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

Percentage Accurate: 69.8% → 99.8%

Time: 15.4s

Alternatives: 17

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: 69.8% 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, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.0%

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

   1. *-commutative 66.0%

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

   2. associate-*l* 66.0%

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

   3. times-frac 91.1%

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

   4. +-commutative 91.1%

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

   5. +-commutative 91.1%

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

   6. associate-+r+ 91.1%

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

   7. +-commutative 91.1%

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

   8. associate-+l+ 91.1%

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

4. Applied egg-rr 91.1%

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

   1. *-un-lft-identity 91.1%

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

   2. distribute-rgt-in 87.9%

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

   3. +-commutative 87.9%

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

   4. distribute-rgt-in 91.1%

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

   5. *-commutative 91.1%

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

   6. times-frac 99.7%

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

   7. +-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 91.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+158}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-162}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.15e+158)
   (* (/ y (+ y x)) (/ 1.0 x))
   (if (<= x -5.5e+21)
     (* (/ x (* (+ y x) (+ y (+ x 1.0)))) (/ y x))
     (if (<= x -1.5e-162)
       (* x (/ y (* (* (+ y x) (+ y x)) (+ x (+ y 1.0)))))
       (/ (/ x y) (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.15e+158) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -5.5e+21) {
		tmp = (x / ((y + x) * (y + (x + 1.0)))) * (y / x);
	} else if (x <= -1.5e-162) {
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.15d+158)) then
        tmp = (y / (y + x)) * (1.0d0 / x)
    else if (x <= (-5.5d+21)) then
        tmp = (x / ((y + x) * (y + (x + 1.0d0)))) * (y / x)
    else if (x <= (-1.5d-162)) then
        tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0d0))))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.15e+158) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -5.5e+21) {
		tmp = (x / ((y + x) * (y + (x + 1.0)))) * (y / x);
	} else if (x <= -1.5e-162) {
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.15e+158:
		tmp = (y / (y + x)) * (1.0 / x)
	elif x <= -5.5e+21:
		tmp = (x / ((y + x) * (y + (x + 1.0)))) * (y / x)
	elif x <= -1.5e-162:
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.15e+158)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / x));
	elseif (x <= -5.5e+21)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0)))) * Float64(y / x));
	elseif (x <= -1.5e-162)
		tmp = Float64(x * Float64(y / Float64(Float64(Float64(y + x) * Float64(y + x)) * Float64(x + Float64(y + 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 (x <= -2.15e+158)
		tmp = (y / (y + x)) * (1.0 / x);
	elseif (x <= -5.5e+21)
		tmp = (x / ((y + x) * (y + (x + 1.0)))) * (y / x);
	elseif (x <= -1.5e-162)
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.15e+158], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.5e+21], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.5e-162], 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[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.15e158

