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

Percentage Accurate: 69.2% → 99.8%

Time: 12.6s

Alternatives: 18

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.8%

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

   1. *-un-lft-identity 66.8%

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

   2. associate-*l* 66.8%

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

   3. times-frac 70.9%

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

   4. +-commutative 70.9%

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

   5. *-commutative 70.9%

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

   6. +-commutative 70.9%

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

   7. associate-+r+ 70.9%

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

   8. +-commutative 70.9%

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

   9. associate-+l+ 70.9%

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

4. Applied egg-rr 70.9%

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

   1. *-commutative 70.9%

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

   2. times-frac 99.7%

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

   3. +-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-*l/ 99.9%

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

   2. *-un-lft-identity 99.9%

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

   3. +-commutative 99.9%

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

   4. associate-+l+ 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 91.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{+32} \lor \neg \left(x \leq -0.185\right):\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{y + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.05e+54)
   (* (/ 1.0 (+ y x)) (/ y x))
   (if (or (<= x -5.5e+32) (not (<= x -0.185)))
     (/ (* (/ x (+ y x)) (/ y (+ y 1.0))) (+ y x))
     (/ (/ y (+ y x)) (+ x 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.05e+54) {
		tmp = (1.0 / (y + x)) * (y / x);
	} else if ((x <= -5.5e+32) || !(x <= -0.185)) {
		tmp = ((x / (y + x)) * (y / (y + 1.0))) / (y + x);
	} else {
		tmp = (y / (y + x)) / (x + 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 <= (-1.05d+54)) then
        tmp = (1.0d0 / (y + x)) * (y / x)
    else if ((x <= (-5.5d+32)) .or. (.not. (x <= (-0.185d0)))) then
        tmp = ((x / (y + x)) * (y / (y + 1.0d0))) / (y + x)
    else
        tmp = (y / (y + x)) / (x + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.05e+54) {
		tmp = (1.0 / (y + x)) * (y / x);
	} else if ((x <= -5.5e+32) || !(x <= -0.185)) {
		tmp = ((x / (y + x)) * (y / (y + 1.0))) / (y + x);
	} else {
		tmp = (y / (y + x)) / (x + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.05e+54:
		tmp = (1.0 / (y + x)) * (y / x)
	elif (x <= -5.5e+32) or not (x <= -0.185):
		tmp = ((x / (y + x)) * (y / (y + 1.0))) / (y + x)
	else:
		tmp = (y / (y + x)) / (x + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.05e+54)
		tmp = Float64(Float64(1.0 / Float64(y + x)) * Float64(y / x));
	elseif ((x <= -5.5e+32) || !(x <= -0.185))
		tmp = Float64(Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(y + 1.0))) / Float64(y + x));
	else
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.05e+54)
		tmp = (1.0 / (y + x)) * (y / x);
	elseif ((x <= -5.5e+32) || ~((x <= -0.185)))
		tmp = ((x / (y + x)) * (y / (y + 1.0))) / (y + x);
	else
		tmp = (y / (y + x)) / (x + 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, -1.05e+54], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -5.5e+32], N[Not[LessEqual[x, -0.185]], $MachinePrecision]], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.04999999999999993e54

   1. Initial program 50.8%

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

      1. *-un-lft-identity 50.8%

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

      2. associate-*l* 50.8%

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

      3. times-frac 60.7%

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

      4. +-commutative 60.7%

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

      5. *-commutative 60.7%

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

      6. +-commutative 60.7%

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

      7. associate-+r+ 60.7%

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

      8. +-commutative 60.7%

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

      9. associate-+l+ 60.7%

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

   4. Applied egg-rr 60.7%

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

      1. *-commutative 60.7%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf 75.5%

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

   if -1.04999999999999993e54 < x < -5.49999999999999984e32 or -0.185 < x

   1. Initial program 70.3%

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

      1. *-un-lft-identity 70.3%

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

      2. associate-*l* 70.3%

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

      3. times-frac 72.7%

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

      4. +-commutative 72.7%

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

      5. *-commutative 72.7%

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

      6. +-commutative 72.7%

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

      7. associate-+r+ 72.7%

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

      8. +-commutative 72.7%

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

      9. associate-+l+ 72.7%

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

   4. Applied egg-rr 72.7%

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

      1. *-commutative 72.7%

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

      2. times-frac 99.7%

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

      3. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.9%

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

      2. *-un-lft-identity 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l+ 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around 0 80.9%

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

       1. +-commutative 80.9%

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

   11. Simplified 80.9%

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

   if -5.49999999999999984e32 < x < -0.185

   1. Initial program 76.6%

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

      1. *-un-lft-identity 76.6%

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

      2. associate-*l* 76.6%

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

      3. times-frac 98.4%

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

      4. +-commutative 98.4%

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

      5. *-commutative 98.4%

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

      6. +-commutative 98.4%

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

      7. associate-+r+ 98.4%

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

      8. +-commutative 98.4%

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

      9. associate-+l+ 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in y around 0 52.4%

