11. Simplified 91.9%

       \[\leadsto \color{blue}{\frac{1 \cdot \frac{1}{x + y}}{\frac{x + y}{y}}} \]

   if -2.64999999999999982e155 < x

   1. Initial program 70.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 70.6%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. associate-*l* 70.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      3. times-frac 96.9%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. +-commutative 96.9%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. +-commutative 96.9%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-+r+ 96.9%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      7. +-commutative 96.9%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      8. associate-+l+ 96.9%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

   4. Applied egg-rr 96.9%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{1}{x + y}}{\frac{x + y}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(y - \left(-1 - x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{\frac{x}{x + y}}{\left(y - \left(-1 - x\right)\right) \cdot \left(\frac{x}{y} + 1\right)} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (/ (/ x (+ x y)) (* (- y (- -1.0 x)) (+ (/ x y) 1.0))))

C

assert(x < y);
double code(double x, double y) {
	return (x / (x + y)) / ((y - (-1.0 - x)) * ((x / y) + 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 / (x + y)) / ((y - ((-1.0d0) - x)) * ((x / y) + 1.0d0))
end function

Java

assert x < y;
public static double code(double x, double y) {
	return (x / (x + y)) / ((y - (-1.0 - x)) * ((x / y) + 1.0));
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return (x / (x + y)) / ((y - (-1.0 - x)) * ((x / y) + 1.0))

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(Float64(x / Float64(x + y)) / Float64(Float64(y - Float64(-1.0 - x)) * Float64(Float64(x / y) + 1.0)))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = (x / (x + y)) / ((y - (-1.0 - x)) * ((x / y) + 1.0));
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(y - N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.3%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative 67.3%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. associate-*l* 67.3%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

   3. times-frac 92.5%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   4. +-commutative 92.5%

      \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   5. +-commutative 92.5%

      \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

   6. associate-+r+ 92.5%

      \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   7. +-commutative 92.5%

      \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

   8. associate-+l+ 92.5%

      \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

4. Applied egg-rr 92.5%

   \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]
5. Step-by-step derivation

   1. clear-num 92.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{y + x}{y}}} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)} \]

   2. associate-/r* 99.8%

      \[\leadsto \frac{1}{\frac{y + x}{y}} \cdot \color{blue}{\frac{\frac{x}{y + x}}{y + \left(1 + x\right)}} \]

   3. frac-times 98.3%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y + x}}{\frac{y + x}{y} \cdot \left(y + \left(1 + x\right)\right)}} \]

   4. *-un-lft-identity 98.3%

      \[\leadsto \frac{\color{blue}{\frac{x}{y + x}}}{\frac{y + x}{y} \cdot \left(y + \left(1 + x\right)\right)} \]

   5. +-commutative 98.3%

      \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{\frac{y + x}{y} \cdot \left(y + \left(1 + x\right)\right)} \]

   6. +-commutative 98.3%

      \[\leadsto \frac{\frac{x}{x + y}}{\frac{\color{blue}{x + y}}{y} \cdot \left(y + \left(1 + x\right)\right)} \]

   7. +-commutative 98.3%

      \[\leadsto \frac{\frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(y + \color{blue}{\left(x + 1\right)}\right)} \]

6. Applied egg-rr 98.3%

   \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(y + \left(x + 1\right)\right)}} \]

7. Taylor expanded in x around 0 98.4%

   \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{\left(1 + \frac{x}{y}\right)} \cdot \left(y + \left(x + 1\right)\right)} \]

8. Final simplification 98.4%

   \[\leadsto \frac{\frac{x}{x + y}}{\left(y - \left(-1 - x\right)\right) \cdot \left(\frac{x}{y} + 1\right)} \]
9. Add Preprocessing

Alternative 19: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{\frac{y}{x + y}}{\left(y - \left(-1 - x\right)\right) \cdot \frac{x + y}{x}} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (/ (/ y (+ x y)) (* (- y (- -1.0 x)) (/ (+ x y) x))))

