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

Percentage Accurate: 69.7% → 99.3%

Time: 15.4s

Alternatives: 18

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.7% 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: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 70.9%

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

   1. *-commutative 70.9%

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

   2. associate-*l* 70.9%

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

   3. times-frac 96.2%

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

   4. +-commutative 96.2%

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

   5. +-commutative 96.2%

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

   6. associate-+r+ 96.2%

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

   7. +-commutative 96.2%

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

   8. associate-+l+ 96.2%

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

4. Applied egg-rr 96.2%

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

   1. clear-num 95.5%

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

   2. un-div-inv 95.5%

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

   3. +-commutative 95.5%

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

   4. *-un-lft-identity 95.5%

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

   5. times-frac 99.0%

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

   6. /-rgt-identity 99.0%

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

   7. +-commutative 99.0%

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

   8. +-commutative 99.0%

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

   9. +-commutative 99.0%

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

   10. associate-+l+ 99.0%

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

   11. +-commutative 99.0%

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

6. Applied egg-rr 99.0%

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

   1. *-un-lft-identity 99.0%

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

   2. times-frac 99.6%

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

   3. +-commutative 99.6%

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

   4. +-commutative 99.6%

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

   5. +-commutative 99.6%

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

   6. associate-+l+ 99.6%

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

8. Applied egg-rr 99.6%

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

   1. associate-*l/ 99.7%

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

   2. *-lft-identity 99.7%

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

   3. associate-/l/ 99.0%

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

   4. *-commutative 99.0%

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

   5. +-commutative 99.0%

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

10. Simplified 99.0%

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

11. Final simplification 99.0%

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

Alternative 2: 94.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{\left(y + x\right) \cdot \frac{x + 1}{x}}\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{+128}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) \cdot \frac{y + 1}{x}}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.7e-22)
   (/ (/ y (+ y x)) (* (+ y x) (/ (+ x 1.0) x)))
   (if (<= y 1.16e+128)
     (* x (/ y (* (* (+ y x) (+ y x)) (+ x (+ y 1.0)))))
     (/ (/ y (* (+ y x) (/ (+ y 1.0) x))) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.7e-22) {
		tmp = (y / (y + x)) / ((y + x) * ((x + 1.0) / x));
	} else if (y <= 1.16e+128) {
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))));
	} else {
		tmp = (y / ((y + x) * ((y + 1.0) / x))) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.7d-22) then
        tmp = (y / (y + x)) / ((y + x) * ((x + 1.0d0) / x))
    else if (y <= 1.16d+128) then
        tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0d0))))
    else
        tmp = (y / ((y + x) * ((y + 1.0d0) / x))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.7e-22) {
		tmp = (y / (y + x)) / ((y + x) * ((x + 1.0) / x));
	} else if (y <= 1.16e+128) {
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))));
	} else {
		tmp = (y / ((y + x) * ((y + 1.0) / x))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.7e-22:
		tmp = (y / (y + x)) / ((y + x) * ((x + 1.0) / x))
	elif y <= 1.16e+128:
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))))
	else:
		tmp = (y / ((y + x) * ((y + 1.0) / x))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.7e-22)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(Float64(y + x) * Float64(Float64(x + 1.0) / x)));
	elseif (y <= 1.16e+128)
		tmp = Float64(x * Float64(y / Float64(Float64(Float64(y + x) * Float64(y + x)) * Float64(x + Float64(y + 1.0)))));
	else
		tmp = Float64(Float64(y / Float64(Float64(y + x) * Float64(Float64(y + 1.0) / x))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.7e-22)
		tmp = (y / (y + x)) / ((y + x) * ((x + 1.0) / x));
	elseif (y <= 1.16e+128)
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))));
	else
		tmp = (y / ((y + x) * ((y + 1.0) / x))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 2.7e-22], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.16e+128], 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[(y / N[(N[(y + x), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.7000000000000002e-22

   1. Initial program 70.4%

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

      1. *-commutative 70.4%

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

      2. associate-*l* 70.4%

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

      3. times-frac 96.6%

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

      4. +-commutative 96.6%

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

      5. +-commutative 96.6%

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

      6. associate-+r+ 96.6%

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

      7. +-commutative 96.6%

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

      8. associate-+l+ 96.6%

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

   4. Applied egg-rr 96.6%

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

      1. clear-num 96.2%

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

      2. un-div-inv 96.2%

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

      3. +-commutative 96.2%

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

      4. *-un-lft-identity 96.2%

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

      5. times-frac 99.3%

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

      6. /-rgt-identity 99.3%

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

      7. +-commutative 99.3%

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

      8. +-commutative 99.3%

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

      9. +-commutative 99.3%

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

      10. associate-+l+ 99.3%

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

      11. +-commutative 99.3%

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

   6. Applied egg-rr 99.3%

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

   7. Taylor expanded in y around 0 79.8%

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

      1. +-commutative 79.8%

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

   9. Simplified 79.8%

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

   if 2.7000000000000002e-22 < y < 1.1600000000000001e128

   1. Initial program 85.2%

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

      1. associate-/l* 94.0%

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

      2. associate-+l+ 94.0%

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

   3. Simplified 94.0%

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

   if 1.1600000000000001e128 < y

   1. Initial program 52.7%

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

      1. *-commutative 52.7%

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

      2. associate-*l* 52.7%

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

      3. times-frac 86.9%

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

      4. +-commutative 86.9%

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

      5. +-commutative 86.9%

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

      6. associate-+r+ 86.9%

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

      7. +-commutative 86.9%

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

      8. associate-+l+ 86.9%

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

   4. Applied egg-rr 86.9%

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

      1. clear-num 86.9%

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

      2. un-div-inv 86.9%

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

      3. +-commutative 86.9%

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

      4. *-un-lft-identity 86.9%

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

      5. times-frac 99.8%

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

      6. /-rgt-identity 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-+l+ 99.8%

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

      11. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. times-frac 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+l+ 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. associate-*l/ 99.9%