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

      1. +-commutative 52.4%

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

   7. Simplified 52.4%

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

      1. associate-*r/ 52.8%

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

      2. associate-*l/ 52.8%

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

      3. *-un-lft-identity 52.8%

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

      4. +-commutative 52.8%

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

   9. Applied egg-rr 52.8%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{+32} \lor \neg \left(x \leq -0.185\right):\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{y + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+64}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-14}:\\ \;\;\;\;\frac{y \cdot x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(1 + \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.1e+64)
   (* (/ 1.0 (+ y x)) (/ y x))
   (if (<= x -1.85e-14)
     (/ (* y x) (* (* (+ y x) (+ y x)) (+ 1.0 (+ y x))))
     (/ (* (/ x (+ y x)) (/ y (+ y 1.0))) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.1e+64) {
		tmp = (1.0 / (y + x)) * (y / x);
	} else if (x <= -1.85e-14) {
		tmp = (y * x) / (((y + x) * (y + x)) * (1.0 + (y + x)));
	} else {
		tmp = ((x / (y + x)) * (y / (y + 1.0))) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.1d+64)) then
        tmp = (1.0d0 / (y + x)) * (y / x)
    else if (x <= (-1.85d-14)) then
        tmp = (y * x) / (((y + x) * (y + x)) * (1.0d0 + (y + x)))
    else
        tmp = ((x / (y + x)) * (y / (y + 1.0d0))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -3.1e+64) {
		tmp = (1.0 / (y + x)) * (y / x);
	} else if (x <= -1.85e-14) {
		tmp = (y * x) / (((y + x) * (y + x)) * (1.0 + (y + x)));
	} else {
		tmp = ((x / (y + x)) * (y / (y + 1.0))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.1e+64:
		tmp = (1.0 / (y + x)) * (y / x)
	elif x <= -1.85e-14:
		tmp = (y * x) / (((y + x) * (y + x)) * (1.0 + (y + x)))
	else:
		tmp = ((x / (y + x)) * (y / (y + 1.0))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.1e+64)
		tmp = Float64(Float64(1.0 / Float64(y + x)) * Float64(y / x));
	elseif (x <= -1.85e-14)
		tmp = Float64(Float64(y * x) / Float64(Float64(Float64(y + x) * Float64(y + x)) * Float64(1.0 + Float64(y + x))));
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(y + 1.0))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.1e+64)
		tmp = (1.0 / (y + x)) * (y / x);
	elseif (x <= -1.85e-14)
		tmp = (y * x) / (((y + x) * (y + x)) * (1.0 + (y + x)));
	else
		tmp = ((x / (y + x)) * (y / (y + 1.0))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -3.1e+64], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.85e-14], N[(N[(y * x), $MachinePrecision] / N[(N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.0999999999999999e64

   1. Initial program 50.8%

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

      1. *-un-lft-identity 50.8%

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

      2. associate-*l* 50.8%

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

      3. times-frac 60.7%

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

      4. +-commutative 60.7%

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

      5. *-commutative 60.7%

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

      6. +-commutative 60.7%

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

      7. associate-+r+ 60.7%

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

      8. +-commutative 60.7%

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

      9. associate-+l+ 60.7%

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

   4. Applied egg-rr 60.7%

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

      1. *-commutative 60.7%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf 75.5%

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

   if -3.0999999999999999e64 < x < -1.85000000000000001e-14

   1. Initial program 85.0%

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

   if -1.85000000000000001e-14 < x

   1. Initial program 69.5%

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

      1. *-un-lft-identity 69.5%

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

      2. associate-*l* 69.5%

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

      3. times-frac 72.0%

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

      4. +-commutative 72.0%

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

      5. *-commutative 72.0%

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

      6. +-commutative 72.0%

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

      7. associate-+r+ 72.0%

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

      8. +-commutative 72.0%

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

      9. associate-+l+ 72.0%

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

   4. Applied egg-rr 72.0%

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

      1. *-commutative 72.0%

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

      2. times-frac 99.7%

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

      3. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.9%

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

      2. *-un-lft-identity 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l+ 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around 0 80.5%

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

       1. +-commutative 80.5%

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

   11. Simplified 80.5%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+64}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-14}:\\ \;\;\;\;\frac{y \cdot x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(1 + \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+64}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -8.8 \cdot 10^{-45}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.6e+64)
   (* (/ 1.0 (+ y x)) (/ y x))
   (if (<= x -8.8e-45)
     (* x (/ (/ y (* (+ y x) (+ y (+ x 1.0)))) (+ y x)))
     (/ (* (/ x (+ y x)) (/ y (+ y 1.0))) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.6e+64) {
		tmp = (1.0 / (y + x)) * (y / x);
	} else if (x <= -8.8e-45) {
		tmp = x * ((y / ((y + x) * (y + (x + 1.0)))) / (y + x));
	} else {
		tmp = ((x / (y + x)) * (y / (y + 1.0))) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.6d+64)) then
        tmp = (1.0d0 / (y + x)) * (y / x)
    else if (x <= (-8.8d-45)) then
        tmp = x * ((y / ((y + x) * (y + (x + 1.0d0)))) / (y + x))
    else
        tmp = ((x / (y + x)) * (y / (y + 1.0d0))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.6e+64) {
		tmp = (1.0 / (y + x)) * (y / x);
	} else if (x <= -8.8e-45) {
		tmp = x * ((y / ((y + x) * (y + (x + 1.0)))) / (y + x));
	} else {
		tmp = ((x / (y + x)) * (y / (y + 1.0))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.6e+64:
		tmp = (1.0 / (y + x)) * (y / x)
	elif x <= -8.8e-45:
		tmp = x * ((y / ((y + x) * (y + (x + 1.0)))) / (y + x))
	else:
		tmp = ((x / (y + x)) * (y / (y + 1.0))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.6e+64)
		tmp = Float64(Float64(1.0 / Float64(y + x)) * Float64(y / x));
	elseif (x <= -8.8e-45)
		tmp = Float64(x * Float64(Float64(y / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0)))) / Float64(y + x)));
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(y + 1.0))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.6e+64)
		tmp = (1.0 / (y + x)) * (y / x);
	elseif (x <= -8.8e-45)
		tmp = x * ((y / ((y + x) * (y + (x + 1.0)))) / (y + x));
	else
		tmp = ((x / (y + x)) * (y / (y + 1.0))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.6e+64], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8.8e-45], N[(x * N[(N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.59999999999999997e64

