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

Percentage Accurate: 68.9% → 99.8%

Time: 13.3s

Alternatives: 23

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: 68.9% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.7%

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

   1. +-commutative 66.7%

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

   2. +-commutative 66.7%

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

   3. +-commutative 66.7%

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

   4. *-commutative 66.7%

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

   5. distribute-rgt1-in 50.5%

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

   6. fma-define 66.7%

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

   7. +-commutative 66.7%

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

   8. +-commutative 66.7%

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

   9. cube-unmult 66.7%

      \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

   10. +-commutative 66.7%

       \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

3. Simplified 66.7%

   \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative 66.7%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

   2. fma-define 50.6%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   3. cube-mult 50.5%

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

   4. distribute-rgt1-in 66.7%

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

   5. *-commutative 66.7%

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

   6. associate-*l* 66.7%

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

   7. times-frac 91.6%

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

   8. associate-+r+ 91.6%

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

6. Applied egg-rr 91.6%

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

   1. clear-num 91.5%

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

   2. associate-/r* 99.8%

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

   3. frac-times 99.0%

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

   4. metadata-eval 99.0%

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

   5. times-frac 99.0%

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

   6. *-un-lft-identity 99.0%

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

   7. *-un-lft-identity 99.0%

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

   8. +-commutative 99.0%

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

   9. +-commutative 99.0%

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

8. Applied egg-rr 99.0%

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

   1. add-exp-log 74.7%

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

   2. associate-/r* 74.8%

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

   3. log-div 36.0%

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

   4. div-inv 35.9%

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

   5. +-commutative 35.9%

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

   6. +-commutative 35.9%

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

   7. clear-num 35.6%

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

   8. +-commutative 35.6%

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

   9. +-commutative 35.6%

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

   10. associate-+l+ 35.6%

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

   11. +-commutative 35.6%

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

   12. log1p-undefine 35.6%

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

10. Applied egg-rr 35.6%

    \[\leadsto \color{blue}{e^{\log \left(\frac{x}{x + y} \cdot \frac{y}{x + y}\right) - \mathsf{log1p}\left(x + y\right)}} \]
11. Step-by-step derivation

    1. exp-diff 35.5%

       \[\leadsto \color{blue}{\frac{e^{\log \left(\frac{x}{x + y} \cdot \frac{y}{x + y}\right)}}{e^{\mathsf{log1p}\left(x + y\right)}}} \]

    2. rem-exp-log 57.3%

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

    3. associate-*r/ 57.3%

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

    4. associate-/r/ 57.3%

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

    5. log1p-undefine 57.3%

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

    6. rem-exp-log 99.8%

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

12. Simplified 99.8%

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

13. Taylor expanded in x around 0 99.8%

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

Alternative 2: 72.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{1 + \left(x + y\right)}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-197}:\\ \;\;\;\;x \cdot \frac{y}{\left(1 + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\left(x + \left(1 + y\right)\right) \cdot \frac{x + y}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3e-11)
   (/ (/ y (+ x y)) (+ 1.0 (+ x y)))
   (if (<= x -3e-197)
     (* x (/ y (* (+ 1.0 y) (* (+ x y) (+ x y)))))
     (/ (/ x y) (* (+ x (+ 1.0 y)) (/ (+ x y) y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3e-11) {
		tmp = (y / (x + y)) / (1.0 + (x + y));
	} else if (x <= -3e-197) {
		tmp = x * (y / ((1.0 + y) * ((x + y) * (x + y))));
	} else {
		tmp = (x / y) / ((x + (1.0 + y)) * ((x + y) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3d-11)) then
        tmp = (y / (x + y)) / (1.0d0 + (x + y))
    else if (x <= (-3d-197)) then
        tmp = x * (y / ((1.0d0 + y) * ((x + y) * (x + y))))
    else
        tmp = (x / y) / ((x + (1.0d0 + y)) * ((x + y) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3e-11) {
		tmp = (y / (x + y)) / (1.0 + (x + y));
	} else if (x <= -3e-197) {
		tmp = x * (y / ((1.0 + y) * ((x + y) * (x + y))));
	} else {
		tmp = (x / y) / ((x + (1.0 + y)) * ((x + y) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3e-11:
		tmp = (y / (x + y)) / (1.0 + (x + y))
	elif x <= -3e-197:
		tmp = x * (y / ((1.0 + y) * ((x + y) * (x + y))))
	else:
		tmp = (x / y) / ((x + (1.0 + y)) * ((x + y) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3e-11)
		tmp = Float64(Float64(y / Float64(x + y)) / Float64(1.0 + Float64(x + y)));
	elseif (x <= -3e-197)
		tmp = Float64(x * Float64(y / Float64(Float64(1.0 + y) * Float64(Float64(x + y) * Float64(x + y)))));
	else
		tmp = Float64(Float64(x / y) / Float64(Float64(x + Float64(1.0 + y)) * Float64(Float64(x + y) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3e-11)
		tmp = (y / (x + y)) / (1.0 + (x + y));
	elseif (x <= -3e-197)
		tmp = x * (y / ((1.0 + y) * ((x + y) * (x + y))));
	else
		tmp = (x / y) / ((x + (1.0 + y)) * ((x + y) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3e-11], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3e-197], N[(x * N[(y / N[(N[(1.0 + y), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(N[(x + N[(1.0 + y), $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3e-11

   1. Initial program 61.4%

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

      1. +-commutative 61.4%

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

      2. +-commutative 61.4%

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

      3. +-commutative 61.4%

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

      4. *-commutative 61.4%

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

      5. distribute-rgt1-in 24.2%

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

      6. fma-define 61.4%

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

      7. +-commutative 61.4%

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

      8. +-commutative 61.4%

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

      9. cube-unmult 61.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 61.4%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 61.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 61.4%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 24.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 24.2%

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

      4. distribute-rgt1-in 61.4%

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

      5. *-commutative 61.4%

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

      6. associate-*l* 61.4%

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

      7. times-frac 87.5%

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

      8. associate-+r+ 87.5%

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

   6. Applied egg-rr 87.5%

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

      1. clear-num 87.5%

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

      2. associate-/r* 99.8%

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

      3. frac-times 99.1%

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

      4. metadata-eval 99.1%

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

      5. times-frac 99.1%

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

      6. *-un-lft-identity 99.1%

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

      7. *-un-lft-identity 99.1%

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

      8. +-commutative 99.1%

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

      9. +-commutative 99.1%

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

   8. Applied egg-rr 99.1%

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

      1. add-exp-log 75.5%

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

      2. associate-/r* 76.2%

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

      3. log-div 0.0%

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

      4. div-inv 0.0%

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

      5. +-commutative 0.0%

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

      6. +-commutative 0.0%

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

      7. clear-num 0.0%

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

      8. +-commutative 0.0%

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

      9. +-commutative 0.0%

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

      10. associate-+l+ 0.0%

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

      11. +-commutative 0.0%

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

      12. log1p-undefine 0.0%

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

   10. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{e^{\log \left(\frac{x}{x + y} \cdot \frac{y}{x + y}\right) - \mathsf{log1p}\left(x + y\right)}} \]
   11. Step-by-step derivation

       1. exp-diff 0.0%

          \[\leadsto \color{blue}{\frac{e^{\log \left(\frac{x}{x + y} \cdot \frac{y}{x + y}\right)}}{e^{\mathsf{log1p}\left(x + y\right)}}} \]

       2. rem-exp-log 11.9%

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

       3. associate-*r/ 11.9%

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

       4. associate-/r/ 11.9%

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

       5. log1p-undefine 11.9%

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

       6. rem-exp-log 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in x around inf 74.9%

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

   if -3e-11 < x < -3.00000000000000026e-197

   1. Initial program 79.7%

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

      1. associate-/l* 92.5%

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

      2. associate-+l+ 92.5%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in x around 0 92.5%

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

      1. +-commutative 92.5%

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

   7. Simplified 92.5%

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

   if -3.00000000000000026e-197 < x

   1. Initial program 66.6%

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

      1. +-commutative 66.6%

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

      2. +-commutative 66.6%

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

      3. +-commutative 66.6%

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

      4. *-commutative 66.6%

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

      5. distribute-rgt1-in 60.1%

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

      6. fma-define 66.6%

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

      7. +-commutative 66.6%

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

      8. +-commutative 66.6%

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

      9. cube-unmult 66.6%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 66.6%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 66.6%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 66.6%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 60.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 60.1%

