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

Percentage Accurate: 68.4% → 99.8%

Time: 13.8s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.6%

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

   1. *-commutative 72.6%

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

   2. times-frac 88.9%

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

   3. pow2 88.9%

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

   4. associate-+r+ 88.9%

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

4. Applied egg-rr 88.9%

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

   1. *-un-lft-identity 88.9%

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

   2. pow2 88.9%

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

   3. times-frac 99.7%

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

   4. +-commutative 99.7%

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

   5. +-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-*l/ 99.8%

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

   2. *-lft-identity 99.8%

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

8. Simplified 99.8%

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

Alternative 2: 68.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-271}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x + 1}\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(y + x\right) \cdot \left(x + 1\right)}}{y + x}\\ \mathbf{elif}\;y \leq 4.16 \cdot 10^{+99}:\\ \;\;\;\;x \cdot \frac{y}{\left(y + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5e-271)
   (* (/ y (+ y x)) (/ 1.0 (+ x 1.0)))
   (if (<= y 1.12e-6)
     (* x (/ (/ y (* (+ y x) (+ x 1.0))) (+ y x)))
     (if (<= y 4.16e+99)
       (* x (/ y (* (+ y 1.0) (* (+ y x) (+ y x)))))
       (/ (/ x (+ y 1.0)) (+ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5e-271) {
		tmp = (y / (y + x)) * (1.0 / (x + 1.0));
	} else if (y <= 1.12e-6) {
		tmp = x * ((y / ((y + x) * (x + 1.0))) / (y + x));
	} else if (y <= 4.16e+99) {
		tmp = x * (y / ((y + 1.0) * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5d-271) then
        tmp = (y / (y + x)) * (1.0d0 / (x + 1.0d0))
    else if (y <= 1.12d-6) then
        tmp = x * ((y / ((y + x) * (x + 1.0d0))) / (y + x))
    else if (y <= 4.16d+99) then
        tmp = x * (y / ((y + 1.0d0) * ((y + x) * (y + x))))
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5e-271) {
		tmp = (y / (y + x)) * (1.0 / (x + 1.0));
	} else if (y <= 1.12e-6) {
		tmp = x * ((y / ((y + x) * (x + 1.0))) / (y + x));
	} else if (y <= 4.16e+99) {
		tmp = x * (y / ((y + 1.0) * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5e-271:
		tmp = (y / (y + x)) * (1.0 / (x + 1.0))
	elif y <= 1.12e-6:
		tmp = x * ((y / ((y + x) * (x + 1.0))) / (y + x))
	elif y <= 4.16e+99:
		tmp = x * (y / ((y + 1.0) * ((y + x) * (y + x))))
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5e-271)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / Float64(x + 1.0)));
	elseif (y <= 1.12e-6)
		tmp = Float64(x * Float64(Float64(y / Float64(Float64(y + x) * Float64(x + 1.0))) / Float64(y + x)));
	elseif (y <= 4.16e+99)
		tmp = Float64(x * Float64(y / Float64(Float64(y + 1.0) * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5e-271)
		tmp = (y / (y + x)) * (1.0 / (x + 1.0));
	elseif (y <= 1.12e-6)
		tmp = x * ((y / ((y + x) * (x + 1.0))) / (y + x));
	elseif (y <= 4.16e+99)
		tmp = x * (y / ((y + 1.0) * ((y + x) * (y + x))));
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5e-271], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.12e-6], N[(x * N[(N[(y / N[(N[(y + x), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.16e+99], N[(x * N[(y / N[(N[(y + 1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 5.0000000000000002e-271

   1. Initial program 75.4%

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

      1. *-commutative 75.4%

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

      2. associate-*l* 75.4%

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

      3. times-frac 95.5%

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

      4. +-commutative 95.5%

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

      5. distribute-lft-in 95.5%

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

      6. *-rgt-identity 95.5%

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

      7. pow2 95.5%

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

   4. Applied egg-rr 95.5%

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

      1. clear-num 94.4%

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

      2. inv-pow 94.4%

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

      3. +-commutative 94.4%

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

      4. associate-+l+ 94.4%

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

      5. +-commutative 94.4%

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

   6. Applied egg-rr 94.4%

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

      1. unpow-1 94.4%

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

      2. associate-+r+ 94.4%

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

      3. *-rgt-identity 94.4%

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

      4. unpow2 94.4%

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

      5. distribute-lft-in 94.4%

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

      6. +-commutative 94.4%

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

      7. associate-+l+ 94.4%

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

   8. Simplified 94.4%

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

   9. Taylor expanded in y around 0 46.2%

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

   if 5.0000000000000002e-271 < y < 1.12000000000000008e-6

   1. Initial program 77.6%

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

      1. associate-/l* 79.7%

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

      2. associate-+l+ 79.7%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in y around 0 79.0%