   1. Initial program 50.8%

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

      1. *-un-lft-identity 50.8%

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

      2. associate-*l* 50.8%

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

      3. times-frac 60.7%

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

      4. +-commutative 60.7%

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

      5. *-commutative 60.7%

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

      6. +-commutative 60.7%

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

      7. associate-+r+ 60.7%

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

      8. +-commutative 60.7%

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

      9. associate-+l+ 60.7%

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

   4. Applied egg-rr 60.7%

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

      1. *-commutative 60.7%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf 75.5%

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

   if -2.59999999999999997e64 < x < -8.79999999999999974e-45

   1. Initial program 85.6%

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

      1. associate-/l* 84.0%

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

      2. associate-+l+ 84.0%

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

   3. Simplified 84.0%

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

      1. *-un-lft-identity 84.0%

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

      2. associate-+r+ 84.0%

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

      3. associate-*l* 84.0%

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

      4. times-frac 93.1%

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

      5. +-commutative 93.1%

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

      6. +-commutative 93.1%

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

      7. associate-+r+ 93.1%

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

      8. +-commutative 93.1%

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

      9. associate-+l+ 93.1%

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

   6. Applied egg-rr 93.1%

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

      1. associate-*l/ 93.1%

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

      2. *-lft-identity 93.1%

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

      3. +-commutative 93.1%

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

   8. Simplified 93.1%

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

   if -8.79999999999999974e-45 < x

   1. Initial program 68.8%

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

      1. *-un-lft-identity 68.8%

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

      2. associate-*l* 68.9%

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

      3. times-frac 71.0%

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

      4. +-commutative 71.0%

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

      5. *-commutative 71.0%

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

      6. +-commutative 71.0%

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

      7. associate-+r+ 71.0%

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

      8. +-commutative 71.0%

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

      9. associate-+l+ 71.0%

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

   4. Applied egg-rr 71.0%

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

      1. *-commutative 71.0%

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

      2. times-frac 99.6%

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

      3. +-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-*l/ 99.9%

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

      2. *-un-lft-identity 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l+ 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around 0 79.8%

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

       1. +-commutative 79.8%

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

   11. Simplified 79.8%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+64}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -8.8 \cdot 10^{-45}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+64}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-20}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.1e+64)
   (* (/ 1.0 (+ y x)) (/ y x))
   (if (<= x -5e-20)
     (* x (/ y (* (* (+ y x) (+ y x)) (+ x (+ y 1.0)))))
     (/ (* (/ x (+ y x)) (/ y (+ y 1.0))) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.1e+64) {
		tmp = (1.0 / (y + x)) * (y / x);
	} else if (x <= -5e-20) {
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))));
	} else {
		tmp = ((x / (y + x)) * (y / (y + 1.0))) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.1d+64)) then
        tmp = (1.0d0 / (y + x)) * (y / x)
    else if (x <= (-5d-20)) then
        tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0d0))))
    else
        tmp = ((x / (y + x)) * (y / (y + 1.0d0))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -3.1e+64) {
		tmp = (1.0 / (y + x)) * (y / x);
	} else if (x <= -5e-20) {
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))));
	} else {
		tmp = ((x / (y + x)) * (y / (y + 1.0))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.1e+64:
		tmp = (1.0 / (y + x)) * (y / x)
	elif x <= -5e-20:
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))))
	else:
		tmp = ((x / (y + x)) * (y / (y + 1.0))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.1e+64)
		tmp = Float64(Float64(1.0 / Float64(y + x)) * Float64(y / x));
	elseif (x <= -5e-20)
		tmp = Float64(x * Float64(y / Float64(Float64(Float64(y + x) * Float64(y + x)) * Float64(x + Float64(y + 1.0)))));
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(y + 1.0))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.1e+64)
		tmp = (1.0 / (y + x)) * (y / x);
	elseif (x <= -5e-20)
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))));
	else
		tmp = ((x / (y + x)) * (y / (y + 1.0))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -3.1e+64], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5e-20], N[(x * N[(y / N[(N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.0999999999999999e64

   1. Initial program 50.8%

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

      1. *-un-lft-identity 50.8%

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

      2. associate-*l* 50.8%

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

      3. times-frac 60.7%

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

      4. +-commutative 60.7%

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

      5. *-commutative 60.7%

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

      6. +-commutative 60.7%

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

      7. associate-+r+ 60.7%

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

      8. +-commutative 60.7%

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

      9. associate-+l+ 60.7%

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

   4. Applied egg-rr 60.7%

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

      1. *-commutative 60.7%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf 75.5%

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

   if -3.0999999999999999e64 < x < -4.9999999999999999e-20

   1. Initial program 86.1%

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

      1. associate-/l* 83.6%

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

      2. associate-+l+ 83.6%

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

   3. Simplified 83.6%

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

   if -4.9999999999999999e-20 < x

   1. Initial program 69.3%

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

      1. *-un-lft-identity 69.3%

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

      2. associate-*l* 69.3%

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

      3. times-frac 71.8%

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

      4. +-commutative 71.8%

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

      5. *-commutative 71.8%

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

      6. +-commutative 71.8%

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

      7. associate-+r+ 71.8%

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

      8. +-commutative 71.8%

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

      9. associate-+l+ 71.8%

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

   4. Applied egg-rr 71.8%

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

      1. *-commutative 71.8%

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

      2. times-frac 99.6%

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

      3. +-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-*l/ 99.9%

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

      2. *-un-lft-identity 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l+ 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around 0 80.4%