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

      4. distribute-rgt1-in 66.6%

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

      5. *-commutative 66.6%

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

      6. associate-*l* 66.6%

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

      7. times-frac 91.9%

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

      8. associate-+r+ 91.9%

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

   6. Applied egg-rr 91.9%

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

      1. clear-num 91.8%

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

      2. associate-/r* 99.7%

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

      3. frac-times 98.7%

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

      4. metadata-eval 98.7%

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

      5. times-frac 98.7%

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

      6. *-un-lft-identity 98.7%

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

      7. *-un-lft-identity 98.7%

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

      8. +-commutative 98.7%

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

      9. +-commutative 98.7%

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

   8. Applied egg-rr 98.7%

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

   9. Taylor expanded in x around 0 74.5%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{1 + \left(x + y\right)}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-197}:\\ \;\;\;\;x \cdot \frac{y}{\left(1 + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\left(x + \left(1 + y\right)\right) \cdot \frac{x + y}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(x + y\right)\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{t\_0}\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{-197}:\\ \;\;\;\;x \cdot \frac{y}{\left(1 + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (+ x y))))
   (if (<= x -3.2e-11)
     (/ (/ y (+ x y)) t_0)
     (if (<= x -2.15e-197)
       (* x (/ y (* (+ 1.0 y) (* (+ x y) (+ x y)))))
       (/ (/ x (+ x y)) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x + y);
	double tmp;
	if (x <= -3.2e-11) {
		tmp = (y / (x + y)) / t_0;
	} else if (x <= -2.15e-197) {
		tmp = x * (y / ((1.0 + y) * ((x + y) * (x + y))));
	} else {
		tmp = (x / (x + y)) / t_0;
	}
	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 = 1.0d0 + (x + y)
    if (x <= (-3.2d-11)) then
        tmp = (y / (x + y)) / t_0
    else if (x <= (-2.15d-197)) then
        tmp = x * (y / ((1.0d0 + y) * ((x + y) * (x + y))))
    else
        tmp = (x / (x + y)) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (x + y);
	double tmp;
	if (x <= -3.2e-11) {
		tmp = (y / (x + y)) / t_0;
	} else if (x <= -2.15e-197) {
		tmp = x * (y / ((1.0 + y) * ((x + y) * (x + y))));
	} else {
		tmp = (x / (x + y)) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x + y)
	tmp = 0
	if x <= -3.2e-11:
		tmp = (y / (x + y)) / t_0
	elif x <= -2.15e-197:
		tmp = x * (y / ((1.0 + y) * ((x + y) * (x + y))))
	else:
		tmp = (x / (x + y)) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x + y))
	tmp = 0.0
	if (x <= -3.2e-11)
		tmp = Float64(Float64(y / Float64(x + y)) / t_0);
	elseif (x <= -2.15e-197)
		tmp = Float64(x * Float64(y / Float64(Float64(1.0 + y) * Float64(Float64(x + y) * Float64(x + y)))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x + y);
	tmp = 0.0;
	if (x <= -3.2e-11)
		tmp = (y / (x + y)) / t_0;
	elseif (x <= -2.15e-197)
		tmp = x * (y / ((1.0 + y) * ((x + y) * (x + y))));
	else
		tmp = (x / (x + y)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.2e-11], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, -2.15e-197], N[(x * N[(y / N[(N[(1.0 + y), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.19999999999999994e-11

   1. Initial program 61.4%

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

      1. +-commutative 61.4%

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

      2. +-commutative 61.4%

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

      3. +-commutative 61.4%

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

      4. *-commutative 61.4%

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

      5. distribute-rgt1-in 24.2%

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

      6. fma-define 61.4%

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

      7. +-commutative 61.4%

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

      8. +-commutative 61.4%

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

      9. cube-unmult 61.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 61.4%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 61.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 61.4%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 24.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 24.2%

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

      4. distribute-rgt1-in 61.4%

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

      5. *-commutative 61.4%

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

      6. associate-*l* 61.4%

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

      7. times-frac 87.5%

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

      8. associate-+r+ 87.5%

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

   6. Applied egg-rr 87.5%

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

      1. clear-num 87.5%

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

      2. associate-/r* 99.8%

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

      3. frac-times 99.1%

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

      4. metadata-eval 99.1%

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

      5. times-frac 99.1%

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

      6. *-un-lft-identity 99.1%

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

      7. *-un-lft-identity 99.1%

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

      8. +-commutative 99.1%

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

      9. +-commutative 99.1%

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

   8. Applied egg-rr 99.1%

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

      1. add-exp-log 75.5%

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

      2. associate-/r* 76.2%

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

      3. log-div 0.0%

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

      4. div-inv 0.0%

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

      5. +-commutative 0.0%

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

      6. +-commutative 0.0%

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

      7. clear-num 0.0%

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

      8. +-commutative 0.0%

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

      9. +-commutative 0.0%

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

      10. associate-+l+ 0.0%

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

      11. +-commutative 0.0%

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

      12. log1p-undefine 0.0%

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

   10. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{e^{\log \left(\frac{x}{x + y} \cdot \frac{y}{x + y}\right) - \mathsf{log1p}\left(x + y\right)}} \]
   11. Step-by-step derivation

       1. exp-diff 0.0%

          \[\leadsto \color{blue}{\frac{e^{\log \left(\frac{x}{x + y} \cdot \frac{y}{x + y}\right)}}{e^{\mathsf{log1p}\left(x + y\right)}}} \]

       2. rem-exp-log 11.9%

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

       3. associate-*r/ 11.9%

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

       4. associate-/r/ 11.9%

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

       5. log1p-undefine 11.9%

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

       6. rem-exp-log 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in x around inf 74.9%

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

   if -3.19999999999999994e-11 < x < -2.15e-197

   1. Initial program 79.7%

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

      1. associate-/l* 92.5%

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

      2. associate-+l+ 92.5%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in x around 0 92.5%

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

      1. +-commutative 92.5%

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

   7. Simplified 92.5%

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

   if -2.15e-197 < x

   1. Initial program 66.6%

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

      1. +-commutative 66.6%

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

      2. +-commutative 66.6%

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

      3. +-commutative 66.6%

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

      4. *-commutative 66.6%

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

      5. distribute-rgt1-in 60.1%

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

      6. fma-define 66.6%

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

      7. +-commutative 66.6%

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

      8. +-commutative 66.6%

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

      9. cube-unmult 66.6%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 66.6%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 66.6%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 66.6%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 60.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 60.1%

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

      4. distribute-rgt1-in 66.6%

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

      5. *-commutative 66.6%

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

      6. associate-*l* 66.6%

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

      7. times-frac 91.9%

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

      8. associate-+r+ 91.9%

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

   6. Applied egg-rr 91.9%

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

      1. clear-num 91.8%

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

      2. associate-/r* 99.7%

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

      3. frac-times 98.7%

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

      4. metadata-eval 98.7%

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

      5. times-frac 98.7%

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

      6. *-un-lft-identity 98.7%

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

      7. *-un-lft-identity 98.7%

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

      8. +-commutative 98.7%

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

      9. +-commutative 98.7%

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

   8. Applied egg-rr 98.7%

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

      1. add-exp-log 79.6%

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

      2. associate-/r* 79.4%

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

      3. log-div 54.5%

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

      4. div-inv 54.5%

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

      5. +-commutative 54.5%

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

      6. +-commutative 54.5%

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

      7. clear-num 53.8%

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

      8. +-commutative 53.8%

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

      9. +-commutative 53.8%

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

      10. associate-+l+ 53.8%

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

      11. +-commutative 53.8%

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

      12. log1p-undefine 53.8%

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

   10. Applied egg-rr 53.8%

       \[\leadsto \color{blue}{e^{\log \left(\frac{x}{x + y} \cdot \frac{y}{x + y}\right) - \mathsf{log1p}\left(x + y\right)}} \]
   11. Step-by-step derivation

       1. exp-diff 53.7%

          \[\leadsto \color{blue}{\frac{e^{\log \left(\frac{x}{x + y} \cdot \frac{y}{x + y}\right)}}{e^{\mathsf{log1p}\left(x + y\right)}}} \]