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

      1. +-commutative 79.0%

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

   7. Simplified 79.0%

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

      1. *-un-lft-identity 79.0%

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

      2. associate-*l* 79.0%

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

      3. times-frac 93.6%

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

      4. +-commutative 93.6%

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

      5. +-commutative 93.6%

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

      6. +-commutative 93.6%

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

   9. Applied egg-rr 93.6%

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

       1. associate-*l/ 93.7%

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

       2. *-lft-identity 93.7%

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

       3. *-commutative 93.7%

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

       4. +-commutative 93.7%

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

   11. Simplified 93.7%

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

   if 1.12000000000000008e-6 < y < 4.1599999999999999e99

   1. Initial program 73.8%

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

      1. associate-/l* 85.4%

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

      2. associate-+l+ 85.4%

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

   3. Simplified 85.4%

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

   5. Taylor expanded in x around 0 76.8%

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

      1. +-commutative 76.8%

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

   7. Simplified 76.8%

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

   if 4.1599999999999999e99 < y

   1. Initial program 52.3%

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

      1. associate-/l* 76.4%

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

      2. associate-+l+ 76.4%

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

   3. Simplified 76.4%

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

   5. Taylor expanded in x around 0 76.4%

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

      1. +-commutative 76.4%

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

   7. Simplified 76.4%

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

      1. *-un-lft-identity 76.4%

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

      2. associate-*l* 76.4%

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

      3. times-frac 84.4%

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

      4. +-commutative 84.4%

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

      5. +-commutative 84.4%

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

   9. Applied egg-rr 84.4%

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

       1. associate-*l/ 84.4%

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

       2. *-lft-identity 84.4%

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

       3. associate-/r* 85.5%

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

   11. Simplified 85.5%

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

       1. associate-*r/ 86.7%

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

       2. associate-/l/ 84.4%

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

   13. Applied egg-rr 84.4%

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

   14. Taylor expanded in x around 0 86.5%

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

       1. +-commutative 86.5%

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

   16. Simplified 86.5%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-271}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x + 1}\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(y + x\right) \cdot \left(x + 1\right)}}{y + x}\\ \mathbf{elif}\;y \leq 4.16 \cdot 10^{+99}:\\ \;\;\;\;x \cdot \frac{y}{\left(y + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+85}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + 1} \cdot \frac{\frac{x}{y + x}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.3e+85)
   (* (/ y (+ y x)) (/ 1.0 x))
   (if (<= x -5e-11)
     (* x (/ y (* (+ x (+ y 1.0)) (* (+ y x) (+ y x)))))
     (* (/ y (+ y 1.0)) (/ (/ x (+ y x)) (+ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.3e+85) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -5e-11) {
		tmp = x * (y / ((x + (y + 1.0)) * ((y + x) * (y + x))));
	} else {
		tmp = (y / (y + 1.0)) * ((x / (y + x)) / (y + x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.3d+85)) then
        tmp = (y / (y + x)) * (1.0d0 / x)
    else if (x <= (-5d-11)) then
        tmp = x * (y / ((x + (y + 1.0d0)) * ((y + x) * (y + x))))
    else
        tmp = (y / (y + 1.0d0)) * ((x / (y + x)) / (y + x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.3e+85) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -5e-11) {
		tmp = x * (y / ((x + (y + 1.0)) * ((y + x) * (y + x))));
	} else {
		tmp = (y / (y + 1.0)) * ((x / (y + x)) / (y + x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.3e+85:
		tmp = (y / (y + x)) * (1.0 / x)
	elif x <= -5e-11:
		tmp = x * (y / ((x + (y + 1.0)) * ((y + x) * (y + x))))
	else:
		tmp = (y / (y + 1.0)) * ((x / (y + x)) / (y + x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.3e+85)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / x));
	elseif (x <= -5e-11)
		tmp = Float64(x * Float64(y / Float64(Float64(x + Float64(y + 1.0)) * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(y / Float64(y + 1.0)) * Float64(Float64(x / Float64(y + x)) / Float64(y + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.3e+85)
		tmp = (y / (y + x)) * (1.0 / x);
	elseif (x <= -5e-11)
		tmp = x * (y / ((x + (y + 1.0)) * ((y + x) * (y + x))));
	else
		tmp = (y / (y + 1.0)) * ((x / (y + x)) / (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.3e+85], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5e-11], N[(x * N[(y / N[(N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000005e85