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

       1. +-commutative 80.4%

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

   11. Simplified 80.4%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+64}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-20}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+66}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+141}:\\ \;\;\;\;x \cdot \frac{\frac{1}{y}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.5e-85)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 1.55e+66)
     (* (/ 1.0 (+ y x)) (/ x (+ y 1.0)))
     (if (<= y 3.4e+141)
       (* x (/ (/ 1.0 y) (+ y x)))
       (/ (/ x (+ y x)) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.5e-85) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 1.55e+66) {
		tmp = (1.0 / (y + x)) * (x / (y + 1.0));
	} else if (y <= 3.4e+141) {
		tmp = x * ((1.0 / y) / (y + x));
	} else {
		tmp = (x / (y + x)) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.5d-85) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 1.55d+66) then
        tmp = (1.0d0 / (y + x)) * (x / (y + 1.0d0))
    else if (y <= 3.4d+141) then
        tmp = x * ((1.0d0 / y) / (y + x))
    else
        tmp = (x / (y + x)) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.5e-85) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 1.55e+66) {
		tmp = (1.0 / (y + x)) * (x / (y + 1.0));
	} else if (y <= 3.4e+141) {
		tmp = x * ((1.0 / y) / (y + x));
	} else {
		tmp = (x / (y + x)) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.5e-85:
		tmp = (y / x) / (x + 1.0)
	elif y <= 1.55e+66:
		tmp = (1.0 / (y + x)) * (x / (y + 1.0))
	elif y <= 3.4e+141:
		tmp = x * ((1.0 / y) / (y + x))
	else:
		tmp = (x / (y + x)) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.5e-85)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 1.55e+66)
		tmp = Float64(Float64(1.0 / Float64(y + x)) * Float64(x / Float64(y + 1.0)));
	elseif (y <= 3.4e+141)
		tmp = Float64(x * Float64(Float64(1.0 / y) / Float64(y + x)));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.5e-85)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 1.55e+66)
		tmp = (1.0 / (y + x)) * (x / (y + 1.0));
	elseif (y <= 3.4e+141)
		tmp = x * ((1.0 / y) / (y + x));
	else
		tmp = (x / (y + x)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 2.5e-85], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.55e+66], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.4e+141], N[(x * N[(N[(1.0 / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 2.5000000000000001e-85

   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. Step-by-step derivation

      1. associate-/l* 78.6%

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

      2. associate-+l+ 78.6%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in y around 0 55.7%