   1. Initial program 54.1%

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

      1. *-commutative 54.1%

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

      2. associate-*l* 54.1%

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

      3. times-frac 71.2%

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

      4. +-commutative 71.2%

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

      5. +-commutative 71.2%

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

      6. associate-+r+ 71.2%

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

      7. +-commutative 71.2%

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

      8. associate-+l+ 71.2%

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

   4. Applied egg-rr 71.2%

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

   5. Taylor expanded in x around inf 85.5%

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

   if -2.15e158 < x < -5.5e21

   1. Initial program 56.5%

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

      1. *-commutative 56.5%

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

      2. associate-*l* 56.5%

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

      3. times-frac 87.0%

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

      4. +-commutative 87.0%

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

      5. +-commutative 87.0%

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

      6. associate-+r+ 87.0%

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

      7. +-commutative 87.0%

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

      8. associate-+l+ 87.0%

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

   4. Applied egg-rr 87.0%

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

   5. Taylor expanded in y around 0 78.3%

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

   if -5.5e21 < x < -1.49999999999999999e-162

   1. Initial program 80.8%

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

      1. associate-/l* 93.7%

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

      2. associate-+l+ 93.7%

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

   3. Simplified 93.7%

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

   if -1.49999999999999999e-162 < x

   1. Initial program 66.5%

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

      1. *-commutative 66.5%

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

      2. associate-*l* 66.4%

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

      3. times-frac 93.7%

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

      4. +-commutative 93.7%

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

      5. +-commutative 93.7%

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

      6. associate-+r+ 93.7%

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

      7. +-commutative 93.7%

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

      8. associate-+l+ 93.7%

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

   4. Applied egg-rr 93.7%

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

      1. *-un-lft-identity 93.7%

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

      2. distribute-rgt-in 92.4%

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

      3. +-commutative 92.4%

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

      4. distribute-rgt-in 93.7%

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

      5. *-commutative 93.7%

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

      6. times-frac 99.8%

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

      7. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 52.1%

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

      1. associate-/r* 50.8%

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

      2. +-commutative 50.8%

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

   9. Simplified 50.8%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+158}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-162}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;y \leq -5 \cdot 10^{-284}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{t\_0}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+123}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(y + x\right) \cdot t\_0}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y + x}}{\frac{y + x}{y}} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= y -5e-284)
     (* (/ y (+ y x)) (/ 1.0 t_0))
     (if (<= y 1.5e+123)
       (* x (/ (/ y (* (+ y x) t_0)) (+ y x)))
       (* (/ (/ 1.0 (+ y x)) (/ (+ y x) y)) (/ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= -5e-284) {
		tmp = (y / (y + x)) * (1.0 / t_0);
	} else if (y <= 1.5e+123) {
		tmp = x * ((y / ((y + x) * t_0)) / (y + x));
	} else {
		tmp = ((1.0 / (y + x)) / ((y + 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) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (y <= (-5d-284)) then
        tmp = (y / (y + x)) * (1.0d0 / t_0)
    else if (y <= 1.5d+123) then
        tmp = x * ((y / ((y + x) * t_0)) / (y + x))
    else
        tmp = ((1.0d0 / (y + x)) / ((y + x) / y)) * (x / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= -5e-284) {
		tmp = (y / (y + x)) * (1.0 / t_0);
	} else if (y <= 1.5e+123) {
		tmp = x * ((y / ((y + x) * t_0)) / (y + x));
	} else {
		tmp = ((1.0 / (y + x)) / ((y + x) / y)) * (x / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if y <= -5e-284:
		tmp = (y / (y + x)) * (1.0 / t_0)
	elif y <= 1.5e+123:
		tmp = x * ((y / ((y + x) * t_0)) / (y + x))
	else:
		tmp = ((1.0 / (y + x)) / ((y + x) / y)) * (x / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (y <= -5e-284)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / t_0));
	elseif (y <= 1.5e+123)
		tmp = Float64(x * Float64(Float64(y / Float64(Float64(y + x) * t_0)) / Float64(y + x)));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(y + x)) / Float64(Float64(y + x) / y)) * Float64(x / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (y <= -5e-284)
		tmp = (y / (y + x)) * (1.0 / t_0);
	elseif (y <= 1.5e+123)
		tmp = x * ((y / ((y + x) * t_0)) / (y + x));
	else
		tmp = ((1.0 / (y + x)) / ((y + 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_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5e-284], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+123], N[(x * N[(N[(y / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.99999999999999973e-284