   1. Initial program 61.0%

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

      1. *-commutative 61.0%

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

      2. associate-*l* 61.0%

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

      3. times-frac 90.3%

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

      4. +-commutative 90.3%

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

      5. distribute-lft-in 90.3%

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

      6. *-rgt-identity 90.3%

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

      7. pow2 90.3%

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

   4. Applied egg-rr 90.3%

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

   5. Taylor expanded in x around inf 84.3%

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

   if -1.30000000000000005e85 < x < -5.00000000000000018e-11

   1. Initial program 87.4%

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

      1. associate-/l* 87.6%

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

      2. associate-+l+ 87.6%

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

   3. Simplified 87.6%

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

   if -5.00000000000000018e-11 < x

   1. Initial program 73.2%

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

      1. associate-/l* 83.1%

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

      2. associate-+l+ 83.1%

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

   3. Simplified 83.1%

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

   5. Taylor expanded in x around 0 79.6%

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

      1. +-commutative 79.6%

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

   7. Simplified 79.6%

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

      1. associate-*r/ 66.2%

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

      2. associate-*l* 66.2%

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

      3. associate-/r* 64.2%

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

      4. *-commutative 64.2%

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

      5. +-commutative 64.2%

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

      6. +-commutative 64.2%

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

   9. Applied egg-rr 64.2%

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

       1. associate-/l* 84.2%

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

       2. *-commutative 84.2%

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

       3. times-frac 84.4%

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

   11. Simplified 84.4%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+85}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + 1} \cdot \frac{\frac{x}{y + x}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-5}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x + 1}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-162}:\\ \;\;\;\;x \cdot \frac{y}{\left(y + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4e-5)
   (* (/ y (+ y x)) (/ 1.0 (+ x 1.0)))
   (if (<= x -5.8e-162)
     (* x (/ y (* (+ y 1.0) (* (+ y x) (+ y x)))))
     (/ (/ x y) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4e-5) {
		tmp = (y / (y + x)) * (1.0 / (x + 1.0));
	} else if (x <= -5.8e-162) {
		tmp = x * (y / ((y + 1.0) * ((y + x) * (y + x))));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4d-5)) then
        tmp = (y / (y + x)) * (1.0d0 / (x + 1.0d0))
    else if (x <= (-5.8d-162)) then
        tmp = x * (y / ((y + 1.0d0) * ((y + x) * (y + x))))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4e-5) {
		tmp = (y / (y + x)) * (1.0 / (x + 1.0));
	} else if (x <= -5.8e-162) {
		tmp = x * (y / ((y + 1.0) * ((y + x) * (y + x))));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4e-5:
		tmp = (y / (y + x)) * (1.0 / (x + 1.0))
	elif x <= -5.8e-162:
		tmp = x * (y / ((y + 1.0) * ((y + x) * (y + x))))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4e-5)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / Float64(x + 1.0)));
	elseif (x <= -5.8e-162)
		tmp = Float64(x * Float64(y / Float64(Float64(y + 1.0) * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4e-5)
		tmp = (y / (y + x)) * (1.0 / (x + 1.0));
	elseif (x <= -5.8e-162)
		tmp = x * (y / ((y + 1.0) * ((y + x) * (y + x))));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4e-5], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.8e-162], N[(x * N[(y / N[(N[(y + 1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.00000000000000033e-5