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

      1. associate-/r* 55.3%

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

      2. +-commutative 55.3%

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

   7. Simplified 55.3%

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

   if 2.5000000000000001e-85 < y < 1.55000000000000009e66

   1. Initial program 91.2%

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

      1. *-un-lft-identity 91.2%

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

      2. associate-*l* 91.2%

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

      3. times-frac 94.2%

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

      4. +-commutative 94.2%

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

      5. *-commutative 94.2%

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

      6. +-commutative 94.2%

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

      7. associate-+r+ 94.2%

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

      8. +-commutative 94.2%

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

      9. associate-+l+ 94.2%

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

   4. Applied egg-rr 94.2%

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

   5. Taylor expanded in x around 0 47.0%

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

      1. +-commutative 47.0%

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

   7. Simplified 47.0%

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

   if 1.55000000000000009e66 < y < 3.3999999999999998e141

   1. Initial program 45.1%

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

      1. associate-/l* 61.2%

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

      2. associate-+l+ 61.2%

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

   3. Simplified 61.2%

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

      1. *-un-lft-identity 61.2%

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

      2. associate-+r+ 61.2%

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

      3. associate-*l* 61.2%

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

      4. times-frac 86.1%

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

      5. +-commutative 86.1%

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

      6. +-commutative 86.1%

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

      7. associate-+r+ 86.1%

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

      8. +-commutative 86.1%

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

      9. associate-+l+ 86.1%

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

   6. Applied egg-rr 86.1%

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

      1. associate-*l/ 85.9%

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

      2. *-lft-identity 85.9%

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

      3. +-commutative 85.9%

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

   8. Simplified 85.9%

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

   9. Taylor expanded in y around inf 78.0%

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

   if 3.3999999999999998e141 < y

   1. Initial program 58.0%

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

      1. *-un-lft-identity 58.0%

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

      2. associate-*l* 58.0%

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

      3. times-frac 60.1%

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

      4. +-commutative 60.1%

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

      5. *-commutative 60.1%

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

      6. +-commutative 60.1%

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

      7. associate-+r+ 60.1%

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

      8. +-commutative 60.1%

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

      9. associate-+l+ 60.1%

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

   4. Applied egg-rr 60.1%

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

      1. *-commutative 60.1%

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

      2. times-frac 99.9%

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

      3. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. associate-*l/ 99.9%

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

      2. *-un-lft-identity 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l+ 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in y around inf 80.9%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+66}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+141}:\\ \;\;\;\;x \cdot \frac{\frac{1}{y}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+65}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{x}{y + 1}\\ \mathbf{elif}\;y \leq 1.76 \cdot 10^{+158}:\\ \;\;\;\;x \cdot \frac{\frac{1}{y}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y + x}{\frac{x}{y}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.8e-85)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 2e+65)
     (* (/ 1.0 (+ y x)) (/ x (+ y 1.0)))
     (if (<= y 1.76e+158)
       (* x (/ (/ 1.0 y) (+ y x)))
       (/ 1.0 (/ (+ y x) (/ x y)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.8e-85) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 2e+65) {
		tmp = (1.0 / (y + x)) * (x / (y + 1.0));
	} else if (y <= 1.76e+158) {
		tmp = x * ((1.0 / y) / (y + x));
	} else {
		tmp = 1.0 / ((y + x) / (x / y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.8d-85) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 2d+65) then
        tmp = (1.0d0 / (y + x)) * (x / (y + 1.0d0))
    else if (y <= 1.76d+158) then
        tmp = x * ((1.0d0 / y) / (y + x))
    else
        tmp = 1.0d0 / ((y + x) / (x / y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.8e-85) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 2e+65) {
		tmp = (1.0 / (y + x)) * (x / (y + 1.0));
	} else if (y <= 1.76e+158) {
		tmp = x * ((1.0 / y) / (y + x));
	} else {
		tmp = 1.0 / ((y + x) / (x / y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.8e-85:
		tmp = (y / x) / (x + 1.0)
	elif y <= 2e+65:
		tmp = (1.0 / (y + x)) * (x / (y + 1.0))
	elif y <= 1.76e+158:
		tmp = x * ((1.0 / y) / (y + x))
	else:
		tmp = 1.0 / ((y + x) / (x / y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.8e-85)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 2e+65)
		tmp = Float64(Float64(1.0 / Float64(y + x)) * Float64(x / Float64(y + 1.0)));
	elseif (y <= 1.76e+158)
		tmp = Float64(x * Float64(Float64(1.0 / y) / Float64(y + x)));
	else
		tmp = Float64(1.0 / Float64(Float64(y + x) / Float64(x / y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.8e-85)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 2e+65)
		tmp = (1.0 / (y + x)) * (x / (y + 1.0));
	elseif (y <= 1.76e+158)
		tmp = x * ((1.0 / y) / (y + x));
	else
		tmp = 1.0 / ((y + x) / (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 2.8e-85], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+65], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.76e+158], N[(x * N[(N[(1.0 / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(y + x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 2.80000000000000017e-85

   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. Step-by-step derivation

      1. associate-/l* 78.6%

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

      2. associate-+l+ 78.6%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in y around 0 55.7%

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

      1. associate-/r* 55.3%

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

      2. +-commutative 55.3%

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

   7. Simplified 55.3%

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

   if 2.80000000000000017e-85 < y < 2e65

   1. Initial program 91.2%

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

      1. *-un-lft-identity 91.2%

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

      2. associate-*l* 91.2%

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

      3. times-frac 94.2%

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

      4. +-commutative 94.2%

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

      5. *-commutative 94.2%

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

      6. +-commutative 94.2%

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

      7. associate-+r+ 94.2%

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

      8. +-commutative 94.2%

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

      9. associate-+l+ 94.2%

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

   4. Applied egg-rr 94.2%

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

   5. Taylor expanded in x around 0 47.0%

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

      1. +-commutative 47.0%

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

   7. Simplified 47.0%

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

   if 2e65 < y < 1.7600000000000001e158

   1. Initial program 47.1%

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

      1. associate-/l* 61.5%

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

      2. associate-+l+ 61.5%

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

   3. Simplified 61.5%

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

      1. *-un-lft-identity 61.5%

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

      2. associate-+r+ 61.5%

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

      3. associate-*l* 61.5%

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

      4. times-frac 79.6%

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

      5. +-commutative 79.6%

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

      6. +-commutative 79.6%

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

      7. associate-+r+ 79.6%

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

      8. +-commutative 79.6%

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

      9. associate-+l+ 79.6%

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

   6. Applied egg-rr 79.6%

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

      1. associate-*l/ 79.6%

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

      2. *-lft-identity 79.6%

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

      3. +-commutative 79.6%

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

   8. Simplified 79.6%

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

   9. Taylor expanded in y around inf 74.1%

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

   if 1.7600000000000001e158 < y

   1. Initial program 60.3%

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

      1. *-un-lft-identity 60.3%

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

      2. associate-*l* 60.3%

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

      3. times-frac 60.3%

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

      4. +-commutative 60.3%

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

      5. *-commutative 60.3%

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

      6. +-commutative 60.3%

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

      7. associate-+r+ 60.3%

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

      8. +-commutative 60.3%

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

      9. associate-+l+ 60.3%

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

   4. Applied egg-rr 60.3%

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

      1. *-commutative 60.3%

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

      2. times-frac 99.9%

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

      3. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around inf 86.5%

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

      1. associate-*l/ 86.6%

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

      2. *-un-lft-identity 86.6%

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

      3. clear-num 86.6%

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

   9. Applied egg-rr 86.6%

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

Alternative 8: 83.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \mathbf{elif}\;y \leq 1.62 \cdot 10^{+159}:\\ \;\;\;\;x \cdot \frac{\frac{1}{y}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y + x}{\frac{x}{y}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.8e-85)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 4.9e+38)
     (/ (/ x (+ y 1.0)) y)
     (if (<= y 1.62e+159)
       (* x (/ (/ 1.0 y) (+ y x)))
       (/ 1.0 (/ (+ y x) (/ x y)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.8e-85) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 4.9e+38) {
		tmp = (x / (y + 1.0)) / y;
	} else if (y <= 1.62e+159) {
		tmp = x * ((1.0 / y) / (y + x));
	} else {
		tmp = 1.0 / ((y + x) / (x / y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.8d-85) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 4.9d+38) then
        tmp = (x / (y + 1.0d0)) / y
    else if (y <= 1.62d+159) then
        tmp = x * ((1.0d0 / y) / (y + x))
    else
        tmp = 1.0d0 / ((y + x) / (x / y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.8e-85) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 4.9e+38) {
		tmp = (x / (y + 1.0)) / y;
	} else if (y <= 1.62e+159) {
		tmp = x * ((1.0 / y) / (y + x));
	} else {
		tmp = 1.0 / ((y + x) / (x / y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.8e-85:
		tmp = (y / x) / (x + 1.0)
	elif y <= 4.9e+38:
		tmp = (x / (y + 1.0)) / y
	elif y <= 1.62e+159:
		tmp = x * ((1.0 / y) / (y + x))
	else:
		tmp = 1.0 / ((y + x) / (x / y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.8e-85)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 4.9e+38)
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	elseif (y <= 1.62e+159)
		tmp = Float64(x * Float64(Float64(1.0 / y) / Float64(y + x)));
	else
		tmp = Float64(1.0 / Float64(Float64(y + x) / Float64(x / y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.8e-85)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 4.9e+38)
		tmp = (x / (y + 1.0)) / y;
	elseif (y <= 1.62e+159)
		tmp = x * ((1.0 / y) / (y + x));
	else
		tmp = 1.0 / ((y + x) / (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 2.8e-85], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.9e+38], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 1.62e+159], N[(x * N[(N[(1.0 / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(y + x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 2.80000000000000017e-85