       2. rem-exp-log 74.9%

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

       3. associate-*r/ 74.9%

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

       4. associate-/r/ 74.9%

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

       5. log1p-undefine 74.9%

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

       6. rem-exp-log 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in x around 0 64.9%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{1 + \left(x + y\right)}\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{-197}:\\ \;\;\;\;x \cdot \frac{y}{\left(1 + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{1 + \left(x + y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(x + y\right)\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{+33}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{t\_0}\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-150}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (+ x y))))
   (if (<= x -8.5e+33)
     (/ (/ y (+ x y)) t_0)
     (if (<= x -6.5e-150)
       (* x (/ y (* (+ x 1.0) (* (+ x y) (+ x y)))))
       (/ (/ x (+ x y)) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x + y);
	double tmp;
	if (x <= -8.5e+33) {
		tmp = (y / (x + y)) / t_0;
	} else if (x <= -6.5e-150) {
		tmp = x * (y / ((x + 1.0) * ((x + y) * (x + y))));
	} else {
		tmp = (x / (x + y)) / t_0;
	}
	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 = 1.0d0 + (x + y)
    if (x <= (-8.5d+33)) then
        tmp = (y / (x + y)) / t_0
    else if (x <= (-6.5d-150)) then
        tmp = x * (y / ((x + 1.0d0) * ((x + y) * (x + y))))
    else
        tmp = (x / (x + y)) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (x + y);
	double tmp;
	if (x <= -8.5e+33) {
		tmp = (y / (x + y)) / t_0;
	} else if (x <= -6.5e-150) {
		tmp = x * (y / ((x + 1.0) * ((x + y) * (x + y))));
	} else {
		tmp = (x / (x + y)) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x + y)
	tmp = 0
	if x <= -8.5e+33:
		tmp = (y / (x + y)) / t_0
	elif x <= -6.5e-150:
		tmp = x * (y / ((x + 1.0) * ((x + y) * (x + y))))
	else:
		tmp = (x / (x + y)) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x + y))
	tmp = 0.0
	if (x <= -8.5e+33)
		tmp = Float64(Float64(y / Float64(x + y)) / t_0);
	elseif (x <= -6.5e-150)
		tmp = Float64(x * Float64(y / Float64(Float64(x + 1.0) * Float64(Float64(x + y) * Float64(x + y)))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x + y);
	tmp = 0.0;
	if (x <= -8.5e+33)
		tmp = (y / (x + y)) / t_0;
	elseif (x <= -6.5e-150)
		tmp = x * (y / ((x + 1.0) * ((x + y) * (x + y))));
	else
		tmp = (x / (x + y)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.5e+33], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, -6.5e-150], N[(x * N[(y / N[(N[(x + 1.0), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.4999999999999998e33

   1. Initial program 59.2%

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

      1. +-commutative 59.2%

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

      2. +-commutative 59.2%

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

      3. +-commutative 59.2%

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

      4. *-commutative 59.2%

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

      5. distribute-rgt1-in 18.5%

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

      6. fma-define 59.2%

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

      7. +-commutative 59.2%

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

      8. +-commutative 59.2%

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

      9. cube-unmult 59.3%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 59.3%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 59.3%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 59.3%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 18.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 18.5%

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

      4. distribute-rgt1-in 59.2%

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

      5. *-commutative 59.2%

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

      6. associate-*l* 59.2%

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

      7. times-frac 87.8%

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

      8. associate-+r+ 87.8%

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

   6. Applied egg-rr 87.8%

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

      1. clear-num 87.7%

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

      2. associate-/r* 99.9%

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

      3. frac-times 99.1%

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

      4. metadata-eval 99.1%

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

      5. times-frac 99.1%

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

      6. *-un-lft-identity 99.1%

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

      7. *-un-lft-identity 99.1%

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

      8. +-commutative 99.1%

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

      9. +-commutative 99.1%

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

   8. Applied egg-rr 99.1%

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

      1. add-exp-log 79.5%

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

      2. associate-/r* 80.4%

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

      3. log-div 0.0%

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

      4. div-inv 0.0%

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

      5. +-commutative 0.0%

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

      6. +-commutative 0.0%

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

      7. clear-num 0.0%

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

      8. +-commutative 0.0%

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

      9. +-commutative 0.0%

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

      10. associate-+l+ 0.0%

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

      11. +-commutative 0.0%

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

      12. log1p-undefine 0.0%

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

   10. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{e^{\log \left(\frac{x}{x + y} \cdot \frac{y}{x + y}\right) - \mathsf{log1p}\left(x + y\right)}} \]
   11. Step-by-step derivation

       1. exp-diff 0.0%

          \[\leadsto \color{blue}{\frac{e^{\log \left(\frac{x}{x + y} \cdot \frac{y}{x + y}\right)}}{e^{\mathsf{log1p}\left(x + y\right)}}} \]

       2. rem-exp-log 10.2%

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

       3. associate-*r/ 10.2%

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

       4. associate-/r/ 10.2%

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

       5. log1p-undefine 10.2%

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

       6. rem-exp-log 99.9%

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

   12. Simplified 99.9%

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

   13. Taylor expanded in x around inf 77.0%

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

   if -8.4999999999999998e33 < x < -6.49999999999999997e-150

   1. Initial program 84.7%

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

      1. associate-/l* 96.0%

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

      2. associate-+l+ 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in y around 0 71.5%

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

      1. +-commutative 71.5%

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

   7. Simplified 71.5%

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

   if -6.49999999999999997e-150 < x

   1. Initial program 66.8%

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

      1. +-commutative 66.8%

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

      2. +-commutative 66.8%

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

      3. +-commutative 66.8%

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

      4. *-commutative 66.8%

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

      5. distribute-rgt1-in 59.5%

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

      6. fma-define 66.8%

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

      7. +-commutative 66.8%

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

      8. +-commutative 66.8%

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

      9. cube-unmult 66.8%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 66.8%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 66.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 66.8%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 59.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 59.5%

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

      4. distribute-rgt1-in 66.8%

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

      5. *-commutative 66.8%

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

      6. associate-*l* 66.8%

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

      7. times-frac 92.3%

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

      8. associate-+r+ 92.3%

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

   6. Applied egg-rr 92.3%

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

      1. clear-num 92.3%

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

      2. associate-/r* 99.7%

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

      3. frac-times 98.8%

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

      4. metadata-eval 98.8%

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

      5. times-frac 98.8%

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

      6. *-un-lft-identity 98.8%

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

      7. *-un-lft-identity 98.8%

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

      8. +-commutative 98.8%

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

      9. +-commutative 98.8%

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

   8. Applied egg-rr 98.8%

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

      1. add-exp-log 77.8%

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

      2. associate-/r* 77.5%

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

      3. log-div 51.7%

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

      4. div-inv 51.7%

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

      5. +-commutative 51.7%

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

      6. +-commutative 51.7%

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

      7. clear-num 51.1%

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

      8. +-commutative 51.1%

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

      9. +-commutative 51.1%

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

      10. associate-+l+ 51.1%

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

      11. +-commutative 51.1%

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

      12. log1p-undefine 51.1%

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

   10. Applied egg-rr 51.1%

       \[\leadsto \color{blue}{e^{\log \left(\frac{x}{x + y} \cdot \frac{y}{x + y}\right) - \mathsf{log1p}\left(x + y\right)}} \]
   11. Step-by-step derivation

       1. exp-diff 51.0%

          \[\leadsto \color{blue}{\frac{e^{\log \left(\frac{x}{x + y} \cdot \frac{y}{x + y}\right)}}{e^{\mathsf{log1p}\left(x + y\right)}}} \]

       2. rem-exp-log 74.0%

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

       3. associate-*r/ 74.0%

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

       4. associate-/r/ 73.9%

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

       5. log1p-undefine 73.9%

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

       6. rem-exp-log 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in x around 0 65.2%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+33}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{1 + \left(x + y\right)}\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-150}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{1 + \left(x + y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(1 + y\right)\\ \mathbf{if}\;y \leq 7 \cdot 10^{+160}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0 \cdot \frac{x + y}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ 1.0 y))))
   (if (<= y 7e+160)
     (* (/ y (+ x y)) (/ x (* (+ x y) t_0)))
     (/ (/ x y) (* t_0 (/ (+ x y) y))))))

C

double code(double x, double y) {
	double t_0 = x + (1.0 + y);
	double tmp;
	if (y <= 7e+160) {
		tmp = (y / (x + y)) * (x / ((x + y) * t_0));
	} else {
		tmp = (x / y) / (t_0 * ((x + y) / y));
	}
	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 = x + (1.0d0 + y)
    if (y <= 7d+160) then
        tmp = (y / (x + y)) * (x / ((x + y) * t_0))
    else
        tmp = (x / y) / (t_0 * ((x + y) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + (1.0 + y);
	double tmp;
	if (y <= 7e+160) {
		tmp = (y / (x + y)) * (x / ((x + y) * t_0));
	} else {
		tmp = (x / y) / (t_0 * ((x + y) / y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (1.0 + y)
	tmp = 0
	if y <= 7e+160:
		tmp = (y / (x + y)) * (x / ((x + y) * t_0))
	else:
		tmp = (x / y) / (t_0 * ((x + y) / y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(1.0 + y))
	tmp = 0.0
	if (y <= 7e+160)
		tmp = Float64(Float64(y / Float64(x + y)) * Float64(x / Float64(Float64(x + y) * t_0)));
	else
		tmp = Float64(Float64(x / y) / Float64(t_0 * Float64(Float64(x + y) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (1.0 + y);
	tmp = 0.0;
	if (y <= 7e+160)
		tmp = (y / (x + y)) * (x / ((x + y) * t_0));
	else
		tmp = (x / y) / (t_0 * ((x + y) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 7e+160], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(t$95$0 * N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 7.00000000000000051e160