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

      2. *-lft-identity 99.9%

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

      3. associate-/l/ 99.9%

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

      4. *-commutative 99.9%

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

      5. +-commutative 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in x around 0 74.3%

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

       1. +-commutative 74.3%

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

   13. Simplified 74.3%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{\left(y + x\right) \cdot \frac{x + 1}{x}}\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{+128}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) \cdot \frac{y + 1}{x}}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{if}\;x \leq -0.0065:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{-13}:\\ \;\;\;\;\frac{-1}{\frac{y \cdot \left(-1 - y\right)}{x}}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-68}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ y (+ y x)) (+ x 1.0))))
   (if (<= x -0.0065)
     t_0
     (if (<= x -3.9e-13)
       (/ -1.0 (/ (* y (- -1.0 y)) x))
       (if (<= x -3e-68) t_0 (/ (/ x (+ y 1.0)) y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y / (y + x)) / (x + 1.0);
	double tmp;
	if (x <= -0.0065) {
		tmp = t_0;
	} else if (x <= -3.9e-13) {
		tmp = -1.0 / ((y * (-1.0 - y)) / x);
	} else if (x <= -3e-68) {
		tmp = t_0;
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y / (y + x)) / (x + 1.0d0)
    if (x <= (-0.0065d0)) then
        tmp = t_0
    else if (x <= (-3.9d-13)) then
        tmp = (-1.0d0) / ((y * ((-1.0d0) - y)) / x)
    else if (x <= (-3d-68)) then
        tmp = t_0
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y / (y + x)) / (x + 1.0);
	double tmp;
	if (x <= -0.0065) {
		tmp = t_0;
	} else if (x <= -3.9e-13) {
		tmp = -1.0 / ((y * (-1.0 - y)) / x);
	} else if (x <= -3e-68) {
		tmp = t_0;
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y / (y + x)) / (x + 1.0)
	tmp = 0
	if x <= -0.0065:
		tmp = t_0
	elif x <= -3.9e-13:
		tmp = -1.0 / ((y * (-1.0 - y)) / x)
	elif x <= -3e-68:
		tmp = t_0
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y / Float64(y + x)) / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -0.0065)
		tmp = t_0;
	elseif (x <= -3.9e-13)
		tmp = Float64(-1.0 / Float64(Float64(y * Float64(-1.0 - y)) / x));
	elseif (x <= -3e-68)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (y / (y + x)) / (x + 1.0);
	tmp = 0.0;
	if (x <= -0.0065)
		tmp = t_0;
	elseif (x <= -3.9e-13)
		tmp = -1.0 / ((y * (-1.0 - y)) / x);
	elseif (x <= -3e-68)
		tmp = t_0;
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0065], t$95$0, If[LessEqual[x, -3.9e-13], N[(-1.0 / N[(N[(y * N[(-1.0 - y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3e-68], t$95$0, N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0064999999999999997 or -3.90000000000000004e-13 < x < -3e-68

   1. Initial program 68.7%

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

      1. *-commutative 68.7%

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

      2. associate-*l* 68.7%

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

      3. times-frac 93.9%

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

      4. +-commutative 93.9%

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

      5. +-commutative 93.9%

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

      6. associate-+r+ 93.9%

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

      7. +-commutative 93.9%

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

      8. associate-+l+ 93.9%

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

   4. Applied egg-rr 93.9%

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

      1. clear-num 93.9%

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

      2. un-div-inv 93.9%

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

      3. +-commutative 93.9%

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

      4. *-un-lft-identity 93.9%

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

      5. times-frac 99.7%

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

      6. /-rgt-identity 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-+l+ 99.7%

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

      11. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around 0 67.3%

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

      1. +-commutative 67.3%

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

   9. Simplified 67.3%

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

   if -0.0064999999999999997 < x < -3.90000000000000004e-13

   1. Initial program 100.0%

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

      1. associate-/l* 99.2%

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

      2. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in x around 0 99.2%

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

      1. associate-/r* 99.2%

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

      2. +-commutative 99.2%

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

   7. Simplified 99.2%

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

      1. associate-/l/ 99.2%

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

      2. *-commutative 99.2%

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

      3. div-inv 99.2%

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

      4. clear-num 100.0%

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

      5. frac-2neg 100.0%

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

      6. metadata-eval 100.0%

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

      7. distribute-rgt-in 100.0%

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

      8. *-un-lft-identity 100.0%

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

      9. fma-define 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. distribute-neg-frac 100.0%

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

       2. neg-mul-1 100.0%

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

       3. fma-undefine 100.0%

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

       4. unpow2 100.0%

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

       5. +-commutative 100.0%

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

       6. *-lft-identity 100.0%

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

       7. unpow2 100.0%

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

       8. distribute-rgt-in 100.0%

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

       9. mul-1-neg 100.0%

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

       10. *-commutative 100.0%

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

       11. distribute-lft-neg-in 100.0%

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

       12. mul-1-neg 100.0%

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

       13. distribute-lft-in 100.0%

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

       14. metadata-eval 100.0%

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

       15. neg-mul-1 100.0%

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

       16. unsub-neg 100.0%

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

   11. Simplified 100.0%

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

   if -3e-68 < x

   1. Initial program 71.5%

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

      1. associate-/l* 86.6%

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

      2. associate-+l+ 86.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in x around 0 57.6%