   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. Step-by-step derivation

      1. associate-/l* 78.6%

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

      2. associate-+l+ 78.6%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in y around 0 55.7%

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

      1. associate-/r* 55.3%

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

      2. +-commutative 55.3%

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

   7. Simplified 55.3%

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

   if 2.80000000000000017e-85 < y < 4.90000000000000002e38

   1. Initial program 93.6%

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

      1. associate-/l* 99.2%

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

      2. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in x around 0 44.5%

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

      1. +-commutative 44.5%

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

   7. Simplified 44.5%

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

      1. *-un-lft-identity 44.5%

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

      2. times-frac 44.3%

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

   9. Applied egg-rr 44.3%

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

       1. associate-*l/ 44.5%

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

       2. *-lft-identity 44.5%

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

   11. Simplified 44.5%

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

   if 4.90000000000000002e38 < y < 1.62e159

   1. Initial program 49.7%

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

      1. associate-/l* 62.3%

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

      2. associate-+l+ 62.3%

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

   3. Simplified 62.3%

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

      1. *-un-lft-identity 62.3%

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

      2. associate-+r+ 62.3%

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

      3. associate-*l* 62.3%

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

      4. times-frac 78.3%

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

      5. +-commutative 78.3%

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

      6. +-commutative 78.3%

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

      7. associate-+r+ 78.3%

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

      8. +-commutative 78.3%

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

      9. associate-+l+ 78.3%

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

   6. Applied egg-rr 78.3%

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

      1. associate-*l/ 78.3%

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

      2. *-lft-identity 78.3%

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

      3. +-commutative 78.3%

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

   8. Simplified 78.3%

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

   9. Taylor expanded in y around inf 73.4%

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

   if 1.62e159 < y

   1. Initial program 60.3%

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

      1. *-un-lft-identity 60.3%

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

      2. associate-*l* 60.3%

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

      3. times-frac 60.3%

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

      4. +-commutative 60.3%

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

      5. *-commutative 60.3%

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

      6. +-commutative 60.3%

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

      7. associate-+r+ 60.3%

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

      8. +-commutative 60.3%

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

      9. associate-+l+ 60.3%

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

   4. Applied egg-rr 60.3%

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

      1. *-commutative 60.3%

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

      2. times-frac 99.9%

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

      3. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around inf 86.5%

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

      1. associate-*l/ 86.6%

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

      2. *-un-lft-identity 86.6%

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

      3. clear-num 86.6%

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

   9. Applied egg-rr 86.6%

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

Alternative 9: 83.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.36 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+134}:\\ \;\;\;\;x \cdot \frac{\frac{1}{y}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.36e-85)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 4.9e+38)
     (/ (/ x (+ y 1.0)) y)
     (if (<= y 3.5e+134) (* x (/ (/ 1.0 y) (+ y x))) (/ (/ x (+ y x)) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.36e-85) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 4.9e+38) {
		tmp = (x / (y + 1.0)) / y;
	} else if (y <= 3.5e+134) {
		tmp = x * ((1.0 / y) / (y + x));
	} else {
		tmp = (x / (y + x)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.36d-85) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 4.9d+38) then
        tmp = (x / (y + 1.0d0)) / y
    else if (y <= 3.5d+134) then
        tmp = x * ((1.0d0 / y) / (y + x))
    else
        tmp = (x / (y + x)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.36e-85) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 4.9e+38) {
		tmp = (x / (y + 1.0)) / y;
	} else if (y <= 3.5e+134) {
		tmp = x * ((1.0 / y) / (y + x));
	} else {
		tmp = (x / (y + x)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.36e-85:
		tmp = (y / x) / (x + 1.0)
	elif y <= 4.9e+38:
		tmp = (x / (y + 1.0)) / y
	elif y <= 3.5e+134:
		tmp = x * ((1.0 / y) / (y + x))
	else:
		tmp = (x / (y + x)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.36e-85)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 4.9e+38)
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	elseif (y <= 3.5e+134)
		tmp = Float64(x * Float64(Float64(1.0 / y) / Float64(y + x)));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.36e-85)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 4.9e+38)
		tmp = (x / (y + 1.0)) / y;
	elseif (y <= 3.5e+134)
		tmp = x * ((1.0 / y) / (y + x));
	else
		tmp = (x / (y + x)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.36e-85], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.9e+38], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 3.5e+134], N[(x * N[(N[(1.0 / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 1.36e-85

   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. Step-by-step derivation

      1. associate-/l* 78.6%

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

      2. associate-+l+ 78.6%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in y around 0 55.7%