   1. Initial program 69.8%

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

      1. +-commutative 69.8%

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

      2. +-commutative 69.8%

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

      3. +-commutative 69.8%

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

      4. *-commutative 69.8%

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

      5. distribute-rgt1-in 50.7%

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

      6. fma-define 69.8%

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

      7. +-commutative 69.8%

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

      8. +-commutative 69.8%

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

      9. cube-unmult 69.8%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 69.8%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 69.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 69.8%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 50.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 50.7%

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

      4. distribute-rgt1-in 69.8%

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

      5. *-commutative 69.8%

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

      6. associate-*l* 69.8%

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

      7. times-frac 95.1%

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

      8. associate-+r+ 95.1%

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

   6. Applied egg-rr 95.1%

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

   if 7.00000000000000051e160 < y

   1. Initial program 49.7%

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

      1. +-commutative 49.7%

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

      2. +-commutative 49.7%

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

      3. +-commutative 49.7%

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

      4. *-commutative 49.7%

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

      5. distribute-rgt1-in 49.7%

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

      6. fma-define 49.7%

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

      7. +-commutative 49.7%

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

      8. +-commutative 49.7%

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

      9. cube-unmult 49.7%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 49.7%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 49.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 49.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 49.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 49.7%

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

      4. distribute-rgt1-in 49.7%

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

      5. *-commutative 49.7%

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

      6. associate-*l* 49.7%

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

      7. times-frac 72.3%

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

      8. associate-+r+ 72.3%

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

   6. Applied egg-rr 72.3%

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

      1. clear-num 72.3%

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

      2. associate-/r* 99.8%

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

      3. frac-times 99.8%

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

      4. metadata-eval 99.8%

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

      5. times-frac 99.8%

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

      6. *-un-lft-identity 99.8%

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

      7. *-un-lft-identity 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in x around 0 91.7%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{+160}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(1 + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\left(x + \left(1 + y\right)\right) \cdot \frac{x + y}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{x}{\frac{x + y}{y}}}{x + y}\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{t\_0}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{1 + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ x (/ (+ x y) y)) (+ x y))))
   (if (<= x -3.2e-11) (/ t_0 (+ x 1.0)) (/ t_0 (+ 1.0 y)))))

C

double code(double x, double y) {
	double t_0 = (x / ((x + y) / y)) / (x + y);
	double tmp;
	if (x <= -3.2e-11) {
		tmp = t_0 / (x + 1.0);
	} else {
		tmp = t_0 / (1.0 + y);
	}
	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 = (x / ((x + y) / y)) / (x + y)
    if (x <= (-3.2d-11)) then
        tmp = t_0 / (x + 1.0d0)
    else
        tmp = t_0 / (1.0d0 + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x / ((x + y) / y)) / (x + y);
	double tmp;
	if (x <= -3.2e-11) {
		tmp = t_0 / (x + 1.0);
	} else {
		tmp = t_0 / (1.0 + y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / ((x + y) / y)) / (x + y)
	tmp = 0
	if x <= -3.2e-11:
		tmp = t_0 / (x + 1.0)
	else:
		tmp = t_0 / (1.0 + y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / Float64(Float64(x + y) / y)) / Float64(x + y))
	tmp = 0.0
	if (x <= -3.2e-11)
		tmp = Float64(t_0 / Float64(x + 1.0));
	else
		tmp = Float64(t_0 / Float64(1.0 + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / ((x + y) / y)) / (x + y);
	tmp = 0.0;
	if (x <= -3.2e-11)
		tmp = t_0 / (x + 1.0);
	else
		tmp = t_0 / (1.0 + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.2e-11], N[(t$95$0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.19999999999999994e-11

   1. Initial program 61.4%

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

      1. +-commutative 61.4%

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

      2. +-commutative 61.4%

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

      3. +-commutative 61.4%

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

      4. *-commutative 61.4%

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

      5. distribute-rgt1-in 24.2%

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

      6. fma-define 61.4%

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

      7. +-commutative 61.4%

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

      8. +-commutative 61.4%

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

      9. cube-unmult 61.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 61.4%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 61.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 61.4%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 24.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 24.2%

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

      4. distribute-rgt1-in 61.4%

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

      5. *-commutative 61.4%

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

      6. associate-*l* 61.4%

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

      7. times-frac 87.5%

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

      8. associate-+r+ 87.5%

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

   6. Applied egg-rr 87.5%

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

      1. clear-num 87.5%

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

      2. associate-/r* 99.8%

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

      3. frac-times 99.1%

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

      4. metadata-eval 99.1%

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

      5. times-frac 99.1%

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

      6. *-un-lft-identity 99.1%

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

      7. *-un-lft-identity 99.1%

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

      8. +-commutative 99.1%

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

      9. +-commutative 99.1%

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

   8. Applied egg-rr 99.1%

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

      1. add-exp-log 75.5%

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

      2. associate-/r* 76.2%

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

      3. log-div 0.0%

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

      4. div-inv 0.0%

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

      5. +-commutative 0.0%

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

      6. +-commutative 0.0%

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

      7. clear-num 0.0%

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

      8. +-commutative 0.0%

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

      9. +-commutative 0.0%

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

      10. associate-+l+ 0.0%

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

      11. +-commutative 0.0%

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

      12. log1p-undefine 0.0%

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

   10. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{e^{\log \left(\frac{x}{x + y} \cdot \frac{y}{x + y}\right) - \mathsf{log1p}\left(x + y\right)}} \]
   11. Step-by-step derivation

       1. exp-diff 0.0%

          \[\leadsto \color{blue}{\frac{e^{\log \left(\frac{x}{x + y} \cdot \frac{y}{x + y}\right)}}{e^{\mathsf{log1p}\left(x + y\right)}}} \]

       2. rem-exp-log 11.9%

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

       3. associate-*r/ 11.9%

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

       4. associate-/r/ 11.9%

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

       5. log1p-undefine 11.9%

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

       6. rem-exp-log 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in x around inf 74.9%

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

   if -3.19999999999999994e-11 < x

   1. Initial program 68.6%

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

      1. +-commutative 68.6%

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

      2. +-commutative 68.6%

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

      3. +-commutative 68.6%

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

      4. *-commutative 68.6%

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

      5. distribute-rgt1-in 60.5%

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

      6. fma-define 68.6%

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

      7. +-commutative 68.6%

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

      8. +-commutative 68.6%

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

      9. cube-unmult 68.7%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 68.7%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 68.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 68.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 60.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 60.5%

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

      4. distribute-rgt1-in 68.6%

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

      5. *-commutative 68.6%

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

      6. associate-*l* 68.7%

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

      7. times-frac 93.1%

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

      8. associate-+r+ 93.1%

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

   6. Applied egg-rr 93.1%

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

      1. clear-num 93.0%

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

      2. associate-/r* 99.7%

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

      3. frac-times 98.9%

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

      4. metadata-eval 98.9%

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

      5. times-frac 98.9%

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

      6. *-un-lft-identity 98.9%

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

      7. *-un-lft-identity 98.9%

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

      8. +-commutative 98.9%

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

      9. +-commutative 98.9%

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

   8. Applied egg-rr 98.9%

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

      1. add-exp-log 74.5%

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

      2. associate-/r* 74.3%

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

      3. log-div 49.5%

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

      4. div-inv 49.5%

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

      5. +-commutative 49.5%

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

      6. +-commutative 49.5%

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

      7. clear-num 48.9%

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

      8. +-commutative 48.9%

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

      9. +-commutative 48.9%

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

      10. associate-+l+ 48.9%

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

      11. +-commutative 48.9%

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

      12. log1p-undefine 48.9%

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

   10. Applied egg-rr 48.9%

       \[\leadsto \color{blue}{e^{\log \left(\frac{x}{x + y} \cdot \frac{y}{x + y}\right) - \mathsf{log1p}\left(x + y\right)}} \]
   11. Step-by-step derivation