   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 92.3%

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

      4. +-commutative 92.3%

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

      5. distribute-lft-in 92.3%

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

      6. *-rgt-identity 92.3%

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

      7. pow2 92.3%

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

   4. Applied egg-rr 92.3%

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

      1. clear-num 92.2%

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

      2. inv-pow 92.2%

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

      3. +-commutative 92.2%

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

      4. associate-+l+ 92.2%

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

      5. +-commutative 92.2%

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

   6. Applied egg-rr 92.2%

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

      1. unpow-1 92.2%

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

      2. associate-+r+ 92.2%

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

      3. *-rgt-identity 92.2%

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

      4. unpow2 92.2%

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

      5. distribute-lft-in 92.2%

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

      6. +-commutative 92.2%

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

      7. associate-+l+ 92.2%

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

   8. Simplified 92.2%

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

   9. Taylor expanded in y around 0 77.8%

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

   if -4.00000000000000033e-5 < x < -5.8000000000000002e-162

   1. Initial program 82.6%

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

      1. associate-/l* 94.6%

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

      2. associate-+l+ 94.6%

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

   3. Simplified 94.6%

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

   5. Taylor expanded in x around 0 94.6%

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

      1. +-commutative 94.6%

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

   7. Simplified 94.6%

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

   if -5.8000000000000002e-162 < x

   1. Initial program 71.4%

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

      1. associate-/l* 80.8%

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

      2. associate-+l+ 80.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in x around 0 76.6%

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

      1. +-commutative 76.6%

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

   7. Simplified 76.6%

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

      1. *-un-lft-identity 76.6%

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

      2. associate-*l* 76.7%

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

      3. times-frac 89.1%

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

      4. +-commutative 89.1%

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

      5. +-commutative 89.1%

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

   9. Applied egg-rr 89.1%

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

       1. associate-*l/ 89.1%

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

       2. *-lft-identity 89.1%

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

       3. associate-/r* 89.2%

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

   11. Simplified 89.2%

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

   12. Taylor expanded in x around 0 57.4%

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

       1. associate-/r* 58.8%

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

   14. Simplified 58.8%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-5}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x + 1}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-162}:\\ \;\;\;\;x \cdot \frac{y}{\left(y + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{+67}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-161}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.65e+67)
   (* (/ y (+ y x)) (/ 1.0 x))
   (if (<= x -3.5e-161)
     (* x (/ y (* (* (+ y x) (+ y x)) (+ x 1.0))))
     (/ (/ x y) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.65e+67) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -3.5e-161) {
		tmp = x * (y / (((y + x) * (y + x)) * (x + 1.0)));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.65d+67)) then
        tmp = (y / (y + x)) * (1.0d0 / x)
    else if (x <= (-3.5d-161)) then
        tmp = x * (y / (((y + x) * (y + x)) * (x + 1.0d0)))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.65e+67) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -3.5e-161) {
		tmp = x * (y / (((y + x) * (y + x)) * (x + 1.0)));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.65e+67:
		tmp = (y / (y + x)) * (1.0 / x)
	elif x <= -3.5e-161:
		tmp = x * (y / (((y + x) * (y + x)) * (x + 1.0)))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.65e+67)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / x));
	elseif (x <= -3.5e-161)
		tmp = Float64(x * Float64(y / Float64(Float64(Float64(y + x) * Float64(y + x)) * Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.65e+67)
		tmp = (y / (y + x)) * (1.0 / x);
	elseif (x <= -3.5e-161)
		tmp = x * (y / (((y + x) * (y + x)) * (x + 1.0)));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.65e+67], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.5e-161], N[(x * N[(y / N[(N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.65e67

   1. Initial program 63.9%

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

      1. *-commutative 63.9%

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

      2. associate-*l* 63.9%

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

      3. times-frac 91.6%

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

      4. +-commutative 91.6%

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

      5. distribute-lft-in 91.6%

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

      6. *-rgt-identity 91.6%

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

      7. pow2 91.6%

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

   4. Applied egg-rr 91.6%

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

   5. Taylor expanded in x around inf 80.3%

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

   if -2.65e67 < x < -3.5000000000000002e-161

   1. Initial program 84.4%

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

      1. associate-/l* 92.4%

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

      2. associate-+l+ 92.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in y around 0 72.2%