   1. Initial program 75.9%

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

      1. *-commutative 75.9%

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

      2. associate-*l* 75.9%

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

      3. times-frac 91.9%

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

      4. +-commutative 91.9%

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

      5. +-commutative 91.9%

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

      6. associate-+r+ 91.9%

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

      7. +-commutative 91.9%

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

      8. associate-+l+ 91.9%

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

   4. Applied egg-rr 91.9%

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

      1. *-un-lft-identity 91.9%

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

      2. distribute-rgt-in 90.3%

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

      3. +-commutative 90.3%

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

      4. distribute-rgt-in 91.9%

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

      5. *-commutative 91.9%

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

      6. times-frac 99.7%

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

      7. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around inf 51.3%

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

   if -4.99999999999999973e-284 < y < 1.50000000000000004e123

   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. Step-by-step derivation

      1. associate-/l* 73.6%

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

      2. associate-+l+ 73.6%

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

   3. Simplified 73.6%

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

      1. *-un-lft-identity 73.6%

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

      2. associate-+r+ 73.6%

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

      3. associate-*l* 73.6%

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

      4. times-frac 93.3%

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

      5. +-commutative 93.3%

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

      6. +-commutative 93.3%

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

      7. associate-+r+ 93.3%

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

      8. +-commutative 93.3%

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

      9. associate-+l+ 93.3%

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

   6. Applied egg-rr 93.3%

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

      1. associate-*l/ 93.3%

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

      2. *-lft-identity 93.3%

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

      3. +-commutative 93.3%

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

   8. Simplified 93.3%

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

   if 1.50000000000000004e123 < y

   1. Initial program 49.8%

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

      1. *-commutative 49.8%

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

      2. associate-*l* 49.8%

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

      3. times-frac 77.2%

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

      4. +-commutative 77.2%

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

      5. +-commutative 77.2%

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

      6. associate-+r+ 77.2%

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

      7. +-commutative 77.2%

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

      8. associate-+l+ 77.2%

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

   4. Applied egg-rr 77.2%

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

      1. clear-num 77.2%

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

      2. associate-/r* 99.8%

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

      3. +-commutative 99.8%

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

      4. frac-times 99.7%

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

      5. metadata-eval 99.7%

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

      6. times-frac 99.7%

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

      7. *-un-lft-identity 99.7%

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

      8. *-un-lft-identity 99.7%

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

      9. +-commutative 99.7%

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

      10. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. div-inv 99.7%

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

      2. *-commutative 99.7%

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

      3. times-frac 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in y around inf 87.2%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-284}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{y + \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+123}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y + x}}{\frac{y + x}{y}} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{y + x}\\ t_1 := y + \left(x + 1\right)\\ \mathbf{if}\;y \leq -1.58 \cdot 10^{+41}:\\ \;\;\;\;t\_0 \cdot \frac{1}{t\_1}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+134}:\\ \;\;\;\;t\_0 \cdot \frac{x}{\left(y + x\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y + x}}{\frac{y + x}{y}} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y x))) (t_1 (+ y (+ x 1.0))))
   (if (<= y -1.58e+41)
     (* t_0 (/ 1.0 t_1))
     (if (<= y 5.8e+134)
       (* t_0 (/ x (* (+ y x) t_1)))
       (* (/ (/ 1.0 (+ y x)) (/ (+ y x) y)) (/ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (y + x);
	double t_1 = y + (x + 1.0);
	double tmp;
	if (y <= -1.58e+41) {
		tmp = t_0 * (1.0 / t_1);
	} else if (y <= 5.8e+134) {
		tmp = t_0 * (x / ((y + x) * t_1));
	} else {
		tmp = ((1.0 / (y + x)) / ((y + 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y / (y + x)
    t_1 = y + (x + 1.0d0)
    if (y <= (-1.58d+41)) then
        tmp = t_0 * (1.0d0 / t_1)
    else if (y <= 5.8d+134) then
        tmp = t_0 * (x / ((y + x) * t_1))
    else
        tmp = ((1.0d0 / (y + x)) / ((y + x) / y)) * (x / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (y + x);
	double t_1 = y + (x + 1.0);
	double tmp;
	if (y <= -1.58e+41) {
		tmp = t_0 * (1.0 / t_1);
	} else if (y <= 5.8e+134) {
		tmp = t_0 * (x / ((y + x) * t_1));
	} else {
		tmp = ((1.0 / (y + x)) / ((y + x) / y)) * (x / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (y + x)
	t_1 = y + (x + 1.0)
	tmp = 0
	if y <= -1.58e+41:
		tmp = t_0 * (1.0 / t_1)
	elif y <= 5.8e+134:
		tmp = t_0 * (x / ((y + x) * t_1))
	else:
		tmp = ((1.0 / (y + x)) / ((y + x) / y)) * (x / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(y + x))
	t_1 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (y <= -1.58e+41)
		tmp = Float64(t_0 * Float64(1.0 / t_1));
	elseif (y <= 5.8e+134)
		tmp = Float64(t_0 * Float64(x / Float64(Float64(y + x) * t_1)));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(y + x)) / Float64(Float64(y + x) / y)) * Float64(x / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (y + x);
	t_1 = y + (x + 1.0);
	tmp = 0.0;
	if (y <= -1.58e+41)
		tmp = t_0 * (1.0 / t_1);
	elseif (y <= 5.8e+134)
		tmp = t_0 * (x / ((y + x) * t_1));
	else
		tmp = ((1.0 / (y + x)) / ((y + 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_] := Block[{t$95$0 = N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.58e+41], N[(t$95$0 * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.8e+134], N[(t$95$0 * N[(x / N[(N[(y + x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.5799999999999999e41

   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. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 67.7%