       1. exp-diff 48.8%

          \[\leadsto \color{blue}{\frac{e^{\log \left(\frac{x}{x + y} \cdot \frac{y}{x + y}\right)}}{e^{\mathsf{log1p}\left(x + y\right)}}} \]

       2. rem-exp-log 74.4%

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

       3. associate-*r/ 74.3%

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

       4. associate-/r/ 74.3%

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

       5. log1p-undefine 74.3%

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

       6. rem-exp-log 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in x around 0 83.9%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{\frac{x}{\frac{x + y}{y}}}{x + y}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{\frac{x + y}{y}}}{x + y}}{1 + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{y}\\ \mathbf{if}\;y \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{\frac{x}{t\_0}}{x + y}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\left(x + \left(1 + y\right)\right) \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x y) y)))
   (if (<= y 5.6e-17)
     (/ (/ (/ x t_0) (+ x y)) (+ x 1.0))
     (/ (/ x y) (* (+ x (+ 1.0 y)) t_0)))))

C

double code(double x, double y) {
	double t_0 = (x + y) / y;
	double tmp;
	if (y <= 5.6e-17) {
		tmp = ((x / t_0) / (x + y)) / (x + 1.0);
	} else {
		tmp = (x / y) / ((x + (1.0 + y)) * t_0);
	}
	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 = (x + y) / y
    if (y <= 5.6d-17) then
        tmp = ((x / t_0) / (x + y)) / (x + 1.0d0)
    else
        tmp = (x / y) / ((x + (1.0d0 + y)) * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x + y) / y;
	double tmp;
	if (y <= 5.6e-17) {
		tmp = ((x / t_0) / (x + y)) / (x + 1.0);
	} else {
		tmp = (x / y) / ((x + (1.0 + y)) * t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + y) / y
	tmp = 0
	if y <= 5.6e-17:
		tmp = ((x / t_0) / (x + y)) / (x + 1.0)
	else:
		tmp = (x / y) / ((x + (1.0 + y)) * t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) / y)
	tmp = 0.0
	if (y <= 5.6e-17)
		tmp = Float64(Float64(Float64(x / t_0) / Float64(x + y)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / y) / Float64(Float64(x + Float64(1.0 + y)) * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + y) / y;
	tmp = 0.0;
	if (y <= 5.6e-17)
		tmp = ((x / t_0) / (x + y)) / (x + 1.0);
	else
		tmp = (x / y) / ((x + (1.0 + y)) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, 5.6e-17], N[(N[(N[(x / t$95$0), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(N[(x + N[(1.0 + y), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 5.5999999999999998e-17

   1. Initial program 68.1%

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

      1. +-commutative 68.1%

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

      2. +-commutative 68.1%

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

      3. +-commutative 68.1%

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

      4. *-commutative 68.1%

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

      5. distribute-rgt1-in 46.9%

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

      6. fma-define 68.1%

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

      7. +-commutative 68.1%

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

      8. +-commutative 68.1%

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

      9. cube-unmult 68.1%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 68.1%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 68.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 68.1%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 46.9%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 46.9%

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

      4. distribute-rgt1-in 68.1%

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

      5. *-commutative 68.1%

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

      6. associate-*l* 68.1%

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

      7. times-frac 95.2%

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

      8. associate-+r+ 95.2%

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

   6. Applied egg-rr 95.2%

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

      1. clear-num 95.2%

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

      2. associate-/r* 99.8%

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

      3. frac-times 98.6%

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

      4. metadata-eval 98.6%

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

      5. times-frac 98.6%

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

      6. *-un-lft-identity 98.6%

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

      7. *-un-lft-identity 98.6%

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

      8. +-commutative 98.6%

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

      9. +-commutative 98.6%

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

   8. Applied egg-rr 98.6%

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

      1. add-exp-log 74.0%

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

      2. associate-/r* 74.0%

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

      3. log-div 25.2%

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

      4. div-inv 25.2%

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

      5. +-commutative 25.2%

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

      6. +-commutative 25.2%

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

      7. clear-num 24.7%

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

      8. +-commutative 24.7%

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

      9. +-commutative 24.7%

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

      10. associate-+l+ 24.7%

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

      11. +-commutative 24.7%

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

      12. log1p-undefine 24.7%

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

   10. Applied egg-rr 24.7%

       \[\leadsto \color{blue}{e^{\log \left(\frac{x}{x + y} \cdot \frac{y}{x + y}\right) - \mathsf{log1p}\left(x + y\right)}} \]
   11. Step-by-step derivation

       1. exp-diff 24.7%

          \[\leadsto \color{blue}{\frac{e^{\log \left(\frac{x}{x + y} \cdot \frac{y}{x + y}\right)}}{e^{\mathsf{log1p}\left(x + y\right)}}} \]

       2. rem-exp-log 46.4%

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

       3. associate-*r/ 46.4%

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

       4. associate-/r/ 46.4%

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

       5. log1p-undefine 46.4%

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

       6. rem-exp-log 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in x around inf 79.1%

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

   if 5.5999999999999998e-17 < y

   1. Initial program 63.2%

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

      1. +-commutative 63.2%

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

      2. +-commutative 63.2%

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

      3. +-commutative 63.2%

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

      4. *-commutative 63.2%

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

      5. distribute-rgt1-in 59.3%

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

      6. fma-define 63.2%

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

      7. +-commutative 63.2%

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

      8. +-commutative 63.2%

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

      9. cube-unmult 63.2%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 63.2%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 63.2%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 63.2%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 59.3%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 59.3%

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

      4. distribute-rgt1-in 63.2%

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

      5. *-commutative 63.2%

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

      6. associate-*l* 63.2%

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

      7. times-frac 82.8%

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

      8. associate-+r+ 82.8%

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

   6. Applied egg-rr 82.8%

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

      1. clear-num 82.8%

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

      2. associate-/r* 99.7%

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

      3. frac-times 99.7%

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

      4. metadata-eval 99.7%

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

      5. times-frac 99.7%

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

      6. *-un-lft-identity 99.7%

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

      7. *-un-lft-identity 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0 90.6%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{\frac{x}{\frac{x + y}{y}}}{x + y}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\left(x + \left(1 + y\right)\right) \cdot \frac{x + y}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{y}\\ \mathbf{if}\;y \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{t\_0 \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\left(x + \left(1 + y\right)\right) \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x y) y)))
   (if (<= y 5.6e-17)
     (/ (/ x (+ x y)) (* t_0 (+ x 1.0)))
     (/ (/ x y) (* (+ x (+ 1.0 y)) t_0)))))

C

double code(double x, double y) {
	double t_0 = (x + y) / y;
	double tmp;
	if (y <= 5.6e-17) {
		tmp = (x / (x + y)) / (t_0 * (x + 1.0));
	} else {
		tmp = (x / y) / ((x + (1.0 + y)) * t_0);
	}
	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 = (x + y) / y
    if (y <= 5.6d-17) then
        tmp = (x / (x + y)) / (t_0 * (x + 1.0d0))
    else
        tmp = (x / y) / ((x + (1.0d0 + y)) * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x + y) / y;
	double tmp;
	if (y <= 5.6e-17) {
		tmp = (x / (x + y)) / (t_0 * (x + 1.0));
	} else {
		tmp = (x / y) / ((x + (1.0 + y)) * t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + y) / y
	tmp = 0
	if y <= 5.6e-17:
		tmp = (x / (x + y)) / (t_0 * (x + 1.0))
	else:
		tmp = (x / y) / ((x + (1.0 + y)) * t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) / y)
	tmp = 0.0
	if (y <= 5.6e-17)
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(t_0 * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(Float64(x + Float64(1.0 + y)) * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + y) / y;
	tmp = 0.0;
	if (y <= 5.6e-17)
		tmp = (x / (x + y)) / (t_0 * (x + 1.0));
	else
		tmp = (x / y) / ((x + (1.0 + y)) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, 5.6e-17], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(N[(x + N[(1.0 + y), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 5.5999999999999998e-17

   1. Initial program 68.1%

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

      1. +-commutative 68.1%

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

      2. +-commutative 68.1%

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

      3. +-commutative 68.1%

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

      4. *-commutative 68.1%

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

      5. distribute-rgt1-in 46.9%

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

      6. fma-define 68.1%

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

      7. +-commutative 68.1%

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

      8. +-commutative 68.1%

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

      9. cube-unmult 68.1%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 68.1%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 68.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 68.1%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 46.9%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 46.9%