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

      1. +-commutative 72.2%

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

   7. Simplified 72.2%

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

   if -3.5000000000000002e-161 < x

   1. Initial program 71.4%

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

      1. associate-/l* 80.8%

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

      2. associate-+l+ 80.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in x around 0 76.6%

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

      1. +-commutative 76.6%

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

   7. Simplified 76.6%

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

      1. *-un-lft-identity 76.6%

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

      2. associate-*l* 76.7%

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

      3. times-frac 89.1%

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

      4. +-commutative 89.1%

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

      5. +-commutative 89.1%

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

   9. Applied egg-rr 89.1%

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

       1. associate-*l/ 89.1%

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

       2. *-lft-identity 89.1%

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

       3. associate-/r* 89.2%

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

   11. Simplified 89.2%

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

   12. Taylor expanded in x around 0 57.4%

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

       1. associate-/r* 58.8%

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

   14. Simplified 58.8%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{+67}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-161}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.000165:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + 1} \cdot \frac{\frac{x}{y + x}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.000165)
   (* (/ y (+ y x)) (/ 1.0 (+ x 1.0)))
   (* (/ y (+ y 1.0)) (/ (/ x (+ y x)) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.000165) {
		tmp = (y / (y + x)) * (1.0 / (x + 1.0));
	} else {
		tmp = (y / (y + 1.0)) * ((x / (y + x)) / (y + x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.000165d0)) then
        tmp = (y / (y + x)) * (1.0d0 / (x + 1.0d0))
    else
        tmp = (y / (y + 1.0d0)) * ((x / (y + x)) / (y + x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.000165) {
		tmp = (y / (y + x)) * (1.0 / (x + 1.0));
	} else {
		tmp = (y / (y + 1.0)) * ((x / (y + x)) / (y + x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.000165:
		tmp = (y / (y + x)) * (1.0 / (x + 1.0))
	else:
		tmp = (y / (y + 1.0)) * ((x / (y + x)) / (y + x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.000165)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / Float64(x + 1.0)));
	else
		tmp = Float64(Float64(y / Float64(y + 1.0)) * Float64(Float64(x / Float64(y + x)) / Float64(y + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.000165)
		tmp = (y / (y + x)) * (1.0 / (x + 1.0));
	else
		tmp = (y / (y + 1.0)) * ((x / (y + x)) / (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.000165], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.65e-4

   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 92.3%

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

      4. +-commutative 92.3%

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

      5. distribute-lft-in 92.3%

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

      6. *-rgt-identity 92.3%

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

      7. pow2 92.3%

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

   4. Applied egg-rr 92.3%

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

      1. clear-num 92.2%

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

      2. inv-pow 92.2%

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

      3. +-commutative 92.2%

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

      4. associate-+l+ 92.2%

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

      5. +-commutative 92.2%

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

   6. Applied egg-rr 92.2%

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

      1. unpow-1 92.2%

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

      2. associate-+r+ 92.2%

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

      3. *-rgt-identity 92.2%

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

      4. unpow2 92.2%

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

      5. distribute-lft-in 92.2%

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

      6. +-commutative 92.2%

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

      7. associate-+l+ 92.2%

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

   8. Simplified 92.2%

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

   9. Taylor expanded in y around 0 77.8%

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

   if -1.65e-4 < x

   1. Initial program 73.3%

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

      1. associate-/l* 83.2%

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

      2. associate-+l+ 83.2%

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

   3. Simplified 83.2%

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

   5. Taylor expanded in x around 0 79.7%

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

      1. +-commutative 79.7%

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

   7. Simplified 79.7%

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

      1. associate-*r/ 66.4%

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

      2. associate-*l* 66.4%

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

      3. associate-/r* 64.4%

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

      4. *-commutative 64.4%

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

      5. +-commutative 64.4%

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

      6. +-commutative 64.4%

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

   9. Applied egg-rr 64.4%

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

       1. associate-/l* 84.2%

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

       2. *-commutative 84.2%

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

       3. times-frac 84.5%

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

   11. Simplified 84.5%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.000165:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + 1} \cdot \frac{\frac{x}{y + x}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{y + x}\\ \mathbf{if}\;x \leq -0.0027:\\ \;\;\;\;t\_0 \cdot \frac{1}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{t\_0}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y x))))
   (if (<= x -0.0027)
     (* t_0 (/ 1.0 (+ x 1.0)))
     (* x (/ (/ t_0 (+ y 1.0)) (+ y x))))))

