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

Percentage Accurate: 69.8% → 99.8%

Time: 11.9s

Alternatives: 14

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

   1. associate-*l* 64.3%

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

   2. times-frac 91.8%

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

   3. +-commutative 91.8%

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

   4. +-commutative 91.8%

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

   5. associate-+r+ 91.8%

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

   6. +-commutative 91.8%

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

   7. associate-+l+ 91.8%

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

4. Applied egg-rr 91.8%

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

   1. clear-num 91.7%

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

   2. associate-/r* 99.7%

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

   3. frac-times 99.0%

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

   4. *-un-lft-identity 99.0%

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

   5. +-commutative 99.0%

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

6. Applied egg-rr 99.0%

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

   1. div-inv 98.9%

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

   2. *-commutative 98.9%

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

   3. times-frac 99.7%

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

8. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. un-div-inv 99.8%

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

   3. +-commutative 99.8%

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

   4. +-commutative 99.8%

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

   5. associate-+l+ 99.8%

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

   6. +-commutative 99.8%

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

10. Applied egg-rr 99.8%

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

Alternative 2: 95.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{+144}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\left(y + x\right) \cdot \frac{y + 1}{y}}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 6.2e+144)
   (* (/ x (+ y x)) (/ y (* (+ y x) (+ y (+ 1.0 x)))))
   (/ (/ x (* (+ y x) (/ (+ y 1.0) y))) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 6.2e+144) {
		tmp = (x / (y + x)) * (y / ((y + x) * (y + (1.0 + x))));
	} else {
		tmp = (x / ((y + x) * ((y + 1.0) / y))) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 6.2d+144) then
        tmp = (x / (y + x)) * (y / ((y + x) * (y + (1.0d0 + x))))
    else
        tmp = (x / ((y + x) * ((y + 1.0d0) / y))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 6.2e+144) {
		tmp = (x / (y + x)) * (y / ((y + x) * (y + (1.0 + x))));
	} else {
		tmp = (x / ((y + x) * ((y + 1.0) / y))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 6.2e+144:
		tmp = (x / (y + x)) * (y / ((y + x) * (y + (1.0 + x))))
	else:
		tmp = (x / ((y + x) * ((y + 1.0) / y))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 6.2e+144)
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(Float64(y + x) * Float64(y + Float64(1.0 + x)))));
	else
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(Float64(y + 1.0) / y))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.2e+144)
		tmp = (x / (y + x)) * (y / ((y + x) * (y + (1.0 + x))));
	else
		tmp = (x / ((y + x) * ((y + 1.0) / y))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 6.2e+144], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.2000000000000003e144