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

      4. distribute-rgt1-in 68.1%

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

      5. *-commutative 68.1%

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

      6. associate-*l* 68.1%

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

      7. times-frac 95.2%

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

      8. associate-+r+ 95.2%

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

   6. Applied egg-rr 95.2%

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

      1. clear-num 95.2%

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

      2. associate-/r* 99.8%

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

      3. frac-times 98.6%

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

      4. metadata-eval 98.6%

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

      5. times-frac 98.6%

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

      6. *-un-lft-identity 98.6%

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

      7. *-un-lft-identity 98.6%

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

      8. +-commutative 98.6%

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

      9. +-commutative 98.6%

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

   8. Applied egg-rr 98.6%

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

   9. Taylor expanded in y around 0 77.8%

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

   if 5.5999999999999998e-17 < y

   1. Initial program 63.2%

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

      1. +-commutative 63.2%

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

      2. +-commutative 63.2%

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

      3. +-commutative 63.2%

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

      4. *-commutative 63.2%

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

      5. distribute-rgt1-in 59.3%

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

      6. fma-define 63.2%

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

      7. +-commutative 63.2%

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

      8. +-commutative 63.2%

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

      9. cube-unmult 63.2%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 63.2%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 63.2%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 63.2%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 59.3%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 59.3%

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

      4. distribute-rgt1-in 63.2%

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

      5. *-commutative 63.2%

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

      6. associate-*l* 63.2%

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

      7. times-frac 82.8%

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

      8. associate-+r+ 82.8%

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

   6. Applied egg-rr 82.8%

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

      1. clear-num 82.8%

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

      2. associate-/r* 99.7%

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

      3. frac-times 99.7%

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

      4. metadata-eval 99.7%

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

      5. times-frac 99.7%

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

      6. *-un-lft-identity 99.7%

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

      7. *-un-lft-identity 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0 90.6%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\left(x + \left(1 + y\right)\right) \cdot \frac{x + y}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.7%

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

   1. +-commutative 66.7%

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

   2. +-commutative 66.7%

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

   3. +-commutative 66.7%

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

   4. *-commutative 66.7%

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

   5. distribute-rgt1-in 50.5%

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

   6. fma-define 66.7%

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

   7. +-commutative 66.7%

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

   8. +-commutative 66.7%

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

   9. cube-unmult 66.7%

      \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

   10. +-commutative 66.7%

       \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

3. Simplified 66.7%

   \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative 66.7%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

   2. fma-define 50.6%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   3. cube-mult 50.5%

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

   4. distribute-rgt1-in 66.7%

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

   5. *-commutative 66.7%

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

   6. associate-*l* 66.7%

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

   7. times-frac 91.6%

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

   8. associate-+r+ 91.6%

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

6. Applied egg-rr 91.6%

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

   1. clear-num 91.5%

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

   2. associate-/r* 99.8%

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

   3. frac-times 99.0%

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

   4. metadata-eval 99.0%

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

   5. times-frac 99.0%

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

   6. *-un-lft-identity 99.0%

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

   7. *-un-lft-identity 99.0%

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

   8. +-commutative 99.0%

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

   9. +-commutative 99.0%

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

8. Applied egg-rr 99.0%

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

9. Taylor expanded in y around inf 99.0%

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

10. Final simplification 99.0%

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

Alternative 10: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.7%

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

   1. +-commutative 66.7%

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

   2. +-commutative 66.7%

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

   3. +-commutative 66.7%

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

   4. *-commutative 66.7%

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

   5. distribute-rgt1-in 50.5%

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

   6. fma-define 66.7%

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

   7. +-commutative 66.7%

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

   8. +-commutative 66.7%

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

   9. cube-unmult 66.7%

      \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

   10. +-commutative 66.7%

       \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

3. Simplified 66.7%

   \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative 66.7%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

   2. fma-define 50.6%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   3. cube-mult 50.5%

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

   4. distribute-rgt1-in 66.7%

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

   5. *-commutative 66.7%

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

   6. associate-*l* 66.7%

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

   7. times-frac 91.6%

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

   8. associate-+r+ 91.6%

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

6. Applied egg-rr 91.6%

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

   1. clear-num 91.5%

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

   2. associate-/r* 99.8%

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

   3. frac-times 99.0%

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

   4. metadata-eval 99.0%

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

   5. times-frac 99.0%

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

   6. *-un-lft-identity 99.0%

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

   7. *-un-lft-identity 99.0%

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

   8. +-commutative 99.0%

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

   9. +-commutative 99.0%

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

8. Applied egg-rr 99.0%

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

   1. add-exp-log 74.7%

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

   2. associate-/r* 74.8%

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

   3. log-div 36.0%

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

   4. div-inv 35.9%

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

   5. +-commutative 35.9%

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

   6. +-commutative 35.9%

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

   7. clear-num 35.6%

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

   8. +-commutative 35.6%

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

   9. +-commutative 35.6%

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

   10. associate-+l+ 35.6%

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

   11. +-commutative 35.6%

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

   12. log1p-undefine 35.6%

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

10. Applied egg-rr 35.6%

    \[\leadsto \color{blue}{e^{\log \left(\frac{x}{x + y} \cdot \frac{y}{x + y}\right) - \mathsf{log1p}\left(x + y\right)}} \]
11. Step-by-step derivation

    1. exp-diff 35.5%

       \[\leadsto \color{blue}{\frac{e^{\log \left(\frac{x}{x + y} \cdot \frac{y}{x + y}\right)}}{e^{\mathsf{log1p}\left(x + y\right)}}} \]

    2. rem-exp-log 57.3%

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

    3. associate-*r/ 57.3%

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

    4. associate-/r/ 57.3%

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

    5. log1p-undefine 57.3%

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

    6. rem-exp-log 99.8%

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

12. Simplified 99.8%

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

    1. associate-/l/ 98.9%

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

    2. div-inv 98.9%

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

14. Applied egg-rr 98.9%

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

    1. associate-*r/ 98.9%

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

    2. *-rgt-identity 98.9%

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

16. Simplified 98.9%

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

Alternative 11: 61.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{1 + \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.4e-67) (/ (/ y (+ x y)) (+ 1.0 (+ x y))) (/ (/ x y) (+ 1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.4e-67) {
		tmp = (y / (x + y)) / (1.0 + (x + y));
	} else {
		tmp = (x / y) / (1.0 + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.4d-67)) then
        tmp = (y / (x + y)) / (1.0d0 + (x + y))
    else
        tmp = (x / y) / (1.0d0 + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.4e-67) {
		tmp = (y / (x + y)) / (1.0 + (x + y));
	} else {
		tmp = (x / y) / (1.0 + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.4e-67:
		tmp = (y / (x + y)) / (1.0 + (x + y))
	else:
		tmp = (x / y) / (1.0 + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.4e-67)
		tmp = Float64(Float64(y / Float64(x + y)) / Float64(1.0 + Float64(x + y)));
	else
		tmp = Float64(Float64(x / y) / Float64(1.0 + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.4e-67)
		tmp = (y / (x + y)) / (1.0 + (x + y));
	else
		tmp = (x / y) / (1.0 + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.4e-67], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.4000000000000002e-67

   1. Initial program 63.2%

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

      1. +-commutative 63.2%

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

      2. +-commutative 63.2%

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

      3. +-commutative 63.2%

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

      4. *-commutative 63.2%

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

      5. distribute-rgt1-in 27.6%

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

      6. fma-define 63.2%

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

      7. +-commutative 63.2%

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

      8. +-commutative 63.2%

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

      9. cube-unmult 63.2%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 63.2%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 63.2%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 63.2%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 27.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 27.6%

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

      4. distribute-rgt1-in 63.2%

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

      5. *-commutative 63.2%

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

      6. associate-*l* 63.2%

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

      7. times-frac 88.5%

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

      8. associate-+r+ 88.5%

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

   6. Applied egg-rr 88.5%

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

      1. clear-num 88.4%

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

      2. associate-/r* 99.8%

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

      3. frac-times 99.2%

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

      4. metadata-eval 99.2%

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

      5. times-frac 99.2%

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

      6. *-un-lft-identity 99.2%

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

      7. *-un-lft-identity 99.2%

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

      8. +-commutative 99.2%

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

      9. +-commutative 99.2%

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

   8. Applied egg-rr 99.2%

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

      1. add-exp-log 75.7%

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

      2. associate-/r* 76.4%

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

      3. log-div 4.9%

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

      4. div-inv 4.9%

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

      5. +-commutative 4.9%

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

      6. +-commutative 4.9%

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

      7. clear-num 4.9%

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

      8. +-commutative 4.9%

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

      9. +-commutative 4.9%

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

      10. associate-+l+ 4.9%

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

      11. +-commutative 4.9%

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

      12. log1p-undefine 4.9%

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

   10. Applied egg-rr 4.9%

       \[\leadsto \color{blue}{e^{\log \left(\frac{x}{x + y} \cdot \frac{y}{x + y}\right) - \mathsf{log1p}\left(x + y\right)}} \]
   11. Step-by-step derivation