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

      2. associate-*l* 67.7%

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

      3. times-frac 82.9%

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

      4. +-commutative 82.9%

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

      5. +-commutative 82.9%

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

      6. associate-+r+ 82.9%

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

      7. +-commutative 82.9%

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

      8. associate-+l+ 82.9%

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

   4. Applied egg-rr 82.9%

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

      1. *-un-lft-identity 82.9%

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

      2. distribute-rgt-in 79.4%

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

      3. +-commutative 79.4%

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

      4. distribute-rgt-in 82.9%

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

      5. *-commutative 82.9%

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

      6. times-frac 99.7%

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

      7. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around inf 26.7%

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

   if -1.5799999999999999e41 < y < 5.80000000000000023e134

   1. Initial program 70.0%

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

      1. *-commutative 70.0%

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

      2. associate-*l* 69.9%

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

      3. times-frac 98.7%

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

      4. +-commutative 98.7%

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

      5. +-commutative 98.7%

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

      6. associate-+r+ 98.7%

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

      7. +-commutative 98.7%

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

      8. associate-+l+ 98.7%

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

   4. Applied egg-rr 98.7%

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

   if 5.80000000000000023e134 < y

   1. Initial program 49.6%

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

      1. *-commutative 49.6%

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

      2. associate-*l* 49.6%

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

      3. times-frac 76.1%

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

      4. +-commutative 76.1%

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

      5. +-commutative 76.1%

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

      6. associate-+r+ 76.1%

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

      7. +-commutative 76.1%

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

      8. associate-+l+ 76.1%

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

   4. Applied egg-rr 76.1%

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

      1. clear-num 76.1%

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

      2. associate-/r* 99.9%

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

      3. +-commutative 99.9%

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

      4. frac-times 99.7%

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

      5. metadata-eval 99.7%

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

      6. times-frac 99.7%

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

      7. *-un-lft-identity 99.7%

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

      8. *-un-lft-identity 99.7%

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

      9. +-commutative 99.7%

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

      10. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. div-inv 99.7%

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

      2. *-commutative 99.7%

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

      3. times-frac 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in y around inf 86.7%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.58 \cdot 10^{+41}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{y + \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+134}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y + x}}{\frac{y + x}{y}} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+159}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-115}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.7e+159)
   (* (/ y (+ y x)) (/ 1.0 x))
   (if (<= x -3.4e-115)
     (* (/ x (* (+ y x) (+ y (+ x 1.0)))) (/ y x))
     (/ (/ x y) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.7e+159) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -3.4e-115) {
		tmp = (x / ((y + x) * (y + (x + 1.0)))) * (y / x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.7d+159)) then
        tmp = (y / (y + x)) * (1.0d0 / x)
    else if (x <= (-3.4d-115)) then
        tmp = (x / ((y + x) * (y + (x + 1.0d0)))) * (y / x)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.7e+159) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -3.4e-115) {
		tmp = (x / ((y + x) * (y + (x + 1.0)))) * (y / x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.7e+159:
		tmp = (y / (y + x)) * (1.0 / x)
	elif x <= -3.4e-115:
		tmp = (x / ((y + x) * (y + (x + 1.0)))) * (y / x)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.7e+159)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / x));
	elseif (x <= -3.4e-115)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0)))) * Float64(y / x));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.7e+159)
		tmp = (y / (y + x)) * (1.0 / x);
	elseif (x <= -3.4e-115)
		tmp = (x / ((y + x) * (y + (x + 1.0)))) * (y / x);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.7e+159], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.4e-115], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.69999999999999996e159

   1. Initial program 54.1%

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

      1. *-commutative 54.1%

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

      2. associate-*l* 54.1%

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

      3. times-frac 71.2%

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

      4. +-commutative 71.2%

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

      5. +-commutative 71.2%

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

      6. associate-+r+ 71.2%

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

      7. +-commutative 71.2%

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

      8. associate-+l+ 71.2%

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

   4. Applied egg-rr 71.2%

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

   5. Taylor expanded in x around inf 85.5%

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

   if -1.69999999999999996e159 < x < -3.3999999999999998e-115

   1. Initial program 70.6%

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

      1. *-commutative 70.6%

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

      2. associate-*l* 70.7%

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

      3. times-frac 94.0%

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

      4. +-commutative 94.0%

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

      5. +-commutative 94.0%

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

      6. associate-+r+ 94.0%

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

      7. +-commutative 94.0%

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

      8. associate-+l+ 94.0%

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

   4. Applied egg-rr 94.0%

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

   5. Taylor expanded in y around 0 73.4%

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

   if -3.3999999999999998e-115 < x

   1. Initial program 66.9%

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

      1. *-commutative 66.9%

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

      2. associate-*l* 66.9%

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

      3. times-frac 94.0%

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

      4. +-commutative 94.0%

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

      5. +-commutative 94.0%

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

      6. associate-+r+ 94.0%

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

      7. +-commutative 94.0%

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

      8. associate-+l+ 94.0%

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

   4. Applied egg-rr 94.0%

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

      1. *-un-lft-identity 94.0%

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

      2. distribute-rgt-in 92.7%

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

      3. +-commutative 92.7%

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

      4. distribute-rgt-in 94.0%

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

      5. *-commutative 94.0%

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

      6. times-frac 99.8%

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

      7. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 53.1%

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

      1. associate-/r* 52.0%

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

      2. +-commutative 52.0%

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

   9. Simplified 52.0%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+159}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-115}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-224}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-167}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.6e-224)
   (/ y x)
   (if (<= y 2.3e-167)
     (/ x y)
     (if (<= y 2.5e-98) (/ y x) (/ x (* y (+ y 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.6e-224) {
		tmp = y / x;
	} else if (y <= 2.3e-167) {
		tmp = x / y;
	} else if (y <= 2.5e-98) {
		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 <= 1.6d-224) then
        tmp = y / x
    else if (y <= 2.3d-167) then
        tmp = x / y
    else if (y <= 2.5d-98) 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 <= 1.6e-224) {
		tmp = y / x;
	} else if (y <= 2.3e-167) {
		tmp = x / y;
	} else if (y <= 2.5e-98) {
		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 <= 1.6e-224:
		tmp = y / x
	elif y <= 2.3e-167:
		tmp = x / y
	elif y <= 2.5e-98:
		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 <= 1.6e-224)
		tmp = Float64(y / x);
	elseif (y <= 2.3e-167)
		tmp = Float64(x / y);
	elseif (y <= 2.5e-98)
		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 <= 1.6e-224)
		tmp = y / x;
	elseif (y <= 2.3e-167)
		tmp = x / y;
	elseif (y <= 2.5e-98)
		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, 1.6e-224], N[(y / x), $MachinePrecision], If[LessEqual[y, 2.3e-167], N[(x / y), $MachinePrecision], If[LessEqual[y, 2.5e-98], N[(y / x), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.5999999999999999e-224 or 2.3000000000000001e-167 < y < 2.50000000000000009e-98