   1. Initial program 67.2%

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

      1. associate-*l* 67.2%

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

      2. times-frac 95.3%

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

      3. +-commutative 95.3%

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

      4. +-commutative 95.3%

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

      5. associate-+r+ 95.3%

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

      6. +-commutative 95.3%

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

      7. associate-+l+ 95.3%

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

   4. Applied egg-rr 95.3%

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

   if 6.2000000000000003e144 < y

   1. Initial program 48.8%

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

      1. associate-*l* 48.8%

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

      2. times-frac 72.6%

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

      3. +-commutative 72.6%

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

      4. +-commutative 72.6%

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

      5. associate-+r+ 72.6%

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

      6. +-commutative 72.6%

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

      7. associate-+l+ 72.6%

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

   4. Applied egg-rr 72.6%

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

      1. associate-*l/ 72.6%

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

      2. clear-num 72.6%

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

      3. un-div-inv 72.6%

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

      4. *-un-lft-identity 72.6%

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

      5. times-frac 99.8%

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

      6. /-rgt-identity 99.8%

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

      7. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 87.7%

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

      1. +-commutative 87.7%

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

   9. Simplified 87.7%

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

Alternative 3: 92.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{x}{\left(y + x\right) \cdot \frac{1 + x}{y}}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\left(y + x\right) \cdot \frac{y + 1}{y}}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 5.4e-8)
   (/ (/ x (* (+ y x) (/ (+ 1.0 x) y))) (+ y x))
   (/ (/ x (* (+ y x) (/ (+ y 1.0) y))) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 5.4e-8) {
		tmp = (x / ((y + x) * ((1.0 + x) / y))) / (y + x);
	} else {
		tmp = (x / ((y + x) * ((y + 1.0) / y))) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5.4d-8) then
        tmp = (x / ((y + x) * ((1.0d0 + x) / y))) / (y + x)
    else
        tmp = (x / ((y + x) * ((y + 1.0d0) / y))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 5.4e-8) {
		tmp = (x / ((y + x) * ((1.0 + x) / y))) / (y + x);
	} else {
		tmp = (x / ((y + x) * ((y + 1.0) / y))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 5.4e-8:
		tmp = (x / ((y + x) * ((1.0 + x) / y))) / (y + x)
	else:
		tmp = (x / ((y + x) * ((y + 1.0) / y))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 5.4e-8)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(Float64(1.0 + x) / y))) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(Float64(y + 1.0) / y))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.4e-8)
		tmp = (x / ((y + x) * ((1.0 + x) / y))) / (y + x);
	else
		tmp = (x / ((y + x) * ((y + 1.0) / y))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 5.4e-8], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(N[(1.0 + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.40000000000000005e-8

   1. Initial program 66.8%

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

      1. associate-*l* 66.8%

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

      2. times-frac 96.1%

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

      3. +-commutative 96.1%

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

      4. +-commutative 96.1%

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

      5. associate-+r+ 96.1%

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

      6. +-commutative 96.1%

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

      7. associate-+l+ 96.1%

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

   4. Applied egg-rr 96.1%

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

      1. associate-*l/ 96.1%

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

      2. clear-num 96.0%

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

      3. un-div-inv 95.9%

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

      4. *-un-lft-identity 95.9%

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

      5. times-frac 99.6%

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

      6. /-rgt-identity 99.6%

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

      7. +-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around 0 80.6%

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

      1. +-commutative 80.6%

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

   9. Simplified 80.6%

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

   if 5.40000000000000005e-8 < y

   1. Initial program 57.0%

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

      1. associate-*l* 57.0%

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

      2. times-frac 79.1%

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

      3. +-commutative 79.1%

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

      4. +-commutative 79.1%

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

      5. associate-+r+ 79.1%

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

      6. +-commutative 79.1%

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

      7. associate-+l+ 79.1%

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

   4. Applied egg-rr 79.1%

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

      1. associate-*l/ 79.1%

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

      2. clear-num 79.1%

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

      3. un-div-inv 79.2%

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

      4. *-un-lft-identity 79.2%

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

      5. times-frac 98.7%

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

      6. /-rgt-identity 98.7%

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

      7. +-commutative 98.7%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in x around 0 81.3%

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

      1. +-commutative 81.3%

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

   9. Simplified 81.3%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{x}{\left(y + x\right) \cdot \frac{1 + x}{y}}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\left(y + x\right) \cdot \frac{y + 1}{y}}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{x}{\left(y + x\right) \cdot \frac{1 + x}{y}}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 5.2e-8)
   (/ (/ x (* (+ y x) (/ (+ 1.0 x) y))) (+ y x))
   (/ (/ x (+ y 1.0)) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 5.2e-8) {
		tmp = (x / ((y + x) * ((1.0 + x) / y))) / (y + x);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5.2d-8) then
        tmp = (x / ((y + x) * ((1.0d0 + x) / y))) / (y + x)
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 5.2e-8) {
		tmp = (x / ((y + x) * ((1.0 + x) / y))) / (y + x);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 5.2e-8:
		tmp = (x / ((y + x) * ((1.0 + x) / y))) / (y + x)
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 5.2e-8)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(Float64(1.0 + x) / y))) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.2e-8)
		tmp = (x / ((y + x) * ((1.0 + x) / y))) / (y + x);
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 5.2e-8], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(N[(1.0 + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.2000000000000002e-8

   1. Initial program 66.8%

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

      1. associate-*l* 66.8%

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

      2. times-frac 96.1%

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

      3. +-commutative 96.1%

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

      4. +-commutative 96.1%

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

      5. associate-+r+ 96.1%

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

      6. +-commutative 96.1%

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

      7. associate-+l+ 96.1%

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

   4. Applied egg-rr 96.1%

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

      1. associate-*l/ 96.1%

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

      2. clear-num 96.0%

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

      3. un-div-inv 95.9%

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

      4. *-un-lft-identity 95.9%

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

      5. times-frac 99.6%

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

      6. /-rgt-identity 99.6%

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

      7. +-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around 0 80.6%

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

      1. +-commutative 80.6%

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

   9. Simplified 80.6%

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

   if 5.2000000000000002e-8 < y

   1. Initial program 57.0%

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

      1. associate-*l* 57.0%

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

      2. times-frac 79.1%

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

      3. +-commutative 79.1%

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

      4. +-commutative 79.1%

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

      5. associate-+r+ 79.1%

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

      6. +-commutative 79.1%

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

      7. associate-+l+ 79.1%

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

   4. Applied egg-rr 79.1%

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

      1. associate-*l/ 79.1%

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

      2. clear-num 79.1%

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

      3. un-div-inv 79.2%

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

      4. *-un-lft-identity 79.2%

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

      5. times-frac 98.7%

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

      6. /-rgt-identity 98.7%

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

      7. +-commutative 98.7%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in x around 0 81.0%