       1. exp-diff 4.9%

          \[\leadsto \color{blue}{\frac{e^{\log \left(\frac{x}{x + y} \cdot \frac{y}{x + y}\right)}}{e^{\mathsf{log1p}\left(x + y\right)}}} \]

       2. rem-exp-log 16.2%

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

       3. associate-*r/ 16.2%

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

       4. associate-/r/ 16.2%

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

       5. log1p-undefine 16.2%

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

       6. rem-exp-log 99.9%

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

   12. Simplified 99.9%

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

   13. Taylor expanded in x around inf 72.0%

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

   if -4.4000000000000002e-67 < x

   1. Initial program 68.1%

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

      1. associate-/l* 80.4%

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

      2. associate-+l+ 80.4%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in x around 0 61.9%

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

      1. associate-/r* 64.6%

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

      2. +-commutative 64.6%

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

   7. Simplified 64.6%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{1 + \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.4e-67) (/ (/ y (+ x 1.0)) (+ x y)) (/ (/ x y) (+ 1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.4e-67) {
		tmp = (y / (x + 1.0)) / (x + y);
	} else {
		tmp = (x / y) / (1.0 + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.4d-67)) then
        tmp = (y / (x + 1.0d0)) / (x + y)
    else
        tmp = (x / y) / (1.0d0 + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.4e-67) {
		tmp = (y / (x + 1.0)) / (x + y);
	} else {
		tmp = (x / y) / (1.0 + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.4e-67:
		tmp = (y / (x + 1.0)) / (x + y)
	else:
		tmp = (x / y) / (1.0 + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.4e-67)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(x + y));
	else
		tmp = Float64(Float64(x / y) / Float64(1.0 + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.4e-67)
		tmp = (y / (x + 1.0)) / (x + y);
	else
		tmp = (x / y) / (1.0 + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.4e-67], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.4000000000000002e-67

   1. Initial program 63.2%

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

      1. +-commutative 63.2%

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

      2. +-commutative 63.2%

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

      3. +-commutative 63.2%

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

      4. *-commutative 63.2%

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

      5. distribute-rgt1-in 27.6%

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

      6. fma-define 63.2%

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

      7. +-commutative 63.2%

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

      8. +-commutative 63.2%

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

      9. cube-unmult 63.2%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 63.2%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 63.2%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 63.2%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 27.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 27.6%

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

      4. distribute-rgt1-in 63.2%

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

      5. *-commutative 63.2%

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

      6. associate-*l* 63.2%

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

      7. times-frac 88.5%

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

      8. associate-+r+ 88.5%

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

   6. Applied egg-rr 88.5%

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

   7. Taylor expanded in y around 0 71.4%

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

      1. +-commutative 71.4%

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

   9. Simplified 71.4%

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

       1. associate-*l/ 71.4%

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

       2. un-div-inv 71.4%

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

   11. Applied egg-rr 71.4%

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

   if -4.4000000000000002e-67 < x

   1. Initial program 68.1%

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

      1. associate-/l* 80.4%

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

      2. associate-+l+ 80.4%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in x around 0 61.9%

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

      1. associate-/r* 64.6%

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

      2. +-commutative 64.6%

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

   7. Simplified 64.6%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.8e-67) (/ (/ y (+ x 1.0)) x) (/ (/ x y) (+ 1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.8e-67) {
		tmp = (y / (x + 1.0)) / x;
	} else {
		tmp = (x / y) / (1.0 + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.8d-67)) then
        tmp = (y / (x + 1.0d0)) / x
    else
        tmp = (x / y) / (1.0d0 + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.8e-67) {
		tmp = (y / (x + 1.0)) / x;
	} else {
		tmp = (x / y) / (1.0 + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.8e-67:
		tmp = (y / (x + 1.0)) / x
	else:
		tmp = (x / y) / (1.0 + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.8e-67)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	else
		tmp = Float64(Float64(x / y) / Float64(1.0 + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.8e-67)
		tmp = (y / (x + 1.0)) / x;
	else
		tmp = (x / y) / (1.0 + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.8e-67], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.8000000000000001e-67

   1. Initial program 63.2%

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

      1. +-commutative 63.2%

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

      2. +-commutative 63.2%

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

      3. +-commutative 63.2%

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

      4. *-commutative 63.2%

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

      5. distribute-rgt1-in 27.6%

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

      6. fma-define 63.2%

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

      7. +-commutative 63.2%

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

      8. +-commutative 63.2%

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

      9. cube-unmult 63.2%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 63.2%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 63.2%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 63.2%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 27.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 27.6%

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

      4. distribute-rgt1-in 63.2%

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

      5. *-commutative 63.2%

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

      6. associate-*l* 63.2%

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

      7. times-frac 88.5%

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

      8. associate-+r+ 88.5%

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

   6. Applied egg-rr 88.5%

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

   7. Taylor expanded in y around 0 71.4%

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

      1. +-commutative 71.4%

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

   9. Simplified 71.4%

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

       1. associate-*l/ 71.4%

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

       2. un-div-inv 71.4%

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

   11. Applied egg-rr 71.4%

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

   12. Taylor expanded in x around inf 70.9%

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

   if -2.8000000000000001e-67 < x

   1. Initial program 68.1%

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

      1. associate-/l* 80.4%

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

      2. associate-+l+ 80.4%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in x around 0 61.9%

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

      1. associate-/r* 64.6%

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

      2. +-commutative 64.6%

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

   7. Simplified 64.6%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.75e-67) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ 1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.75e-67) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (1.0 + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.75d-67)) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / y) / (1.0d0 + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.75e-67) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (1.0 + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.75e-67:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / y) / (1.0 + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.75e-67)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / y) / Float64(1.0 + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.75e-67)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / y) / (1.0 + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.75e-67], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7500000000000001e-67

   1. Initial program 63.2%

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

      1. +-commutative 63.2%

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

      2. +-commutative 63.2%

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

      3. +-commutative 63.2%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 63.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 27.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 63.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 63.2%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 63.2%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 63.2%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 63.2%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 63.2%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 63.2%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 27.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 27.6%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 63.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 63.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 63.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 88.5%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 88.5%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 88.5%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   7. Step-by-step derivation

      1. clear-num 88.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + y}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)} \]

      2. associate-/r* 99.8%

         \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + \left(y + 1\right)}} \]

      3. frac-times 99.2%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(x + \left(y + 1\right)\right)}} \]

      4. metadata-eval 99.2%

         \[\leadsto \frac{\color{blue}{\frac{1}{1}} \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(x + \left(y + 1\right)\right)} \]

      5. times-frac 99.2%

         \[\leadsto \frac{\color{blue}{\frac{1 \cdot x}{1 \cdot \left(x + y\right)}}}{\frac{x + y}{y} \cdot \left(x + \left(y + 1\right)\right)} \]

      6. *-un-lft-identity 99.2%

         \[\leadsto \frac{\frac{\color{blue}{x}}{1 \cdot \left(x + y\right)}}{\frac{x + y}{y} \cdot \left(x + \left(y + 1\right)\right)} \]

      7. *-un-lft-identity 99.2%

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{\frac{x + y}{y} \cdot \left(x + \left(y + 1\right)\right)} \]

      8. +-commutative 99.2%

         \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{\frac{x + y}{y} \cdot \left(x + \left(y + 1\right)\right)} \]

      9. +-commutative 99.2%

         \[\leadsto \frac{\frac{x}{y + x}}{\frac{\color{blue}{y + x}}{y} \cdot \left(x + \left(y + 1\right)\right)} \]

   8. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{y + x}{y} \cdot \left(x + \left(y + 1\right)\right)}} \]

   9. Taylor expanded in y around 0 67.3%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   10. Step-by-step derivation

       1. associate-/r* 70.9%

          \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

       2. +-commutative 70.9%

          \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   11. Simplified 70.9%

       \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if -2.7500000000000001e-67 < x

   1. Initial program 68.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 80.4%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 80.4%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 80.4%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 61.9%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 64.6%

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

      2. +-commutative 64.6%

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y + 1}} \]