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

      1. +-commutative 57.6%

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

   7. Simplified 57.6%

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

      1. *-un-lft-identity 57.6%

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

      2. times-frac 56.9%

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

   9. Applied egg-rr 56.9%

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

       1. associate-*l/ 57.0%

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

       2. *-lft-identity 57.0%

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

   11. Simplified 57.0%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0065:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{-13}:\\ \;\;\;\;\frac{-1}{\frac{y \cdot \left(-1 - y\right)}{x}}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-12} \lor \neg \left(x \leq -5.6 \cdot 10^{-67}\right):\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.05e+30)
   (/ (/ y x) x)
   (if (or (<= x -2e-12) (not (<= x -5.6e-67)))
     (/ x (* y (+ y 1.0)))
     (/ y (* x (+ x 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.05e+30) {
		tmp = (y / x) / x;
	} else if ((x <= -2e-12) || !(x <= -5.6e-67)) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = y / (x * (x + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.05d+30)) then
        tmp = (y / x) / x
    else if ((x <= (-2d-12)) .or. (.not. (x <= (-5.6d-67)))) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = y / (x * (x + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.05e+30) {
		tmp = (y / x) / x;
	} else if ((x <= -2e-12) || !(x <= -5.6e-67)) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = y / (x * (x + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.05e+30:
		tmp = (y / x) / x
	elif (x <= -2e-12) or not (x <= -5.6e-67):
		tmp = x / (y * (y + 1.0))
	else:
		tmp = y / (x * (x + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.05e+30)
		tmp = Float64(Float64(y / x) / x);
	elseif ((x <= -2e-12) || !(x <= -5.6e-67))
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.05e+30)
		tmp = (y / x) / x;
	elseif ((x <= -2e-12) || ~((x <= -5.6e-67)))
		tmp = x / (y * (y + 1.0));
	else
		tmp = y / (x * (x + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.05e+30], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[Or[LessEqual[x, -2e-12], N[Not[LessEqual[x, -5.6e-67]], $MachinePrecision]], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05e30

   1. Initial program 67.3%

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

      1. *-commutative 67.3%

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

      2. associate-*l* 67.3%

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

      3. times-frac 92.2%

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

      4. +-commutative 92.2%

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

      5. +-commutative 92.2%

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

      6. associate-+r+ 92.2%

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

      7. +-commutative 92.2%

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

      8. associate-+l+ 92.2%

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

   4. Applied egg-rr 92.2%

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

      1. clear-num 92.2%

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

      2. un-div-inv 92.2%

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

      3. +-commutative 92.2%

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

      4. *-un-lft-identity 92.2%

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

      5. times-frac 99.8%

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

      6. /-rgt-identity 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-+l+ 99.8%

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

      11. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf 73.3%

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

   8. Taylor expanded in y around 0 72.9%

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

   if -1.05e30 < x < -1.99999999999999996e-12 or -5.60000000000000021e-67 < x

   1. Initial program 72.0%

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

      1. associate-/l* 86.3%

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

      2. associate-+l+ 86.3%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in x around 0 58.8%

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

      1. +-commutative 58.8%

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

   7. Simplified 58.8%

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

   if -1.99999999999999996e-12 < x < -5.60000000000000021e-67

   1. Initial program 71.6%

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

      1. associate-/l* 90.0%

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

      2. associate-+l+ 90.0%

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

   3. Simplified 90.0%

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

   5. Taylor expanded in y around 0 62.9%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-12} \lor \neg \left(x \leq -5.6 \cdot 10^{-67}\right):\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+34}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{-69}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -4.5e+34)
   (/ (/ y x) x)
   (if (<= x -5.8e-13)
     (/ (/ x y) (+ y x))
     (if (<= x -3.7e-69) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4.5e+34) {
		tmp = (y / x) / x;
	} else if (x <= -5.8e-13) {
		tmp = (x / y) / (y + x);
	} else if (x <= -3.7e-69) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.5d+34)) then
        tmp = (y / x) / x
    else if (x <= (-5.8d-13)) then
        tmp = (x / y) / (y + x)
    else if (x <= (-3.7d-69)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -4.5e+34) {
		tmp = (y / x) / x;
	} else if (x <= -5.8e-13) {
		tmp = (x / y) / (y + x);
	} else if (x <= -3.7e-69) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -4.5e+34:
		tmp = (y / x) / x
	elif x <= -5.8e-13:
		tmp = (x / y) / (y + x)
	elif x <= -3.7e-69:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -4.5e+34)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -5.8e-13)
		tmp = Float64(Float64(x / y) / Float64(y + x));
	elseif (x <= -3.7e-69)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.5e+34)
		tmp = (y / x) / x;
	elseif (x <= -5.8e-13)
		tmp = (x / y) / (y + x);
	elseif (x <= -3.7e-69)
		tmp = y / (x * (x + 1.0));
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -4.5e+34], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -5.8e-13], N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.7e-69], 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 4 regimes

2. if x < -4.5e34

   1. Initial program 67.3%

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

      1. *-commutative 67.3%

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

      2. associate-*l* 67.3%

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

      3. times-frac 91.8%

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

      4. +-commutative 91.8%

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

      5. +-commutative 91.8%

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

      6. associate-+r+ 91.8%

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

      7. +-commutative 91.8%

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

      8. associate-+l+ 91.8%

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

   4. Applied egg-rr 91.8%

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

      1. clear-num 91.8%

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

      2. un-div-inv 91.8%

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

      3. +-commutative 91.8%

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

      4. *-un-lft-identity 91.8%

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

      5. times-frac 99.8%

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

      6. /-rgt-identity 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-+l+ 99.8%

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

      11. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf 76.9%

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

   8. Taylor expanded in y around 0 76.5%

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

   if -4.5e34 < x < -5.7999999999999995e-13

   1. Initial program 77.4%

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

      1. *-commutative 77.4%

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

      2. associate-*l* 77.5%

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

      3. times-frac 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+r+ 99.5%

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

      7. +-commutative 99.5%

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

      8. associate-+l+ 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. clear-num 99.5%

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

      2. un-div-inv 99.5%

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

      3. +-commutative 99.5%

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

      4. *-un-lft-identity 99.5%

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

      5. times-frac 99.5%

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

      6. /-rgt-identity 99.5%

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

      7. +-commutative 99.5%

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

      8. +-commutative 99.5%

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

      9. +-commutative 99.5%

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

      10. associate-+l+ 99.5%

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

      11. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. times-frac 99.3%

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

      3. +-commutative 99.3%

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

      4. +-commutative 99.3%

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

      5. +-commutative 99.3%

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

      6. associate-+l+ 99.3%

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

   8. Applied egg-rr 99.3%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

      3. associate-/l/ 99.6%

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

      4. *-commutative 99.6%

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

      5. +-commutative 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in y around inf 85.0%