C

double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (x <= -0.0027) {
		tmp = t_0 * (1.0 / (x + 1.0));
	} else {
		tmp = x * ((t_0 / (y + 1.0)) / (y + x));
	}
	return tmp;
}

Fortran

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)
    if (x <= (-0.0027d0)) then
        tmp = t_0 * (1.0d0 / (x + 1.0d0))
    else
        tmp = x * ((t_0 / (y + 1.0d0)) / (y + x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (x <= -0.0027) {
		tmp = t_0 * (1.0 / (x + 1.0));
	} else {
		tmp = x * ((t_0 / (y + 1.0)) / (y + x));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (y + x)
	tmp = 0
	if x <= -0.0027:
		tmp = t_0 * (1.0 / (x + 1.0))
	else:
		tmp = x * ((t_0 / (y + 1.0)) / (y + x))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(y + x))
	tmp = 0.0
	if (x <= -0.0027)
		tmp = Float64(t_0 * Float64(1.0 / Float64(x + 1.0)));
	else
		tmp = Float64(x * Float64(Float64(t_0 / Float64(y + 1.0)) / Float64(y + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (y + x);
	tmp = 0.0;
	if (x <= -0.0027)
		tmp = t_0 * (1.0 / (x + 1.0));
	else
		tmp = x * ((t_0 / (y + 1.0)) / (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0027], N[(t$95$0 * N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(t$95$0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0027000000000000001

   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 92.3%

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

      4. +-commutative 92.3%

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

      5. distribute-lft-in 92.3%

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

      6. *-rgt-identity 92.3%

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

      7. pow2 92.3%

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

   4. Applied egg-rr 92.3%

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

      1. clear-num 92.2%

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

      2. inv-pow 92.2%

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

      3. +-commutative 92.2%

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

      4. associate-+l+ 92.2%

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

      5. +-commutative 92.2%

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

   6. Applied egg-rr 92.2%

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

      1. unpow-1 92.2%

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

      2. associate-+r+ 92.2%

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

      3. *-rgt-identity 92.2%

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

      4. unpow2 92.2%

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

      5. distribute-lft-in 92.2%

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

      6. +-commutative 92.2%

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

      7. associate-+l+ 92.2%

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

   8. Simplified 92.2%

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

   9. Taylor expanded in y around 0 77.8%

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

   if -0.0027000000000000001 < x

   1. Initial program 73.3%

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

      1. associate-/l* 83.2%

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

      2. associate-+l+ 83.2%

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

   3. Simplified 83.2%

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

   5. Taylor expanded in x around 0 79.7%

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

      1. +-commutative 79.7%

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

   7. Simplified 79.7%

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

      1. *-un-lft-identity 79.7%

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

      2. associate-*l* 79.7%

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

      3. times-frac 90.9%

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

      4. +-commutative 90.9%

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

      5. +-commutative 90.9%

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

   9. Applied egg-rr 90.9%

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

       1. associate-*l/ 90.9%

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

       2. *-lft-identity 90.9%

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

       3. associate-/r* 91.0%

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

   11. Simplified 91.0%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0027:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\frac{y}{y + x}}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-182}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.05e-182)
   (* (/ y (+ y x)) (/ 1.0 (+ x 1.0)))
   (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.05e-182) {
		tmp = (y / (y + x)) * (1.0 / (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.05d-182)) then
        tmp = (y / (y + x)) * (1.0d0 / (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.05e-182) {
		tmp = (y / (y + x)) * (1.0 / (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.05e-182:
		tmp = (y / (y + x)) * (1.0 / (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.05e-182)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.05e-182)
		tmp = (y / (y + x)) * (1.0 / (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.05e-182], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0500000000000001e-182