   1. Initial program 72.7%

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

      1. associate-/l* 81.2%

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

      2. associate-+l+ 81.2%

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

   3. Simplified 81.2%

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

      1. associate-*r/ 72.7%

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

      2. associate-+r+ 72.7%

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

      3. associate-/r* 75.0%

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

      4. clear-num 74.9%

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

      5. associate-+r+ 74.9%

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

      6. +-commutative 74.9%

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

      7. associate-+l+ 74.9%

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

      8. *-commutative 74.9%

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

      9. associate-/l* 84.5%

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

      10. pow2 84.5%

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

      11. +-commutative 84.5%

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

   6. Applied egg-rr 84.5%

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

      1. associate-/r/ 84.6%

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

      2. *-commutative 84.6%

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

      3. associate-*l* 82.6%

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

      4. +-commutative 82.6%

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

   8. Simplified 82.6%

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

   9. Taylor expanded in y around 0 55.1%

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

   10. Taylor expanded in x around 0 33.5%

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

   if 1.5999999999999999e-224 < y < 2.3000000000000001e-167

   1. Initial program 45.8%

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

      1. associate-/l* 52.6%

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

      2. associate-+l+ 52.6%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in x around 0 38.8%

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

      1. +-commutative 38.8%

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

   7. Simplified 38.8%

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

   8. Taylor expanded in y around 0 38.8%

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

   if 2.50000000000000009e-98 < y

   1. Initial program 54.7%

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

      1. associate-/l* 75.6%

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

      2. associate-+l+ 75.6%

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

   3. Simplified 75.6%

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

   5. Taylor expanded in x around 0 59.1%

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

      1. +-commutative 59.1%

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

   7. Simplified 59.1%

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

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-224}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-167}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-97}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.1e-97)
   (/ (/ y (+ x 1.0)) x)
   (/ (/ x y) (* (+ y (+ x 1.0)) (/ (+ y x) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-97) {
		tmp = (y / (x + 1.0)) / x;
	} else {
		tmp = (x / y) / ((y + (x + 1.0)) * ((y + x) / y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.1d-97) then
        tmp = (y / (x + 1.0d0)) / x
    else
        tmp = (x / y) / ((y + (x + 1.0d0)) * ((y + x) / y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-97) {
		tmp = (y / (x + 1.0)) / x;
	} else {
		tmp = (x / y) / ((y + (x + 1.0)) * ((y + x) / y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.1e-97:
		tmp = (y / (x + 1.0)) / x
	else:
		tmp = (x / y) / ((y + (x + 1.0)) * ((y + x) / y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.1e-97)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	else
		tmp = Float64(Float64(x / y) / Float64(Float64(y + Float64(x + 1.0)) * Float64(Float64(y + x) / y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.1e-97)
		tmp = (y / (x + 1.0)) / x;
	else
		tmp = (x / y) / ((y + (x + 1.0)) * ((y + x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.1e-97], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.10000000000000002e-97

   1. Initial program 71.0%

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

      1. associate-/l* 79.4%

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

      2. associate-+l+ 79.4%

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

   3. Simplified 79.4%

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

      1. associate-*r/ 71.0%

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

      2. associate-+r+ 71.0%

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

      3. associate-/r* 73.2%

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

      4. clear-num 73.1%

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

      5. associate-+r+ 73.1%

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

      6. +-commutative 73.1%

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

      7. associate-+l+ 73.1%

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

      8. *-commutative 73.1%

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

      9. associate-/l* 82.5%

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

      10. pow2 82.5%

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

      11. +-commutative 82.5%

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

   6. Applied egg-rr 82.5%

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

      1. associate-/r/ 82.6%

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

      2. *-commutative 82.6%

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

      3. associate-*l* 80.7%

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

      4. +-commutative 80.7%

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

   8. Simplified 80.7%

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

   9. Taylor expanded in y around 0 55.6%

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

       1. un-div-inv 55.7%

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

       2. +-commutative 55.7%

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

       3. *-commutative 55.7%

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

       4. associate-/r* 59.3%

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

   11. Applied egg-rr 59.3%

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

   if 3.10000000000000002e-97 < y

   1. Initial program 54.7%

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

      1. *-commutative 54.7%

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

      2. associate-*l* 54.7%

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

      3. times-frac 84.8%

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

      4. +-commutative 84.8%

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

      5. +-commutative 84.8%

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

      6. associate-+r+ 84.8%

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

      7. +-commutative 84.8%

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

      8. associate-+l+ 84.8%

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

   4. Applied egg-rr 84.8%

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

      1. clear-num 84.8%

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

      2. associate-/r* 99.7%

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

      3. +-commutative 99.7%

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

      4. frac-times 98.8%

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

      5. metadata-eval 98.8%

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

      6. times-frac 98.8%

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

      7. *-un-lft-identity 98.8%

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

      8. *-un-lft-identity 98.8%

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

      9. +-commutative 98.8%

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

      10. +-commutative 98.8%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in x around 0 79.0%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-97}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.0%