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

   if -5.7999999999999995e-13 < x < -3.7000000000000002e-69

   1. Initial program 71.6%

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

      1. associate-/l* 90.0%

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

      2. associate-+l+ 90.0%

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

   3. Simplified 90.0%

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

   5. Taylor expanded in y around 0 62.9%

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

   if -3.7000000000000002e-69 < x

   1. Initial program 71.5%

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

      1. associate-/l* 86.6%

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

      2. associate-+l+ 86.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in x around 0 57.6%

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

      1. +-commutative 57.6%

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

   7. Simplified 57.6%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+34}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{-69}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{+34}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-66}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -4.1e+34)
   (/ (/ y x) x)
   (if (<= x -6e-12)
     (/ (/ x y) (+ y x))
     (if (<= x -4.6e-66) (/ y (* x (+ x 1.0))) (/ (/ x (+ y 1.0)) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4.1e+34) {
		tmp = (y / x) / x;
	} else if (x <= -6e-12) {
		tmp = (x / y) / (y + x);
	} else if (x <= -4.6e-66) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.1d+34)) then
        tmp = (y / x) / x
    else if (x <= (-6d-12)) then
        tmp = (x / y) / (y + x)
    else if (x <= (-4.6d-66)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -4.1e+34) {
		tmp = (y / x) / x;
	} else if (x <= -6e-12) {
		tmp = (x / y) / (y + x);
	} else if (x <= -4.6e-66) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -4.1e+34:
		tmp = (y / x) / x
	elif x <= -6e-12:
		tmp = (x / y) / (y + x)
	elif x <= -4.6e-66:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -4.1e+34)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -6e-12)
		tmp = Float64(Float64(x / y) / Float64(y + x));
	elseif (x <= -4.6e-66)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.1e+34)
		tmp = (y / x) / x;
	elseif (x <= -6e-12)
		tmp = (x / y) / (y + x);
	elseif (x <= -4.6e-66)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -4.1e+34], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -6e-12], N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.6e-66], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.0999999999999998e34

   1. Initial program 67.3%

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

      1. *-commutative 67.3%

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

      2. associate-*l* 67.3%

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

      3. times-frac 91.8%

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

      4. +-commutative 91.8%

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

      5. +-commutative 91.8%

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

      6. associate-+r+ 91.8%

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

      7. +-commutative 91.8%

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

      8. associate-+l+ 91.8%

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

   4. Applied egg-rr 91.8%

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

      1. clear-num 91.8%

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

      2. un-div-inv 91.8%

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

      3. +-commutative 91.8%

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

      4. *-un-lft-identity 91.8%

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

      5. times-frac 99.8%

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

      6. /-rgt-identity 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-+l+ 99.8%

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

      11. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf 76.9%

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

   8. Taylor expanded in y around 0 76.5%

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

   if -4.0999999999999998e34 < x < -6.0000000000000003e-12

   1. Initial program 77.4%

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

      1. *-commutative 77.4%

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

      2. associate-*l* 77.5%

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

      3. times-frac 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+r+ 99.5%

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

      7. +-commutative 99.5%

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

      8. associate-+l+ 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. clear-num 99.5%

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

      2. un-div-inv 99.5%

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

      3. +-commutative 99.5%

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

      4. *-un-lft-identity 99.5%

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

      5. times-frac 99.5%

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

      6. /-rgt-identity 99.5%

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

      7. +-commutative 99.5%

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

      8. +-commutative 99.5%

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

      9. +-commutative 99.5%

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

      10. associate-+l+ 99.5%

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

      11. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. times-frac 99.3%

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

      3. +-commutative 99.3%

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

      4. +-commutative 99.3%

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

      5. +-commutative 99.3%

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

      6. associate-+l+ 99.3%

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

   8. Applied egg-rr 99.3%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

      3. associate-/l/ 99.6%

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

      4. *-commutative 99.6%

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

      5. +-commutative 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in y around inf 85.0%

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

   if -6.0000000000000003e-12 < x < -4.59999999999999984e-66

   1. Initial program 71.6%

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

      1. associate-/l* 90.0%

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

      2. associate-+l+ 90.0%

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

   3. Simplified 90.0%

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

   5. Taylor expanded in y around 0 62.9%

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

   if -4.59999999999999984e-66 < x

   1. Initial program 71.5%

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

      1. associate-/l* 86.6%

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

      2. associate-+l+ 86.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in x around 0 57.6%

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

      1. +-commutative 57.6%

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

   7. Simplified 57.6%

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

      1. *-un-lft-identity 57.6%

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

      2. times-frac 56.9%

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

   9. Applied egg-rr 56.9%

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

       1. associate-*l/ 57.0%

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

       2. *-lft-identity 57.0%

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

   11. Simplified 57.0%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{+34}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-66}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0035:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-66}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -0.0035)
   (/ (/ y x) (+ x 1.0))
   (if (<= x -1e-12)
     (/ x (* y (+ y 1.0)))
     (if (<= x -1.7e-66) (/ y (* x (+ x 1.0))) (/ (/ x (+ y 1.0)) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -0.0035) {
		tmp = (y / x) / (x + 1.0);
	} else if (x <= -1e-12) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -1.7e-66) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.0035d0)) then
        tmp = (y / x) / (x + 1.0d0)
    else if (x <= (-1d-12)) then
        tmp = x / (y * (y + 1.0d0))
    else if (x <= (-1.7d-66)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -0.0035) {
		tmp = (y / x) / (x + 1.0);
	} else if (x <= -1e-12) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -1.7e-66) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -0.0035:
		tmp = (y / x) / (x + 1.0)
	elif x <= -1e-12:
		tmp = x / (y * (y + 1.0))
	elif x <= -1.7e-66:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -0.0035)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (x <= -1e-12)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	elseif (x <= -1.7e-66)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.0035)
		tmp = (y / x) / (x + 1.0);
	elseif (x <= -1e-12)
		tmp = x / (y * (y + 1.0));
	elseif (x <= -1.7e-66)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -0.0035], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1e-12], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.7e-66], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -0.00350000000000000007