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

      1. +-commutative 81.0%

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

   9. Simplified 81.0%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{x}{\left(y + x\right) \cdot \frac{1 + x}{y}}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

   1. associate-*l* 64.3%

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

   2. times-frac 91.8%

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

   3. +-commutative 91.8%

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

   4. +-commutative 91.8%

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

   5. associate-+r+ 91.8%

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

   6. +-commutative 91.8%

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

   7. associate-+l+ 91.8%

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

4. Applied egg-rr 91.8%

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

   1. clear-num 91.7%

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

   2. associate-/r* 99.7%

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

   3. frac-times 99.0%

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

   4. *-un-lft-identity 99.0%

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

   5. +-commutative 99.0%

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

6. Applied egg-rr 99.0%

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

7. Final simplification 99.0%

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

Alternative 6: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 64.3%

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

   1. associate-*l* 64.3%

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

   2. times-frac 91.8%

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

   3. +-commutative 91.8%

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

   4. +-commutative 91.8%

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

   5. associate-+r+ 91.8%

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

   6. +-commutative 91.8%

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

   7. associate-+l+ 91.8%

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

4. Applied egg-rr 91.8%

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

   1. associate-*l/ 91.8%

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

   2. clear-num 91.7%

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

   3. un-div-inv 91.7%

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

   4. *-un-lft-identity 91.7%

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

   5. times-frac 99.3%

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

   6. /-rgt-identity 99.3%

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

   7. +-commutative 99.3%

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

6. Applied egg-rr 99.3%

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

7. Final simplification 99.3%

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

Alternative 7: 83.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{y}{x}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.55e-140) (/ (/ y x) (+ 1.0 x)) (/ (/ x (+ y 1.0)) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.55e-140) {
		tmp = (y / x) / (1.0 + x);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.55d-140) then
        tmp = (y / x) / (1.0d0 + x)
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.55e-140) {
		tmp = (y / x) / (1.0 + x);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.55e-140:
		tmp = (y / x) / (1.0 + x)
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.55e-140)
		tmp = Float64(Float64(y / x) / Float64(1.0 + x));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.55e-140)
		tmp = (y / x) / (1.0 + x);
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.55e-140], N[(N[(y / x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.55e-140

   1. Initial program 64.3%

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

      1. associate-/l* 78.2%

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

      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 y around 0 59.1%

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

      1. associate-/r* 56.7%

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

      2. +-commutative 56.7%

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

   7. Simplified 56.7%

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

   if 1.55e-140 < y

   1. Initial program 64.4%

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

      1. associate-*l* 64.4%

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

      2. times-frac 85.2%

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

      3. +-commutative 85.2%

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

      4. +-commutative 85.2%

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

      5. associate-+r+ 85.2%

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

      6. +-commutative 85.2%

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

      7. associate-+l+ 85.2%

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

   4. Applied egg-rr 85.2%

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

      1. associate-*l/ 85.2%

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

      2. clear-num 85.2%

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

      3. un-div-inv 85.2%

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

      4. *-un-lft-identity 85.2%

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

      5. times-frac 98.9%

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

      6. /-rgt-identity 98.9%

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

      7. +-commutative 98.9%

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in x around 0 67.8%

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

      1. +-commutative 67.8%

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

   9. Simplified 67.8%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{y}{x}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{y}{x}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.55e-140) (/ (/ y x) (+ 1.0 x)) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.55e-140) {
		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 <= 1.55d-140) 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 <= 1.55e-140) {
		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 <= 1.55e-140:
		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 <= 1.55e-140)
		tmp = Float64(Float64(y / x) / Float64(1.0 + x));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.55e-140)
		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, 1.55e-140], N[(N[(y / x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.55e-140