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

   1. *-commutative 66.0%

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

   2. associate-*l* 66.0%

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

   3. times-frac 91.1%

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

   4. +-commutative 91.1%

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

   5. +-commutative 91.1%

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

   6. associate-+r+ 91.1%

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

   7. +-commutative 91.1%

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

   8. associate-+l+ 91.1%

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

4. Applied egg-rr 91.1%

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

   1. clear-num 91.1%

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

   2. associate-/r* 99.7%

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

   3. +-commutative 99.7%

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

   4. frac-times 99.4%

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

   5. metadata-eval 99.4%

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

   6. times-frac 99.4%

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

   7. *-un-lft-identity 99.4%

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

   8. *-un-lft-identity 99.4%

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

   9. +-commutative 99.4%

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

   10. +-commutative 99.4%

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

6. Applied egg-rr 99.4%

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

   1. div-inv 99.4%

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

   2. *-commutative 99.4%

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

   3. times-frac 99.7%

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

   4. +-commutative 99.7%

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

   5. +-commutative 99.7%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 9: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.0%

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

   1. *-commutative 66.0%

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

   2. associate-*l* 66.0%

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

   3. times-frac 91.1%

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

   4. +-commutative 91.1%

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

   5. +-commutative 91.1%

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

   6. associate-+r+ 91.1%

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

   7. +-commutative 91.1%

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

   8. associate-+l+ 91.1%

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

4. Applied egg-rr 91.1%

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

   1. *-un-lft-identity 91.1%

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

   2. distribute-rgt-in 87.9%

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

   3. +-commutative 87.9%

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

   4. distribute-rgt-in 91.1%

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

   5. *-commutative 91.1%

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

   6. times-frac 99.7%

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

   7. +-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

   1. clear-num 99.7%

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

   2. +-commutative 99.7%

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

   3. associate-*l/ 99.7%

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

   4. *-un-lft-identity 99.7%

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

   5. frac-times 99.4%

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

   6. *-un-lft-identity 99.4%

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

   7. associate-/r* 99.7%

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

   8. associate-/l/ 99.4%

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

   9. +-commutative 99.4%

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

   10. +-commutative 99.4%

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

8. Applied egg-rr 99.4%

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

9. Final simplification 99.4%

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

Alternative 10: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.0%

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

   1. *-commutative 66.0%

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

   2. associate-*l* 66.0%

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

   3. times-frac 91.1%

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

   4. +-commutative 91.1%

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

   5. +-commutative 91.1%

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

   6. associate-+r+ 91.1%

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

   7. +-commutative 91.1%

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

   8. associate-+l+ 91.1%

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

4. Applied egg-rr 91.1%

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

   1. clear-num 91.1%

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

   2. associate-/r* 99.7%

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

   3. +-commutative 99.7%

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

   4. frac-times 99.4%

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

   5. metadata-eval 99.4%

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

   6. times-frac 99.4%

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

   7. *-un-lft-identity 99.4%

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

   8. *-un-lft-identity 99.4%

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

   9. +-commutative 99.4%

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

   10. +-commutative 99.4%

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

6. Applied egg-rr 99.4%

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

7. Taylor expanded in x around 0 99.4%

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

8. Final simplification 99.4%

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

Alternative 11: 82.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-97}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.1e-97) (/ (/ y (+ x 1.0)) x) (/ (/ x (+ y x)) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-97) {
		tmp = (y / (x + 1.0)) / x;
	} else {
		tmp = (x / (y + x)) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.1d-97) then
        tmp = (y / (x + 1.0d0)) / x
    else
        tmp = (x / (y + x)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-97) {
		tmp = (y / (x + 1.0)) / x;
	} else {
		tmp = (x / (y + x)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.1e-97:
		tmp = (y / (x + 1.0)) / x
	else:
		tmp = (x / (y + x)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.1e-97)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.1e-97)
		tmp = (y / (x + 1.0)) / x;
	else
		tmp = (x / (y + x)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.1e-97], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.10000000000000002e-97