C

assert(x < y);
double code(double x, double y) {
	return (y / (x + y)) / ((y - (-1.0 - x)) * ((x + 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)) / ((y - ((-1.0d0) - x)) * ((x + y) / x))
end function

Java

assert x < y;
public static double code(double x, double y) {
	return (y / (x + y)) / ((y - (-1.0 - x)) * ((x + y) / x));
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return (y / (x + y)) / ((y - (-1.0 - x)) * ((x + y) / x))

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(Float64(y / Float64(x + y)) / Float64(Float64(y - Float64(-1.0 - x)) * Float64(Float64(x + y) / x)))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = (y / (x + y)) / ((y - (-1.0 - x)) * ((x + 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[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(y - N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.3%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative 67.3%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. associate-*l* 67.3%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

   3. times-frac 92.5%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   4. +-commutative 92.5%

      \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   5. +-commutative 92.5%

      \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

   6. associate-+r+ 92.5%

      \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   7. +-commutative 92.5%

      \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

   8. associate-+l+ 92.5%

      \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

4. Applied egg-rr 92.5%

   \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]
5. Step-by-step derivation

   1. frac-times 67.3%

      \[\leadsto \color{blue}{\frac{y \cdot x}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)\right)}} \]

   2. *-commutative 67.3%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)\right)} \]

   3. frac-times 92.4%

      \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]

   4. clear-num 92.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{y + x}{x}}} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)} \]

   5. associate-/r* 99.7%

      \[\leadsto \frac{1}{\frac{y + x}{x}} \cdot \color{blue}{\frac{\frac{y}{y + x}}{y + \left(1 + x\right)}} \]

   6. frac-times 99.6%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{y}{y + x}}{\frac{y + x}{x} \cdot \left(y + \left(1 + x\right)\right)}} \]

   7. *-un-lft-identity 99.6%

      \[\leadsto \frac{\color{blue}{\frac{y}{y + x}}}{\frac{y + x}{x} \cdot \left(y + \left(1 + x\right)\right)} \]

   8. +-commutative 99.6%

      \[\leadsto \frac{\frac{y}{\color{blue}{x + y}}}{\frac{y + x}{x} \cdot \left(y + \left(1 + x\right)\right)} \]

   9. +-commutative 99.6%

      \[\leadsto \frac{\frac{y}{x + y}}{\frac{\color{blue}{x + y}}{x} \cdot \left(y + \left(1 + x\right)\right)} \]

   10. +-commutative 99.6%

       \[\leadsto \frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \left(y + \color{blue}{\left(x + 1\right)}\right)} \]

6. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{\frac{x + y}{x} \cdot \left(y + \left(x + 1\right)\right)}} \]

7. Final simplification 99.6%

   \[\leadsto \frac{\frac{y}{x + y}}{\left(y - \left(-1 - x\right)\right) \cdot \frac{x + y}{x}} \]
8. Add Preprocessing

Alternative 20: 85.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y - \left(-1 - x\right)\\ \mathbf{if}\;y \leq 1.7 \cdot 10^{-149}:\\ \;\;\;\;\frac{y \cdot \frac{1}{x}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- y (- -1.0 x))))
   (if (<= y 1.7e-149) (/ (* y (/ 1.0 x)) t_0) (/ x (* (+ x y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y - (-1.0 - x);
	double tmp;
	if (y <= 1.7e-149) {
		tmp = (y * (1.0 / x)) / t_0;
	} else {
		tmp = x / ((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 = y - ((-1.0d0) - x)
    if (y <= 1.7d-149) then
        tmp = (y * (1.0d0 / x)) / t_0
    else
        tmp = x / ((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 = y - (-1.0 - x);
	double tmp;
	if (y <= 1.7e-149) {
		tmp = (y * (1.0 / x)) / t_0;
	} else {
		tmp = x / ((x + y) * t_0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y - (-1.0 - x)
	tmp = 0
	if y <= 1.7e-149:
		tmp = (y * (1.0 / x)) / t_0
	else:
		tmp = x / ((x + y) * t_0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y - Float64(-1.0 - x))
	tmp = 0.0
	if (y <= 1.7e-149)
		tmp = Float64(Float64(y * Float64(1.0 / x)) / t_0);
	else
		tmp = Float64(x / Float64(Float64(x + y) * t_0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y - (-1.0 - x);
	tmp = 0.0;
	if (y <= 1.7e-149)
		tmp = (y * (1.0 / x)) / t_0;
	else
		tmp = x / ((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[(y - N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.7e-149], N[(N[(y * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(x / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.6999999999999999e-149