   1. Initial program 68.3%

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

      1. associate-/l* 80.6%

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

      2. associate-+l+ 80.6%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in y around 0 68.9%

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

      1. associate-/r* 67.4%

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

      2. +-commutative 67.4%

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

   7. Simplified 67.4%

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

   if -0.00350000000000000007 < x < -9.9999999999999998e-13

   1. Initial program 100.0%

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

      1. associate-/l* 99.2%

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

      2. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in x around 0 99.2%

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

      1. +-commutative 99.2%

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

   7. Simplified 99.2%

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

   if -9.9999999999999998e-13 < x < -1.69999999999999999e-66

   1. Initial program 71.6%

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

      1. associate-/l* 90.0%

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

      2. associate-+l+ 90.0%

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

   3. Simplified 90.0%

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

   5. Taylor expanded in y around 0 62.9%

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

   if -1.69999999999999999e-66 < x

   1. Initial program 71.5%

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

      1. associate-/l* 86.6%

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

      2. associate-+l+ 86.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in x around 0 57.6%

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

      1. +-commutative 57.6%

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

   7. Simplified 57.6%

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

      1. *-un-lft-identity 57.6%

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

      2. times-frac 56.9%

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

   9. Applied egg-rr 56.9%

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

       1. associate-*l/ 57.0%

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

       2. *-lft-identity 57.0%

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

   11. Simplified 57.0%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0035:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-66}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -0.145:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{\frac{y \cdot \left(-1 - y\right)}{x}}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-66}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -0.145)
   (/ (/ y x) (+ x 1.0))
   (if (<= x -9e-12)
     (/ -1.0 (/ (* y (- -1.0 y)) x))
     (if (<= x -3e-66) (/ y (* x (+ x 1.0))) (/ (/ x (+ y 1.0)) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -0.145) {
		tmp = (y / x) / (x + 1.0);
	} else if (x <= -9e-12) {
		tmp = -1.0 / ((y * (-1.0 - y)) / x);
	} else if (x <= -3e-66) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.145d0)) then
        tmp = (y / x) / (x + 1.0d0)
    else if (x <= (-9d-12)) then
        tmp = (-1.0d0) / ((y * ((-1.0d0) - y)) / x)
    else if (x <= (-3d-66)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -0.145) {
		tmp = (y / x) / (x + 1.0);
	} else if (x <= -9e-12) {
		tmp = -1.0 / ((y * (-1.0 - y)) / x);
	} else if (x <= -3e-66) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -0.145:
		tmp = (y / x) / (x + 1.0)
	elif x <= -9e-12:
		tmp = -1.0 / ((y * (-1.0 - y)) / x)
	elif x <= -3e-66:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -0.145)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (x <= -9e-12)
		tmp = Float64(-1.0 / Float64(Float64(y * Float64(-1.0 - y)) / x));
	elseif (x <= -3e-66)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.145)
		tmp = (y / x) / (x + 1.0);
	elseif (x <= -9e-12)
		tmp = -1.0 / ((y * (-1.0 - y)) / x);
	elseif (x <= -3e-66)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -0.145], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -9e-12], N[(-1.0 / N[(N[(y * N[(-1.0 - y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3e-66], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -0.14499999999999999

   1. Initial program 68.3%

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

      1. associate-/l* 80.6%

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

      2. associate-+l+ 80.6%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in y around 0 68.9%

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

      1. associate-/r* 67.4%

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

      2. +-commutative 67.4%

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

   7. Simplified 67.4%

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

   if -0.14499999999999999 < x < -8.99999999999999962e-12

   1. Initial program 100.0%

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

      1. associate-/l* 99.2%

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

      2. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in x around 0 99.2%

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

      1. associate-/r* 99.2%

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

      2. +-commutative 99.2%

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

   7. Simplified 99.2%

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

      1. associate-/l/ 99.2%

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

      2. *-commutative 99.2%

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

      3. div-inv 99.2%

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

      4. clear-num 100.0%

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

      5. frac-2neg 100.0%

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

      6. metadata-eval 100.0%

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

      7. distribute-rgt-in 100.0%

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

      8. *-un-lft-identity 100.0%

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

      9. fma-define 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. distribute-neg-frac 100.0%

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

       2. neg-mul-1 100.0%

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

       3. fma-undefine 100.0%

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

       4. unpow2 100.0%

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

       5. +-commutative 100.0%

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

       6. *-lft-identity 100.0%

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

       7. unpow2 100.0%

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

       8. distribute-rgt-in 100.0%

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

       9. mul-1-neg 100.0%

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

       10. *-commutative 100.0%

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

       11. distribute-lft-neg-in 100.0%

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

       12. mul-1-neg 100.0%

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

       13. distribute-lft-in 100.0%

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

       14. metadata-eval 100.0%

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

       15. neg-mul-1 100.0%

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

       16. unsub-neg 100.0%

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

   11. Simplified 100.0%

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

   if -8.99999999999999962e-12 < x < -3.0000000000000002e-66

   1. Initial program 71.6%

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

      1. associate-/l* 90.0%

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

      2. associate-+l+ 90.0%

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

   3. Simplified 90.0%

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

   5. Taylor expanded in y around 0 62.9%

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

   if -3.0000000000000002e-66 < x

   1. Initial program 71.5%

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

      1. associate-/l* 86.6%

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

      2. associate-+l+ 86.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in x around 0 57.6%