   1. Initial program 71.0%

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

      1. associate-/l* 79.4%

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

      2. associate-+l+ 79.4%

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

   3. Simplified 79.4%

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

      1. associate-*r/ 71.0%

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

      2. associate-+r+ 71.0%

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

      3. associate-/r* 73.2%

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

      4. clear-num 73.1%

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

      5. associate-+r+ 73.1%

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

      6. +-commutative 73.1%

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

      7. associate-+l+ 73.1%

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

      8. *-commutative 73.1%

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

      9. associate-/l* 82.5%

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

      10. pow2 82.5%

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

      11. +-commutative 82.5%

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

   6. Applied egg-rr 82.5%

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

      1. associate-/r/ 82.6%

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

      2. *-commutative 82.6%

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

      3. associate-*l* 80.7%

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

      4. +-commutative 80.7%

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

   8. Simplified 80.7%

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

   9. Taylor expanded in y around 0 55.6%

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

       1. un-div-inv 55.7%

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

       2. +-commutative 55.7%

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

       3. *-commutative 55.7%

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

       4. associate-/r* 59.3%

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

   11. Applied egg-rr 59.3%

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

   if 3.10000000000000002e-97 < y

   1. Initial program 54.7%

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

      1. *-commutative 54.7%

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

      2. associate-*l* 54.7%

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

      3. times-frac 84.8%

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

      4. +-commutative 84.8%

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

      5. +-commutative 84.8%

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

      6. associate-+r+ 84.8%

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

      7. +-commutative 84.8%

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

      8. associate-+l+ 84.8%

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

   4. Applied egg-rr 84.8%

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

      1. clear-num 84.8%

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

      2. associate-/r* 99.7%

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

      3. +-commutative 99.7%

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

      4. frac-times 98.8%

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

      5. metadata-eval 98.8%

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

      6. times-frac 98.8%

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

      7. *-un-lft-identity 98.8%

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

      8. *-un-lft-identity 98.8%

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

      9. +-commutative 98.8%

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

      10. +-commutative 98.8%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in x around 0 60.4%

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

      1. +-commutative 60.4%

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

   9. Simplified 60.4%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-97}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 78.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.1e-97) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-97) {
		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 <= 3.1d-97) 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 <= 3.1e-97) {
		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 <= 3.1e-97:
		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 <= 3.1e-97)
		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 <= 3.1e-97)
		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, 3.1e-97], 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 < 3.10000000000000002e-97

   1. Initial program 71.0%

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

      1. associate-/l* 79.4%

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

      2. associate-+l+ 79.4%

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

   3. Simplified 79.4%

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

   5. Taylor expanded in y around 0 55.7%

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

   if 3.10000000000000002e-97 < y

   1. Initial program 54.7%

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

      1. associate-/l* 75.6%

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

      2. associate-+l+ 75.6%

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

   3. Simplified 75.6%

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

   5. Taylor expanded in x around 0 59.1%

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

      1. +-commutative 59.1%

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

   7. Simplified 59.1%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 80.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.1e-97) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-97) {
		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 <= 3.1d-97) 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 <= 3.1e-97) {
		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 <= 3.1e-97:
		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 <= 3.1e-97)
		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 <= 3.1e-97)
		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, 3.1e-97], 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 < 3.10000000000000002e-97

   1. Initial program 71.0%

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

      1. associate-/l* 79.4%

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

      2. associate-+l+ 79.4%

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

   3. Simplified 79.4%

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

   5. Taylor expanded in y around 0 55.7%

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

   if 3.10000000000000002e-97 < y

   1. Initial program 54.7%

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

      1. *-commutative 54.7%

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

      2. associate-*l* 54.7%

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

      3. times-frac 84.8%

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

      4. +-commutative 84.8%

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

      5. +-commutative 84.8%

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

      6. associate-+r+ 84.8%

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

      7. +-commutative 84.8%

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

      8. associate-+l+ 84.8%

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

   4. Applied egg-rr 84.8%

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

      1. *-un-lft-identity 84.8%

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

      2. distribute-rgt-in 77.0%

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

      3. +-commutative 77.0%

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

      4. distribute-rgt-in 84.8%

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

      5. *-commutative 84.8%

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

      6. times-frac 99.7%

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

      7. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0 59.1%

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

      1. associate-/r* 59.8%

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

      2. +-commutative 59.8%

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

   9. Simplified 59.8%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 82.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-97}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.1e-97) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-97) {
		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 <= 3.1d-97) 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 <= 3.1e-97) {
		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 <= 3.1e-97:
		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 <= 3.1e-97)
		tmp = Float64(Float64(y / 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 <= 3.1e-97)
		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, 3.1e-97], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.10000000000000002e-97