   1. Initial program 64.3%

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

      1. associate-/l* 78.2%

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

      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 y around 0 59.1%

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

      1. associate-/r* 56.7%

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

      2. +-commutative 56.7%

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

   7. Simplified 56.7%

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

   if 1.55e-140 < y

   1. Initial program 64.4%

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

      1. associate-/l* 79.1%

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

      2. associate-+l+ 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in x around 0 61.1%

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

      1. associate-/r* 61.1%

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

      2. +-commutative 61.1%

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

   7. Simplified 61.1%

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

      1. associate-*r/ 67.2%

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

      2. div-inv 67.3%

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

   9. Applied egg-rr 67.3%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{y}{x}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-140}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.4e-140) (/ y (* x (+ 1.0 x))) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.4e-140) {
		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 <= 1.4d-140) 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 <= 1.4e-140) {
		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 <= 1.4e-140:
		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 <= 1.4e-140)
		tmp = Float64(y / Float64(x * Float64(1.0 + x)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.4e-140)
		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, 1.4e-140], N[(y / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.4000000000000001e-140

   1. Initial program 64.3%

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

      1. associate-/l* 78.2%

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

      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 y around 0 59.1%

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

   if 1.4000000000000001e-140 < y

   1. Initial program 64.4%

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

      1. associate-/l* 79.1%

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

      2. associate-+l+ 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in x around 0 61.1%

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

      1. associate-/r* 61.1%

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

      2. +-commutative 61.1%

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

   7. Simplified 61.1%

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

      1. associate-*r/ 67.2%

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

      2. div-inv 67.3%

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

   9. Applied egg-rr 67.3%

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

Alternative 10: 79.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-140}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.55e-140) (/ y (* x (+ 1.0 x))) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.55e-140) {
		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 <= 1.55d-140) 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 <= 1.55e-140) {
		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 <= 1.55e-140:
		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 <= 1.55e-140)
		tmp = Float64(y / Float64(x * Float64(1.0 + x)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.55e-140)
		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, 1.55e-140], N[(y / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.55e-140

   1. Initial program 64.3%

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

      1. associate-/l* 78.2%

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

      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 y around 0 59.1%

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

   if 1.55e-140 < y

   1. Initial program 64.4%

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

      1. associate-/l* 79.1%

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

      2. associate-+l+ 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in x around 0 61.0%

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

      1. +-commutative 61.0%

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

   7. Simplified 61.0%

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

Alternative 11: 49.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

   1. associate-/l* 78.5%

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

   2. associate-+l+ 78.5%

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

3. Simplified 78.5%

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

5. Taylor expanded in x around 0 49.4%

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

   1. +-commutative 49.4%

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

7. Simplified 49.4%

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

Alternative 12: 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 64.3%

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

   1. associate-/l* 78.5%

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

   2. associate-+l+ 78.5%

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

3. Simplified 78.5%

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

5. Taylor expanded in x around 0 49.4%

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

   1. +-commutative 49.4%

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

7. Simplified 49.4%

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

8. Taylor expanded in y around 0 24.8%

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

Alternative 13: 4.3% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

   1. *-un-lft-identity 64.3%

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

   2. associate-*l* 64.3%

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

   3. times-frac 68.6%

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

   4. +-commutative 68.6%

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

   5. *-commutative 68.6%

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

   6. +-commutative 68.6%

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

   7. associate-+r+ 68.6%

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

   8. +-commutative 68.6%

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

   9. associate-+l+ 68.6%

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

4. Applied egg-rr 68.6%

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

5. Taylor expanded in x around inf 36.4%

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

6. Taylor expanded in y around inf 4.4%

   \[\leadsto \color{blue}{\frac{1}{x}} \]
7. Add Preprocessing

Alternative 14: 3.5% accurate, 17.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

   1. associate-*l* 64.3%

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

   2. times-frac 91.8%

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

   3. +-commutative 91.8%

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

   4. +-commutative 91.8%

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

   5. associate-+r+ 91.8%

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

   6. +-commutative 91.8%

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

   7. associate-+l+ 91.8%

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

4. Applied egg-rr 91.8%

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

   1. clear-num 91.7%

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

   2. associate-/r* 99.7%

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

   3. frac-times 99.0%

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

   4. *-un-lft-identity 99.0%

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

   5. +-commutative 99.0%

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in y around 0 49.2%

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

   1. +-commutative 49.2%

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

9. Simplified 49.2%

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

10. Taylor expanded in x around 0 3.5%

    \[\leadsto \color{blue}{1} \]
11. 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 2024119 
(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))))