   1. Initial program 73.4%

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

      1. *-commutative 73.4%

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

      2. associate-*l* 73.4%

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

      3. times-frac 95.0%

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

      4. +-commutative 95.0%

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

      5. distribute-lft-in 95.0%

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

      6. *-rgt-identity 95.0%

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

      7. pow2 95.0%

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

   4. Applied egg-rr 95.0%

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

      1. clear-num 94.0%

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

      2. inv-pow 94.0%

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

      3. +-commutative 94.0%

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

      4. associate-+l+ 94.0%

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

      5. +-commutative 94.0%

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

   6. Applied egg-rr 94.0%

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

      1. unpow-1 94.0%

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

      2. associate-+r+ 94.0%

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

      3. *-rgt-identity 94.0%

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

      4. unpow2 94.0%

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

      5. distribute-lft-in 94.0%

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

      6. +-commutative 94.0%

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

      7. associate-+l+ 94.0%

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

   8. Simplified 94.0%

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

   9. Taylor expanded in y around 0 65.3%

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

   if -2.0500000000000001e-182 < x

   1. Initial program 72.1%

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

      1. associate-/l* 81.7%

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

      2. associate-+l+ 81.7%

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

   3. Simplified 81.7%

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

   5. Taylor expanded in x around 0 77.4%

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

      1. +-commutative 77.4%

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

   7. Simplified 77.4%

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

      1. *-un-lft-identity 77.4%

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

      2. associate-*l* 77.5%

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

      3. times-frac 88.9%

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

      4. +-commutative 88.9%

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

      5. +-commutative 88.9%

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

   9. Applied egg-rr 88.9%

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

       1. associate-*l/ 88.9%

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

       2. *-lft-identity 88.9%

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

       3. associate-/r* 89.0%

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

   11. Simplified 89.0%

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

   12. Taylor expanded in x around 0 57.8%

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

       1. associate-/r* 59.2%

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

   14. Simplified 59.2%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-182}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-182}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.05e-182) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.05e-182) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.05d-182)) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.05e-182) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.05e-182:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.05e-182)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.05e-182)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.05e-182], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0500000000000001e-182

   1. Initial program 73.4%

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

   3. Taylor expanded in y around 0 64.0%

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

      1. associate-/r* 64.9%

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

      2. +-commutative 64.9%

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

   5. Simplified 64.9%

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

   if -2.0500000000000001e-182 < x

   1. Initial program 72.1%

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

      1. associate-/l* 81.7%

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

      2. associate-+l+ 81.7%

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

   3. Simplified 81.7%

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

   5. Taylor expanded in x around 0 77.4%

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

      1. +-commutative 77.4%

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

   7. Simplified 77.4%

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

      1. *-un-lft-identity 77.4%

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

      2. associate-*l* 77.5%

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

      3. times-frac 88.9%

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

      4. +-commutative 88.9%

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

      5. +-commutative 88.9%

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

   9. Applied egg-rr 88.9%

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

       1. associate-*l/ 88.9%

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

       2. *-lft-identity 88.9%

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

       3. associate-/r* 89.0%

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

   11. Simplified 89.0%

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

   12. Taylor expanded in x around 0 57.8%

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

       1. associate-/r* 59.2%

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

   14. Simplified 59.2%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-182}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-182}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.05e-182) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.05e-182) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.05d-182)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.05e-182) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.05e-182:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.05e-182)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.05e-182)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.05e-182], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0500000000000001e-182

   1. Initial program 73.4%

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

   3. Taylor expanded in y around 0 64.0%

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

   if -2.0500000000000001e-182 < x

   1. Initial program 72.1%

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

      1. associate-/l* 81.7%

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

      2. associate-+l+ 81.7%

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

   3. Simplified 81.7%

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

   5. Taylor expanded in x around 0 77.4%

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

      1. +-commutative 77.4%

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

   7. Simplified 77.4%

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

      1. *-un-lft-identity 77.4%

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

      2. associate-*l* 77.5%

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

      3. times-frac 88.9%

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

      4. +-commutative 88.9%

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

      5. +-commutative 88.9%

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

   9. Applied egg-rr 88.9%

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

       1. associate-*l/ 88.9%

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

       2. *-lft-identity 88.9%

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

       3. associate-/r* 89.0%

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

   11. Simplified 89.0%

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

   12. Taylor expanded in x around 0 57.8%

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

       1. associate-/r* 59.2%

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

   14. Simplified 59.2%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-182}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-182}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.05e-182) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.05e-182) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.05d-182)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.05e-182) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.05e-182:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.05e-182)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.05e-182)
		tmp = y / (x * (x + 1.0));
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.05e-182], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0500000000000001e-182