   1. Initial program 71.0%

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

      1. associate-/l* 79.4%

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

      2. associate-+l+ 79.4%

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

   3. Simplified 79.4%

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

   5. Taylor expanded in y around 0 55.7%

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

      1. associate-/r* 59.3%

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

      2. +-commutative 59.3%

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

   7. Simplified 59.3%

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

   if 3.10000000000000002e-97 < y

   1. Initial program 54.7%

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

      1. *-commutative 54.7%

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

      2. associate-*l* 54.7%

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

      3. times-frac 84.8%

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

      4. +-commutative 84.8%

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

      5. +-commutative 84.8%

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

      6. associate-+r+ 84.8%

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

      7. +-commutative 84.8%

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

      8. associate-+l+ 84.8%

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

   4. Applied egg-rr 84.8%

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

      1. *-un-lft-identity 84.8%

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

      2. distribute-rgt-in 77.0%

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

      3. +-commutative 77.0%

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

      4. distribute-rgt-in 84.8%

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

      5. *-commutative 84.8%

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

      6. times-frac 99.7%

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

      7. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0 59.1%

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

      1. associate-/r* 59.8%

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

      2. +-commutative 59.8%

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

   9. Simplified 59.8%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-97}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 82.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-97}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.1e-97) (/ (/ y (+ x 1.0)) x) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-97) {
		tmp = (y / (x + 1.0)) / x;
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.1d-97) then
        tmp = (y / (x + 1.0d0)) / x
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-97) {
		tmp = (y / (x + 1.0)) / x;
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.1e-97:
		tmp = (y / (x + 1.0)) / x
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.1e-97)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.1e-97)
		tmp = (y / (x + 1.0)) / x;
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.1e-97], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.10000000000000002e-97

   1. Initial program 71.0%

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

      1. associate-/l* 79.4%

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

      2. associate-+l+ 79.4%

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

   3. Simplified 79.4%

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

      1. associate-*r/ 71.0%

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

      2. associate-+r+ 71.0%

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

      3. associate-/r* 73.2%

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

      4. clear-num 73.1%

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

      5. associate-+r+ 73.1%

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

      6. +-commutative 73.1%

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

      7. associate-+l+ 73.1%

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

      8. *-commutative 73.1%

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

      9. associate-/l* 82.5%

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

      10. pow2 82.5%

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

      11. +-commutative 82.5%

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

   6. Applied egg-rr 82.5%

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

      1. associate-/r/ 82.6%

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

      2. *-commutative 82.6%

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

      3. associate-*l* 80.7%

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

      4. +-commutative 80.7%

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

   8. Simplified 80.7%

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

   9. Taylor expanded in y around 0 55.6%

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

       1. un-div-inv 55.7%

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

       2. +-commutative 55.7%

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

       3. *-commutative 55.7%

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

       4. associate-/r* 59.3%

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

   11. Applied egg-rr 59.3%

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

   if 3.10000000000000002e-97 < y

   1. Initial program 54.7%

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

      1. *-commutative 54.7%

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

      2. associate-*l* 54.7%

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

      3. times-frac 84.8%

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

      4. +-commutative 84.8%

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

      5. +-commutative 84.8%

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

      6. associate-+r+ 84.8%

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

      7. +-commutative 84.8%

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

      8. associate-+l+ 84.8%

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

   4. Applied egg-rr 84.8%

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

      1. *-un-lft-identity 84.8%

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

      2. distribute-rgt-in 77.0%

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

      3. +-commutative 77.0%

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

      4. distribute-rgt-in 84.8%

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

      5. *-commutative 84.8%

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

      6. times-frac 99.7%

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

      7. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0 59.1%

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

      1. associate-/r* 59.8%

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

      2. +-commutative 59.8%

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

   9. Simplified 59.8%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-97}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-197}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -1.02e-197) (/ y x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.02e-197) {
		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 (x <= (-1.02d-197)) 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 (x <= -1.02e-197) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.02e-197:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.02e-197)
		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 (x <= -1.02e-197)
		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[x, -1.02e-197], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0199999999999999e-197

   1. Initial program 63.2%

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

      1. associate-/l* 76.1%

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

      2. associate-+l+ 76.1%

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

   3. Simplified 76.1%

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

      1. associate-*r/ 63.2%

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

      2. associate-+r+ 63.2%

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

      3. associate-/r* 69.0%

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

      4. clear-num 68.9%

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

      5. associate-+r+ 68.9%

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

      6. +-commutative 68.9%

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

      7. associate-+l+ 68.9%

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

      8. *-commutative 68.9%

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

      9. associate-/l* 83.6%

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

      10. pow2 83.6%

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

      11. +-commutative 83.6%

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

   6. Applied egg-rr 83.6%

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

      1. associate-/r/ 83.8%

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

      2. *-commutative 83.8%

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

      3. associate-*l* 80.9%

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

      4. +-commutative 80.9%

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

   8. Simplified 80.9%

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

   9. Taylor expanded in y around 0 54.3%

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

   10. Taylor expanded in x around 0 26.7%

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

   if -1.0199999999999999e-197 < x

   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* 79.5%

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

      2. associate-+l+ 79.5%

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

   3. Simplified 79.5%

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

   5. Taylor expanded in x around 0 52.0%

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

      1. +-commutative 52.0%

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

   7. Simplified 52.0%

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

   8. Taylor expanded in y around 0 31.1%

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

4. Final simplification 29.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-197}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 27.0% 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 66.0%

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

   1. associate-/l* 78.2%

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

   2. associate-+l+ 78.2%

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

3. Simplified 78.2%

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

5. Taylor expanded in x around 0 46.9%

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

   1. +-commutative 46.9%

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

7. Simplified 46.9%

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

8. Taylor expanded in y around 0 22.4%

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

9. Final simplification 22.4%

   \[\leadsto \frac{x}{y} \]
10. 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 2024079 
(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))))