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

      1. +-commutative 57.6%

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

   7. Simplified 57.6%

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

      1. *-un-lft-identity 57.6%

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

      2. times-frac 56.9%

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

   9. Applied egg-rr 56.9%

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

       1. associate-*l/ 57.0%

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

       2. *-lft-identity 57.0%

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

   11. Simplified 57.0%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.145:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{\frac{y \cdot \left(-1 - y\right)}{x}}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-66}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -0.055:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) \cdot \frac{y + 1}{x}}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -0.055)
   (* (/ x (* (+ y x) (+ y (+ x 1.0)))) (/ y x))
   (/ (/ y (* (+ y x) (/ (+ y 1.0) x))) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -0.055) {
		tmp = (x / ((y + x) * (y + (x + 1.0)))) * (y / x);
	} else {
		tmp = (y / ((y + x) * ((y + 1.0) / x))) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.055d0)) then
        tmp = (x / ((y + x) * (y + (x + 1.0d0)))) * (y / x)
    else
        tmp = (y / ((y + x) * ((y + 1.0d0) / x))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -0.055) {
		tmp = (x / ((y + x) * (y + (x + 1.0)))) * (y / x);
	} else {
		tmp = (y / ((y + x) * ((y + 1.0) / x))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -0.055:
		tmp = (x / ((y + x) * (y + (x + 1.0)))) * (y / x)
	else:
		tmp = (y / ((y + x) * ((y + 1.0) / x))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -0.055)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0)))) * Float64(y / x));
	else
		tmp = Float64(Float64(y / Float64(Float64(y + x) * Float64(Float64(y + 1.0) / x))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.055)
		tmp = (x / ((y + x) * (y + (x + 1.0)))) * (y / x);
	else
		tmp = (y / ((y + x) * ((y + 1.0) / x))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -0.055], 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[(y / N[(N[(y + x), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0550000000000000003

   1. Initial program 68.3%

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

      1. *-commutative 68.3%

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

      2. associate-*l* 68.3%

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

      3. times-frac 93.1%

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

      4. +-commutative 93.1%

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

      5. +-commutative 93.1%

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

      6. associate-+r+ 93.1%

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

      7. +-commutative 93.1%

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

      8. associate-+l+ 93.1%

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

   4. Applied egg-rr 93.1%

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

   5. Taylor expanded in y around 0 79.3%

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

   if -0.0550000000000000003 < x

   1. Initial program 71.8%

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

      1. *-commutative 71.8%

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

      2. associate-*l* 71.9%

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

      3. times-frac 97.4%

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

      4. +-commutative 97.4%

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

      5. +-commutative 97.4%

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

      6. associate-+r+ 97.4%

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

      7. +-commutative 97.4%

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

      8. associate-+l+ 97.4%

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

   4. Applied egg-rr 97.4%

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

      1. clear-num 96.4%

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

      2. un-div-inv 96.4%

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

      3. +-commutative 96.4%

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

      4. *-un-lft-identity 96.4%

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

      5. times-frac 98.7%

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

      6. /-rgt-identity 98.7%

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

      7. +-commutative 98.7%

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

      8. +-commutative 98.7%

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

      9. +-commutative 98.7%

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

      10. associate-+l+ 98.7%

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

      11. +-commutative 98.7%

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

   6. Applied egg-rr 98.7%

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

      1. *-un-lft-identity 98.7%

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

      2. times-frac 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+l+ 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. associate-/l/ 98.8%

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

      4. *-commutative 98.8%

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

      5. +-commutative 98.8%

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

   10. Simplified 98.8%

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

   11. Taylor expanded in x around 0 81.0%

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

       1. +-commutative 81.0%

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

   13. Simplified 81.0%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.055:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) \cdot \frac{y + 1}{x}}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 92.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 0.013:\\ \;\;\;\;\frac{\frac{y}{y + x}}{\left(y + x\right) \cdot \frac{x + 1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 0.013)
   (/ (/ y (+ y x)) (* (+ y x) (/ (+ x 1.0) x)))
   (/ x (* (+ y x) (+ y (+ x 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 0.013) {
		tmp = (y / (y + x)) / ((y + x) * ((x + 1.0) / x));
	} else {
		tmp = x / ((y + x) * (y + (x + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.013d0) then
        tmp = (y / (y + x)) / ((y + x) * ((x + 1.0d0) / x))
    else
        tmp = x / ((y + x) * (y + (x + 1.0d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 0.013) {
		tmp = (y / (y + x)) / ((y + x) * ((x + 1.0) / x));
	} else {
		tmp = x / ((y + x) * (y + (x + 1.0)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 0.013:
		tmp = (y / (y + x)) / ((y + x) * ((x + 1.0) / x))
	else:
		tmp = x / ((y + x) * (y + (x + 1.0)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 0.013)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(Float64(y + x) * Float64(Float64(x + 1.0) / x)));
	else
		tmp = Float64(x / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.013)
		tmp = (y / (y + x)) / ((y + x) * ((x + 1.0) / x));
	else
		tmp = x / ((y + x) * (y + (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 0.013], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.0129999999999999994