   1. Initial program 63.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 75.7%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 75.7%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 75.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 63.2%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      2. associate-+r+ 63.2%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      3. times-frac 82.6%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      4. associate-*r/ 82.6%

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1}} \]

      5. pow2 82.6%

         \[\leadsto \frac{\frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot y}{\left(x + y\right) + 1} \]

      6. +-commutative 82.6%

         \[\leadsto \frac{\frac{x}{{\color{blue}{\left(y + x\right)}}^{2}} \cdot y}{\left(x + y\right) + 1} \]

      7. associate-+r+ 82.6%

         \[\leadsto \frac{\frac{x}{{\left(y + x\right)}^{2}} \cdot y}{\color{blue}{x + \left(y + 1\right)}} \]

      8. +-commutative 82.6%

         \[\leadsto \frac{\frac{x}{{\left(y + x\right)}^{2}} \cdot y}{\color{blue}{\left(y + 1\right) + x}} \]

      9. associate-+l+ 82.6%

         \[\leadsto \frac{\frac{x}{{\left(y + x\right)}^{2}} \cdot y}{\color{blue}{y + \left(1 + x\right)}} \]

   6. Applied egg-rr 82.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{{\left(y + x\right)}^{2}} \cdot y}{y + \left(1 + x\right)}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 82.6%

         \[\leadsto \frac{\frac{\color{blue}{1 \cdot x}}{{\left(y + x\right)}^{2}} \cdot y}{y + \left(1 + x\right)} \]

      2. unpow2 82.6%

         \[\leadsto \frac{\frac{1 \cdot x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right)}} \cdot y}{y + \left(1 + x\right)} \]

      3. times-frac 99.7%

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{y + x} \cdot \frac{x}{y + x}\right)} \cdot y}{y + \left(1 + x\right)} \]

      4. +-commutative 99.7%

         \[\leadsto \frac{\left(\frac{1}{\color{blue}{x + y}} \cdot \frac{x}{y + x}\right) \cdot y}{y + \left(1 + x\right)} \]

      5. +-commutative 99.7%

         \[\leadsto \frac{\left(\frac{1}{x + y} \cdot \frac{x}{\color{blue}{x + y}}\right) \cdot y}{y + \left(1 + x\right)} \]

   8. Applied egg-rr 99.7%

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{x + y} \cdot \frac{x}{x + y}\right)} \cdot y}{y + \left(1 + x\right)} \]

   9. Taylor expanded in x around inf 56.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{x}} \cdot y}{y + \left(1 + x\right)} \]

   if 1.6999999999999999e-149 < y

   1. Initial program 73.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 73.8%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. associate-*l* 73.9%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      3. times-frac 93.4%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. +-commutative 93.4%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. +-commutative 93.4%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-+r+ 93.4%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      7. +-commutative 93.4%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      8. associate-+l+ 93.4%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

   4. Applied egg-rr 93.4%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]

   5. Taylor expanded in y around inf 79.4%

      \[\leadsto \color{blue}{1} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{-149}:\\ \;\;\;\;\frac{y \cdot \frac{1}{x}}{y - \left(-1 - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y - \left(-1 - x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 61.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -60000000:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -60000000.0) (* (/ 1.0 x) (/ y x)) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -60000000.0) {
		tmp = (1.0 / x) * (y / 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 <= (-60000000.0d0)) then
        tmp = (1.0d0 / x) * (y / 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 <= -60000000.0) {
		tmp = (1.0 / x) * (y / x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -60000000.0:
		tmp = (1.0 / x) * (y / x)
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -60000000.0)
		tmp = Float64(Float64(1.0 / x) * Float64(y / 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 <= -60000000.0)
		tmp = (1.0 / x) * (y / 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, -60000000.0], N[(N[(1.0 / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6e7