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

      1. associate-/r* 55.3%

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

      2. +-commutative 55.3%

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

   7. Simplified 55.3%

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

   if 1.36e-85 < y < 4.90000000000000002e38

   1. Initial program 93.6%

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

      1. associate-/l* 99.2%

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

      2. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in x around 0 44.5%

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

      1. +-commutative 44.5%

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

   7. Simplified 44.5%

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

      1. *-un-lft-identity 44.5%

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

      2. times-frac 44.3%

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

   9. Applied egg-rr 44.3%

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

       1. associate-*l/ 44.5%

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

       2. *-lft-identity 44.5%

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

   11. Simplified 44.5%

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

   if 4.90000000000000002e38 < y < 3.50000000000000003e134

   1. Initial program 46.3%

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

      1. associate-/l* 60.1%

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

      2. associate-+l+ 60.1%

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

   3. Simplified 60.1%

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

      1. *-un-lft-identity 60.1%

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

      2. associate-+r+ 60.1%

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

      3. associate-*l* 60.1%

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

      4. times-frac 81.6%

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

      5. +-commutative 81.6%

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

      6. +-commutative 81.6%

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

      7. associate-+r+ 81.6%

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

      8. +-commutative 81.6%

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

      9. associate-+l+ 81.6%

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

   6. Applied egg-rr 81.6%

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

      1. associate-*l/ 81.5%

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

      2. *-lft-identity 81.5%

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

      3. +-commutative 81.5%

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

   8. Simplified 81.5%

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

   9. Taylor expanded in y around inf 74.5%

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

   if 3.50000000000000003e134 < y

   1. Initial program 58.9%

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

      1. *-un-lft-identity 58.9%

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

      2. associate-*l* 58.9%

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

      3. times-frac 60.9%

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

      4. +-commutative 60.9%

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

      5. *-commutative 60.9%

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

      6. +-commutative 60.9%

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

      7. associate-+r+ 60.9%

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

      8. +-commutative 60.9%

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

      9. associate-+l+ 60.9%

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

   4. Applied egg-rr 60.9%

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

      1. *-commutative 60.9%

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

      2. times-frac 99.9%

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

      3. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around inf 80.9%

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

      1. *-commutative 80.9%

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

      2. associate-*l/ 80.9%

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

      3. div-inv 80.9%

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

   9. Applied egg-rr 80.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 10: 82.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.02 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.02e-85) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ y 1.0)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.02e-85) {
		tmp = (y / x) / (x + 1.0);
	} 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) :: tmp
    if (y <= 2.02d-85) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.02e-85) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.02e-85:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.02e-85)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	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)
	tmp = 0.0;
	if (y <= 2.02e-85)
		tmp = (y / x) / (x + 1.0);
	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_] := If[LessEqual[y, 2.02e-85], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.02000000000000013e-85

   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. Step-by-step derivation

      1. associate-/l* 78.6%

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

      2. associate-+l+ 78.6%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in y around 0 55.7%

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

      1. associate-/r* 55.3%

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

      2. +-commutative 55.3%

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

   7. Simplified 55.3%

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

   if 2.02000000000000013e-85 < y

   1. Initial program 67.7%

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

      1. associate-/l* 82.1%

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

      2. associate-+l+ 82.1%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in x around 0 65.6%

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

      1. +-commutative 65.6%

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

   7. Simplified 65.6%

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

      1. *-un-lft-identity 65.6%

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

      2. times-frac 65.9%

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

   9. Applied egg-rr 65.9%

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

       1. associate-*l/ 65.9%

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

       2. *-lft-identity 65.9%

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

   11. Simplified 65.9%

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

Alternative 11: 81.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.36 \cdot 10^{-85}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.36e-85) (/ y (* x (+ x 1.0))) (/ (/ x (+ y 1.0)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.36e-85) {
		tmp = y / (x * (x + 1.0));
	} 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) :: tmp
    if (y <= 1.36d-85) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.36e-85) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.36e-85:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.36e-85)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	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)
	tmp = 0.0;
	if (y <= 1.36e-85)
		tmp = y / (x * (x + 1.0));
	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_] := If[LessEqual[y, 1.36e-85], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.36e-85

   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. Step-by-step derivation

      1. associate-/l* 78.6%

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

      2. associate-+l+ 78.6%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in y around 0 55.7%

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

   if 1.36e-85 < y

   1. Initial program 67.7%

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

      1. associate-/l* 82.1%

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

      2. associate-+l+ 82.1%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in x around 0 65.6%

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

      1. +-commutative 65.6%

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

   7. Simplified 65.6%

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

      1. *-un-lft-identity 65.6%

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

      2. times-frac 65.9%

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

   9. Applied egg-rr 65.9%

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

       1. associate-*l/ 65.9%

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

       2. *-lft-identity 65.9%

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

   11. Simplified 65.9%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.36 \cdot 10^{-85}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-85}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.5e-85) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.5e-85) {
		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 (y <= 2.5d-85) 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 (y <= 2.5e-85) {
		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 y <= 2.5e-85:
		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 (y <= 2.5e-85)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.5e-85)
		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[y, 2.5e-85], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.5000000000000001e-85

   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. Step-by-step derivation

      1. associate-/l* 78.6%

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

      2. associate-+l+ 78.6%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in y around 0 55.7%

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

   if 2.5000000000000001e-85 < y

   1. Initial program 67.7%

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

      1. associate-/l* 82.1%

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

      2. associate-+l+ 82.1%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in x around 0 65.6%

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

      1. +-commutative 65.6%

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

   7. Simplified 65.6%

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

      1. add-sqr-sqrt 31.1%

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

      2. *-commutative 31.1%

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

      3. times-frac 29.9%

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

   9. Applied egg-rr 29.9%

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

       1. associate-*l/ 29.9%

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

       2. associate-*r/ 29.9%

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

       3. add-sqr-sqrt 65.9%

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

       4. +-commutative 65.9%

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

   11. Applied egg-rr 65.9%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-85}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.7 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4.7e-86) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.7e-86) {
		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 (y <= 4.7d-86) 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 (y <= 4.7e-86) {
		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 y <= 4.7e-86:
		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 (y <= 4.7e-86)
		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 (y <= 4.7e-86)
		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[y, 4.7e-86], 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 2 regimes