   1. Initial program 71.1%

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

      1. *-commutative 71.1%

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

      2. associate-*l* 71.1%

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

      3. times-frac 96.7%

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

      4. +-commutative 96.7%

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

      5. +-commutative 96.7%

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

      6. associate-+r+ 96.7%

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

      7. +-commutative 96.7%

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

      8. associate-+l+ 96.7%

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

   4. Applied egg-rr 96.7%

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

      1. clear-num 96.3%

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

      2. un-div-inv 96.3%

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

      3. +-commutative 96.3%

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

      4. *-un-lft-identity 96.3%

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

      5. times-frac 99.3%

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

      6. /-rgt-identity 99.3%

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

      7. +-commutative 99.3%

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

      8. +-commutative 99.3%

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

      9. +-commutative 99.3%

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

      10. associate-+l+ 99.3%

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

      11. +-commutative 99.3%

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

   6. Applied egg-rr 99.3%

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

   7. Taylor expanded in y around 0 80.3%

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

      1. +-commutative 80.3%

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

   9. Simplified 80.3%

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

   if 0.0129999999999999994 < y

   1. Initial program 70.1%

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

      1. *-commutative 70.1%

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

      2. associate-*l* 70.1%

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

      3. times-frac 94.4%

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

      4. +-commutative 94.4%

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

      5. +-commutative 94.4%

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

      6. associate-+r+ 94.4%

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

      7. +-commutative 94.4%

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

      8. associate-+l+ 94.4%

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

   4. Applied egg-rr 94.4%

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

   5. Taylor expanded in y around inf 90.5%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.013:\\ \;\;\;\;\frac{\frac{y}{y + x}}{\left(y + x\right) \cdot \frac{x + 1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -0.39:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-123}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-1 + \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -0.39)
   (/ (/ y x) x)
   (if (<= x 7e-123) (/ 1.0 (/ y x)) (+ 1.0 (+ -1.0 (/ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -0.39) {
		tmp = (y / x) / x;
	} else if (x <= 7e-123) {
		tmp = 1.0 / (y / x);
	} else {
		tmp = 1.0 + (-1.0 + (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 <= (-0.39d0)) then
        tmp = (y / x) / x
    else if (x <= 7d-123) then
        tmp = 1.0d0 / (y / x)
    else
        tmp = 1.0d0 + ((-1.0d0) + (x / y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -0.39) {
		tmp = (y / x) / x;
	} else if (x <= 7e-123) {
		tmp = 1.0 / (y / x);
	} else {
		tmp = 1.0 + (-1.0 + (x / y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -0.39:
		tmp = (y / x) / x
	elif x <= 7e-123:
		tmp = 1.0 / (y / x)
	else:
		tmp = 1.0 + (-1.0 + (x / y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -0.39)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= 7e-123)
		tmp = Float64(1.0 / Float64(y / x));
	else
		tmp = Float64(1.0 + Float64(-1.0 + 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 <= -0.39)
		tmp = (y / x) / x;
	elseif (x <= 7e-123)
		tmp = 1.0 / (y / x);
	else
		tmp = 1.0 + (-1.0 + (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, -0.39], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 7e-123], N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.39000000000000001

   1. Initial program 68.3%

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

      1. *-commutative 68.3%

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

      2. associate-*l* 68.3%

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

      3. times-frac 93.1%

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

      4. +-commutative 93.1%

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

      5. +-commutative 93.1%

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

      6. associate-+r+ 93.1%

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

      7. +-commutative 93.1%

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

      8. associate-+l+ 93.1%

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

   4. Applied egg-rr 93.1%

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

      1. clear-num 93.0%

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

      2. un-div-inv 93.0%

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

      3. +-commutative 93.0%

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

      4. *-un-lft-identity 93.0%

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

      5. times-frac 99.7%

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

      6. /-rgt-identity 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-+l+ 99.7%

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

      11. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around inf 66.5%

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

   8. Taylor expanded in y around 0 66.0%

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

   if -0.39000000000000001 < x < 6.9999999999999997e-123

   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* 82.4%

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

      2. associate-+l+ 82.4%

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

   3. Simplified 82.4%

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

   5. Taylor expanded in x around 0 74.6%

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

      1. associate-/r* 74.7%

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

      2. +-commutative 74.7%

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

   7. Simplified 74.7%

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

   8. Taylor expanded in y around 0 52.1%

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

      1. div-inv 52.2%

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

      2. clear-num 52.6%

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

   10. Applied egg-rr 52.6%

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

   if 6.9999999999999997e-123 < x

   1. Initial program 80.6%

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

      1. associate-/l* 92.2%

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

      2. associate-+l+ 92.2%

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

   3. Simplified 92.2%

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

   5. Taylor expanded in x around 0 35.8%

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

      1. associate-/r* 35.8%

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

      2. +-commutative 35.8%

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

   7. Simplified 35.8%

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

   8. Taylor expanded in y around 0 7.0%

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

      1. div-inv 7.0%

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

      2. expm1-log1p-u 6.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{y}\right)\right)} \]

      3. expm1-undefine 14.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{y}\right)} - 1} \]

      4. log1p-undefine 14.4%

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

      5. add-exp-log 15.4%

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

   10. Applied egg-rr 15.4%

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

       1. associate--l+ 15.4%

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

   12. Simplified 15.4%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.39:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-123}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-1 + \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 93.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.9%

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

   1. *-commutative 70.9%

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

   2. associate-*l* 70.9%

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

   3. times-frac 96.2%

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

   4. +-commutative 96.2%

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

   5. +-commutative 96.2%

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

   6. associate-+r+ 96.2%

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

   7. +-commutative 96.2%

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

   8. associate-+l+ 96.2%

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

4. Applied egg-rr 96.2%

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

5. Final simplification 96.2%

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

Alternative 13: 83.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-246}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.4e-246) (/ (/ y x) (+ x 1.0)) (/ x (* (+ y x) (+ y (+ x 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.4e-246) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = x / ((y + x) * (y + (x + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.4d-246) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = x / ((y + x) * (y + (x + 1.0d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.4e-246) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = x / ((y + x) * (y + (x + 1.0)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.4e-246:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = x / ((y + x) * (y + (x + 1.0)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.4e-246)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(x / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.4e-246)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = x / ((y + x) * (y + (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.4e-246], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.4000000000000001e-246

   1. Initial program 67.8%

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

      1. associate-/l* 82.6%

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

      2. associate-+l+ 82.6%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in y around 0 52.5%