   1. Initial program 57.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 57.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. associate-*l* 57.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      3. times-frac 80.2%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. +-commutative 80.2%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. +-commutative 80.2%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-+r+ 80.2%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      7. +-commutative 80.2%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      8. associate-+l+ 80.2%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

   4. Applied egg-rr 80.2%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]

   5. Taylor expanded in x around inf 86.5%

      \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{x}} \]

   6. Taylor expanded in y around 0 86.3%

      \[\leadsto \color{blue}{\frac{y}{x}} \cdot \frac{1}{x} \]

   if -6e7 < x

   1. Initial program 70.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 82.0%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 82.0%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 82.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 64.4%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 64.4%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   7. Simplified 64.4%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   8. Taylor expanded in y around 0 42.4%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -60000000:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 78.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+17}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.9e+17) (* (/ 1.0 x) (/ y x)) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.9e+17) {
		tmp = (1.0 / x) * (y / 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 <= (-2.9d+17)) then
        tmp = (1.0d0 / x) * (y / 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 <= -2.9e+17) {
		tmp = (1.0 / x) * (y / x);
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.9e+17:
		tmp = (1.0 / x) * (y / 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 <= -2.9e+17)
		tmp = Float64(Float64(1.0 / x) * Float64(y / 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 <= -2.9e+17)
		tmp = (1.0 / x) * (y / 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, -2.9e+17], N[(N[(1.0 / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.9e17

   1. Initial program 55.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 55.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. associate-*l* 55.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      3. times-frac 79.2%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. +-commutative 79.2%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. +-commutative 79.2%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-+r+ 79.2%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      7. +-commutative 79.2%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      8. associate-+l+ 79.2%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

   4. Applied egg-rr 79.2%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]

   5. Taylor expanded in x around inf 88.1%

      \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{x}} \]

   6. Taylor expanded in y around 0 88.0%

      \[\leadsto \color{blue}{\frac{y}{x}} \cdot \frac{1}{x} \]

   if -2.9e17 < x

   1. Initial program 70.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 82.2%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 82.2%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 82.2%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 64.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 64.0%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   7. Simplified 64.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+17}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 28.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -2.85e+18) (/ 1.0 x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.85e+18) {
		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 <= (-2.85d+18)) 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 <= -2.85e+18) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.85e+18:
		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 <= -2.85e+18)
		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 <= -2.85e+18)
		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, -2.85e+18], N[(1.0 / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.85e18

   1. Initial program 55.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 55.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. associate-*l* 55.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      3. times-frac 79.2%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. +-commutative 79.2%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. +-commutative 79.2%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-+r+ 79.2%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      7. +-commutative 79.2%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      8. associate-+l+ 79.2%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

   4. Applied egg-rr 79.2%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]

   5. Taylor expanded in x around inf 88.1%

      \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{x}} \]

   6. Taylor expanded in y around inf 6.2%

      \[\leadsto \color{blue}{\frac{1}{x}} \]

   if -2.85e18 < x

   1. Initial program 70.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 82.2%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 82.2%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 82.2%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 64.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 64.0%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   7. Simplified 64.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   8. Taylor expanded in y around 0 41.8%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 4.3% 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 67.3%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative 67.3%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. associate-*l* 67.3%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

   3. times-frac 92.5%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   4. +-commutative 92.5%

      \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   5. +-commutative 92.5%

      \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

   6. associate-+r+ 92.5%

      \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   7. +-commutative 92.5%

      \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

   8. associate-+l+ 92.5%

      \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

4. Applied egg-rr 92.5%

   \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]

5. Taylor expanded in x around inf 38.6%

   \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{x}} \]

6. Taylor expanded in y around inf 4.3%

   \[\leadsto \color{blue}{\frac{1}{x}} \]

7. Final simplification 4.3%

   \[\leadsto \frac{1}{x} \]
8. Add Preprocessing

Developer target: 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 2024078 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))