   1. Initial program 73.4%

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

   3. Taylor expanded in y around 0 64.0%

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

   if -2.0500000000000001e-182 < x

   1. Initial program 72.1%

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

   3. Taylor expanded in x around 0 57.8%

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

      1. +-commutative 57.8%

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

   5. Simplified 57.8%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-182}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 44.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-127}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.4e-127) (/ y x) (/ x (* y (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.4e-127) {
		tmp = y / x;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.4d-127) then
        tmp = y / x
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.4e-127) {
		tmp = y / x;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.4e-127:
		tmp = y / x
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.4e-127)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.4e-127)
		tmp = y / x;
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.4e-127], N[(y / x), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.39999999999999982e-127

   1. Initial program 72.3%

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

      1. associate-/l* 81.7%

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

      2. associate-+l+ 81.7%

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

   3. Simplified 81.7%

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

   5. Taylor expanded in x around 0 76.1%

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

      1. +-commutative 76.1%

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

   7. Simplified 76.1%

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

   8. Taylor expanded in y around 0 35.1%

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

   if 2.39999999999999982e-127 < y

   1. Initial program 73.2%

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

   3. Taylor expanded in x around 0 54.3%

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

      1. +-commutative 54.3%

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

   5. Simplified 54.3%

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

Alternative 13: 32.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-128}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 7.2e-128) (/ y x) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.2e-128) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 7.2d-128) then
        tmp = y / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7.2e-128) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7.2e-128:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.2e-128)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7.2e-128)
		tmp = y / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.2e-128], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.20000000000000049e-128

   1. Initial program 72.3%

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

      1. associate-/l* 81.7%

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

      2. associate-+l+ 81.7%

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

   3. Simplified 81.7%

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

   5. Taylor expanded in x around 0 76.1%

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

      1. +-commutative 76.1%

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

   7. Simplified 76.1%

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

   8. Taylor expanded in y around 0 35.1%

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

   if 7.20000000000000049e-128 < y

   1. Initial program 73.2%

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

      1. associate-/l* 84.8%

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

      2. associate-+l+ 84.8%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in x around 0 54.3%

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

      1. +-commutative 54.3%

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

   7. Simplified 54.3%

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

   8. Taylor expanded in y around 0 26.3%

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

Alternative 14: 26.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ x y))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / y
end function

Java

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

Python

def code(x, y):
	return x / y

Julia

function code(x, y)
	return Float64(x / y)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.6%

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

   1. associate-/l* 82.8%

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

   2. associate-+l+ 82.8%

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

3. Simplified 82.8%

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

5. Taylor expanded in x around 0 49.9%

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

   1. +-commutative 49.9%

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

7. Simplified 49.9%

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

8. Taylor expanded in y around 0 24.8%

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

Alternative 15: 3.6% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ -x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- x))

C

double code(double x, double y) {
	return -x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -x
end function

Java

public static double code(double x, double y) {
	return -x;
}

Python

def code(x, y):
	return -x

Julia

function code(x, y)
	return Float64(-x)
end

MATLAB

function tmp = code(x, y)
	tmp = -x;
end

Wolfram

code[x_, y_] := (-x)

Derivation

1. Initial program 72.6%

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

   1. associate-/l* 82.8%

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

   2. associate-+l+ 82.8%

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

3. Simplified 82.8%

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

5. Taylor expanded in x around 0 49.9%

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

   1. +-commutative 49.9%

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

7. Simplified 49.9%

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

8. Taylor expanded in y around 0 14.3%

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

   1. mul-1-neg 14.3%

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

   2. distribute-lft-neg-out 14.3%

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

   3. *-commutative 14.3%

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

10. Simplified 14.3%

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

11. Taylor expanded in y around inf 3.5%

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

    1. mul-1-neg 3.5%

       \[\leadsto \color{blue}{-x} \]

13. Simplified 3.5%

    \[\leadsto \color{blue}{-x} \]
14. 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 2024111 
(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))))