2. if y < 4.7000000000000001e-86

   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. Step-by-step derivation

      1. associate-/l* 78.6%

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

      2. associate-+l+ 78.6%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in y around 0 55.7%

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

   if 4.7000000000000001e-86 < y

   1. Initial program 67.7%

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

      1. associate-/l* 82.1%

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

      2. associate-+l+ 82.1%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in x around 0 65.6%

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

      1. +-commutative 65.6%

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

   7. Simplified 65.6%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.7 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-135}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4.8e-135) (/ y x) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.8e-135) {
		tmp = 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 (y <= 4.8d-135) then
        tmp = 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 (y <= 4.8e-135) {
		tmp = y / x;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 4.8e-135:
		tmp = 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 (y <= 4.8e-135)
		tmp = 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 (y <= 4.8e-135)
		tmp = 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[y, 4.8e-135], N[(y / x), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.7999999999999997e-135

   1. Initial program 65.4%

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

      1. associate-/l* 77.4%

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

      2. associate-+l+ 77.4%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in y around 0 54.1%

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

      1. associate-/r* 53.7%

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

      2. +-commutative 53.7%

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

   7. Simplified 53.7%

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

   8. Taylor expanded in x around 0 34.4%

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

   if 4.7999999999999997e-135 < y

   1. Initial program 69.0%

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

      1. associate-/l* 83.6%

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

      2. associate-+l+ 83.6%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in x around 0 61.1%

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

      1. +-commutative 61.1%

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

   7. Simplified 61.1%

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

Alternative 15: 44.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-135}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4.5e-135) (/ y x) (* x (/ 1.0 (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.5e-135) {
		tmp = y / x;
	} else {
		tmp = x * (1.0 / (y + x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.5d-135) then
        tmp = y / x
    else
        tmp = x * (1.0d0 / (y + x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 4.5e-135) {
		tmp = y / x;
	} else {
		tmp = x * (1.0 / (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 4.5e-135:
		tmp = y / x
	else:
		tmp = x * (1.0 / (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4.5e-135)
		tmp = Float64(y / x);
	else
		tmp = Float64(x * Float64(1.0 / Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.5e-135)
		tmp = y / x;
	else
		tmp = x * (1.0 / (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 4.5e-135], N[(y / x), $MachinePrecision], N[(x * N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.49999999999999987e-135

   1. Initial program 65.4%

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

      1. associate-/l* 77.4%

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

      2. associate-+l+ 77.4%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in y around 0 54.1%

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

      1. associate-/r* 53.7%

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

      2. +-commutative 53.7%

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

   7. Simplified 53.7%

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

   8. Taylor expanded in x around 0 34.4%

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

   if 4.49999999999999987e-135 < y

   1. Initial program 69.0%

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

      1. *-un-lft-identity 69.0%

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

      2. associate-*l* 69.0%

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

      3. times-frac 74.1%

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

      4. +-commutative 74.1%

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

      5. *-commutative 74.1%

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

      6. +-commutative 74.1%

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

      7. associate-+r+ 74.1%

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

      8. +-commutative 74.1%

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

      9. associate-+l+ 74.1%

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

   4. Applied egg-rr 74.1%

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

   5. Taylor expanded in x around 0 61.9%

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

      1. +-commutative 61.9%

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

   7. Simplified 61.9%

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

   8. Taylor expanded in y around 0 36.2%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-135}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-135}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 4.4e-135) (/ y x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.4e-135) {
		tmp = 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 (y <= 4.4d-135) then
        tmp = 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 (y <= 4.4e-135) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 4.4e-135:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4.4e-135)
		tmp = 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 (y <= 4.4e-135)
		tmp = 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[y, 4.4e-135], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.3999999999999999e-135

   1. Initial program 65.4%

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

      1. associate-/l* 77.4%

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

      2. associate-+l+ 77.4%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in y around 0 54.1%

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

      1. associate-/r* 53.7%

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

      2. +-commutative 53.7%

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

   7. Simplified 53.7%

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

   8. Taylor expanded in x around 0 34.4%

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

   if 4.3999999999999999e-135 < y

   1. Initial program 69.0%

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

      1. associate-/l* 83.6%

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

      2. associate-+l+ 83.6%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in x around 0 61.1%

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

      1. +-commutative 61.1%

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

   7. Simplified 61.1%

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

   8. Taylor expanded in y around 0 35.9%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 26.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{x}{y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.8%

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

   1. associate-/l* 79.8%

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

   2. associate-+l+ 79.8%

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

3. Simplified 79.8%

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

5. Taylor expanded in x around 0 51.9%

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

   1. +-commutative 51.9%

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

7. Simplified 51.9%

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

8. Taylor expanded in y around 0 30.7%

   \[\leadsto \color{blue}{\frac{x}{y}} \]
9. Add Preprocessing

Alternative 18: 4.0% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

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 / y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.8%

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

   1. *-un-lft-identity 66.8%

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

   2. associate-*l* 66.8%

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

   3. times-frac 70.9%

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

   4. +-commutative 70.9%

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

   5. *-commutative 70.9%

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

   6. +-commutative 70.9%

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

   7. associate-+r+ 70.9%

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

   8. +-commutative 70.9%

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

   9. associate-+l+ 70.9%

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

4. Applied egg-rr 70.9%

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

   1. *-commutative 70.9%

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

   2. times-frac 99.7%

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

   3. +-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in y around inf 38.6%

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

8. Taylor expanded in y around 0 4.1%

   \[\leadsto \color{blue}{\frac{1}{y}} \]
9. 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 2024103 
(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))))