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

      1. associate-/r* 53.1%

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

      2. +-commutative 53.1%

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

   7. Simplified 53.1%

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

   if 3.4000000000000001e-246 < y

   1. Initial program 74.9%

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

      1. *-commutative 74.9%

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

      2. associate-*l* 74.9%

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

      3. times-frac 97.3%

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

      4. +-commutative 97.3%

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

      5. +-commutative 97.3%

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

      6. associate-+r+ 97.3%

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

      7. +-commutative 97.3%

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

      8. associate-+l+ 97.3%

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

   4. Applied egg-rr 97.3%

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

   5. Taylor expanded in y around inf 75.5%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-246}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 78.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+28}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.55e+28) (/ (/ y x) x) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.55e+28) {
		tmp = (y / x) / x;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.55d+28)) then
        tmp = (y / x) / x
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.55e+28) {
		tmp = (y / x) / x;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.55e+28:
		tmp = (y / x) / x
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.55e+28)
		tmp = Float64(Float64(y / x) / x);
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.55e+28)
		tmp = (y / x) / x;
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.55e+28], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55e28

   1. Initial program 67.3%

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

      1. *-commutative 67.3%

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

      2. associate-*l* 67.3%

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

      3. times-frac 92.2%

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

      4. +-commutative 92.2%

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

      5. +-commutative 92.2%

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

      6. associate-+r+ 92.2%

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

      7. +-commutative 92.2%

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

      8. associate-+l+ 92.2%

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

   4. Applied egg-rr 92.2%

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

      1. clear-num 92.2%

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

      2. un-div-inv 92.2%

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

      3. +-commutative 92.2%

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

      4. *-un-lft-identity 92.2%

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

      5. times-frac 99.8%

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

      6. /-rgt-identity 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. associate-+l+ 99.8%

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

      11. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf 73.3%

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

   8. Taylor expanded in y around 0 72.9%

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

   if -1.55e28 < x

   1. Initial program 72.0%

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

      1. associate-/l* 86.5%

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

      2. associate-+l+ 86.5%

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

   3. Simplified 86.5%

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

   5. Taylor expanded in x around 0 57.5%

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

      1. +-commutative 57.5%

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

   7. Simplified 57.5%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+28}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -0.39:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -0.39) (/ (/ y x) x) (/ 1.0 (/ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -0.39) {
		tmp = (y / x) / x;
	} else {
		tmp = 1.0 / (y / x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.39d0)) then
        tmp = (y / x) / x
    else
        tmp = 1.0d0 / (y / x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -0.39) {
		tmp = (y / x) / x;
	} else {
		tmp = 1.0 / (y / x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -0.39:
		tmp = (y / x) / x
	else:
		tmp = 1.0 / (y / x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -0.39)
		tmp = Float64(Float64(y / x) / x);
	else
		tmp = Float64(1.0 / Float64(y / x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.39)
		tmp = (y / x) / x;
	else
		tmp = 1.0 / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -0.39], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.39000000000000001

   1. Initial program 68.3%

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

      1. *-commutative 68.3%

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

      2. associate-*l* 68.3%

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

      3. times-frac 93.1%

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

      4. +-commutative 93.1%

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

      5. +-commutative 93.1%

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

      6. associate-+r+ 93.1%

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

      7. +-commutative 93.1%

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

      8. associate-+l+ 93.1%

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

   4. Applied egg-rr 93.1%

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

      1. clear-num 93.0%

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

      2. un-div-inv 93.0%

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

      3. +-commutative 93.0%

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

      4. *-un-lft-identity 93.0%

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

      5. times-frac 99.7%

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

      6. /-rgt-identity 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. associate-+l+ 99.7%

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

      11. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around inf 66.5%

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

   8. Taylor expanded in y around 0 66.0%

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

   if -0.39000000000000001 < x

   1. Initial program 71.8%

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

      1. associate-/l* 87.0%

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

      2. associate-+l+ 87.0%

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

   3. Simplified 87.0%

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

   5. Taylor expanded in x around 0 56.6%

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

      1. associate-/r* 56.7%

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

      2. +-commutative 56.7%

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

   7. Simplified 56.7%

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

   8. Taylor expanded in y around 0 31.2%

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

      1. div-inv 31.3%

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

      2. clear-num 31.6%

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

   10. Applied egg-rr 31.6%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.39:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 26.8% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.9%

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

   1. associate-/l* 85.3%

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

   2. associate-+l+ 85.3%

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

3. Simplified 85.3%

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

5. Taylor expanded in x around 0 50.1%

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

   1. associate-/r* 50.1%

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

   2. +-commutative 50.1%

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

7. Simplified 50.1%

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

8. Taylor expanded in y around 0 23.7%

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

   1. div-inv 23.7%

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

   2. clear-num 23.9%

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

10. Applied egg-rr 23.9%

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

11. Final simplification 23.9%

    \[\leadsto \frac{1}{\frac{y}{x}} \]
12. Add Preprocessing

Alternative 17: 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 70.9%

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

   1. *-commutative 70.9%

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

   2. associate-*l* 70.9%

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

   3. times-frac 96.2%

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

   4. +-commutative 96.2%

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

   5. +-commutative 96.2%

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

   6. associate-+r+ 96.2%

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

   7. +-commutative 96.2%

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

   8. associate-+l+ 96.2%

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

4. Applied egg-rr 96.2%

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

   1. clear-num 95.5%

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

   2. un-div-inv 95.5%

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

   3. +-commutative 95.5%

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

   4. *-un-lft-identity 95.5%

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

   5. times-frac 99.0%

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

   6. /-rgt-identity 99.0%

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

   7. +-commutative 99.0%

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

   8. +-commutative 99.0%

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

   9. +-commutative 99.0%

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

   10. associate-+l+ 99.0%

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

   11. +-commutative 99.0%

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in x around inf 36.7%

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

8. Taylor expanded in y around inf 4.2%

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

9. Final simplification 4.2%

   \[\leadsto \frac{1}{x} \]
10. Add Preprocessing

Alternative 18: 26.5% 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 70.9%

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

   1. associate-/l* 85.3%

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

   2. associate-+l+ 85.3%

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

3. Simplified 85.3%

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

5. Taylor expanded in x around 0 50.2%

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

   1. +-commutative 50.2%

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

7. Simplified 50.2%

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

8. Taylor expanded in y around 0 23.7%

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

9. Final simplification 23.7%

   \[\leadsto \frac{x}{y} \]
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 2024066 
(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))))