   7. Simplified 64.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 60.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-67}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.4e-67) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ 1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.4e-67) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (1.0 + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.4d-67)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (1.0d0 + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.4e-67) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (1.0 + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.4e-67:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (1.0 + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.4e-67)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(1.0 + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.4e-67)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (1.0 + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.4e-67], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.4000000000000002e-67

   1. Initial program 63.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 84.1%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 84.1%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 84.1%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 67.3%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 67.3%

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   7. Simplified 67.3%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]

   if -4.4000000000000002e-67 < x

   1. Initial program 68.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 80.4%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 80.4%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 80.4%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 61.9%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 64.6%

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

      2. +-commutative 64.6%

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y + 1}} \]

   7. Simplified 64.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-67}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 60.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-67}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(1 + y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.6e-67) (/ y (* x (+ x 1.0))) (/ x (* y (+ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.6e-67) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (1.0 + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.6d-67)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = x / (y * (1.0d0 + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.6e-67) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (1.0 + y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.6e-67:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = x / (y * (1.0 + y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.6e-67)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(y * Float64(1.0 + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.6e-67)
		tmp = y / (x * (x + 1.0));
	else
		tmp = x / (y * (1.0 + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.6e-67], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.59999999999999999e-67

   1. Initial program 63.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 84.1%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 84.1%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 84.1%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 67.3%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 67.3%

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   7. Simplified 67.3%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]

   if -3.59999999999999999e-67 < x

   1. Initial program 68.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 80.4%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 80.4%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 80.4%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 61.9%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 31.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.75:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 0.75) (- (/ x y) x) (/ x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.75) {
		tmp = (x / y) - x;
	} else {
		tmp = x / (y * 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 <= 0.75d0) then
        tmp = (x / y) - x
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.75) {
		tmp = (x / y) - x;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.75:
		tmp = (x / y) - x
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.75)
		tmp = Float64(Float64(x / y) - x);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.75)
		tmp = (x / y) - x;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.75], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.75

   1. Initial program 69.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 83.6%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 83.6%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 83.6%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 46.6%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 49.3%

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

      2. +-commutative 49.3%

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y + 1}} \]

   7. Simplified 49.3%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]

   8. Taylor expanded in y around 0 20.4%

      \[\leadsto \color{blue}{\frac{x + -1 \cdot \left(x \cdot y\right)}{y}} \]
   9. Step-by-step derivation

      1. neg-mul-1 20.4%

         \[\leadsto \frac{x + \color{blue}{\left(-x \cdot y\right)}}{y} \]

      2. unsub-neg 20.4%

         \[\leadsto \frac{\color{blue}{x - x \cdot y}}{y} \]

   10. Simplified 20.4%

       \[\leadsto \color{blue}{\frac{x - x \cdot y}{y}} \]

   11. Taylor expanded in x around 0 20.4%

       \[\leadsto \color{blue}{\frac{x \cdot \left(1 - y\right)}{y}} \]
   12. Step-by-step derivation

       1. sub-neg 20.4%

          \[\leadsto \frac{x \cdot \color{blue}{\left(1 + \left(-y\right)\right)}}{y} \]

       2. distribute-lft-in 20.4%

          \[\leadsto \frac{\color{blue}{x \cdot 1 + x \cdot \left(-y\right)}}{y} \]

       3. distribute-rgt-neg-in 20.4%

          \[\leadsto \frac{x \cdot 1 + \color{blue}{\left(-x \cdot y\right)}}{y} \]

       4. unsub-neg 20.4%

          \[\leadsto \frac{\color{blue}{x \cdot 1 - x \cdot y}}{y} \]

       5. *-rgt-identity 20.4%

          \[\leadsto \frac{\color{blue}{x} - x \cdot y}{y} \]

       6. div-sub 20.4%

          \[\leadsto \color{blue}{\frac{x}{y} - \frac{x \cdot y}{y}} \]

       7. associate-/l* 20.5%

          \[\leadsto \frac{x}{y} - \color{blue}{x \cdot \frac{y}{y}} \]

       8. *-inverses 20.5%

          \[\leadsto \frac{x}{y} - x \cdot \color{blue}{1} \]

       9. *-rgt-identity 20.5%

          \[\leadsto \frac{x}{y} - \color{blue}{x} \]

   13. Simplified 20.5%

       \[\leadsto \color{blue}{\frac{x}{y} - x} \]

   if 0.75 < y

   1. Initial program 59.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 75.7%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 75.7%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 75.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 67.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   6. Taylor expanded in y around inf 66.1%

      \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 26.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.2e-11) (/ 1.0 (+ x 1.0)) (/ 1.0 (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.2e-11) {
		tmp = 1.0 / (x + 1.0);
	} else {
		tmp = 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 (x <= (-3.2d-11)) then
        tmp = 1.0d0 / (x + 1.0d0)
    else
        tmp = 1.0d0 / (y / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.2e-11) {
		tmp = 1.0 / (x + 1.0);
	} else {
		tmp = 1.0 / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.2e-11:
		tmp = 1.0 / (x + 1.0)
	else:
		tmp = 1.0 / (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.2e-11)
		tmp = Float64(1.0 / Float64(x + 1.0));
	else
		tmp = Float64(1.0 / Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.2e-11)
		tmp = 1.0 / (x + 1.0);
	else
		tmp = 1.0 / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.2e-11], N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.19999999999999994e-11

   1. Initial program 61.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 61.4%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 61.4%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 61.4%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 61.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 24.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 61.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 61.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 61.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 61.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 61.4%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 61.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 61.4%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 24.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 24.2%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 61.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 61.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 61.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 87.5%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 87.5%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 87.5%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

   7. Taylor expanded in y around 0 74.4%

      \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{1 + x}} \]
   8. Step-by-step derivation

      1. +-commutative 74.4%

         \[\leadsto \frac{y}{x + y} \cdot \frac{1}{\color{blue}{x + 1}} \]

   9. Simplified 74.4%

      \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{x + 1}} \]

   10. Taylor expanded in y around inf 6.4%

       \[\leadsto \color{blue}{\frac{1}{1 + x}} \]
   11. Step-by-step derivation

       1. +-commutative 6.4%

          \[\leadsto \frac{1}{\color{blue}{x + 1}} \]

   12. Simplified 6.4%

       \[\leadsto \color{blue}{\frac{1}{x + 1}} \]

   if -3.19999999999999994e-11 < x

   1. Initial program 68.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 81.0%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 81.0%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 81.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 62.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   6. Taylor expanded in y around 0 36.6%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
   7. Step-by-step derivation

      1. clear-num 36.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]

      2. inv-pow 36.5%

         \[\leadsto \color{blue}{{\left(\frac{y}{x}\right)}^{-1}} \]

   8. Applied egg-rr 36.5%

      \[\leadsto \color{blue}{{\left(\frac{y}{x}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. unpow-1 36.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]

   10. Simplified 36.5%

       \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 26.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.2e-11) (/ 1.0 (+ x 1.0)) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.2e-11) {
		tmp = 1.0 / (x + 1.0);
	} 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 (x <= (-3.2d-11)) then
        tmp = 1.0d0 / (x + 1.0d0)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.2e-11) {
		tmp = 1.0 / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.2e-11:
		tmp = 1.0 / (x + 1.0)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.2e-11)
		tmp = Float64(1.0 / Float64(x + 1.0));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.2e-11)
		tmp = 1.0 / (x + 1.0);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.2e-11], N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.19999999999999994e-11

   1. Initial program 61.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 61.4%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 61.4%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 61.4%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 61.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 24.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 61.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 61.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 61.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 61.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 61.4%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 61.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 61.4%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 24.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 24.2%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 61.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 61.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 61.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 87.5%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 87.5%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 87.5%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

   7. Taylor expanded in y around 0 74.4%

      \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{1 + x}} \]
   8. Step-by-step derivation

      1. +-commutative 74.4%

         \[\leadsto \frac{y}{x + y} \cdot \frac{1}{\color{blue}{x + 1}} \]

   9. Simplified 74.4%

      \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{x + 1}} \]

   10. Taylor expanded in y around inf 6.4%

       \[\leadsto \color{blue}{\frac{1}{1 + x}} \]
   11. Step-by-step derivation

       1. +-commutative 6.4%

          \[\leadsto \frac{1}{\color{blue}{x + 1}} \]

   12. Simplified 6.4%

       \[\leadsto \color{blue}{\frac{1}{x + 1}} \]

   if -3.19999999999999994e-11 < x

   1. Initial program 68.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 81.0%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 81.0%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 81.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 62.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   6. Taylor expanded in y around 0 36.6%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 26.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3800:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -3800.0) (/ 1.0 x) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3800.0) {
		tmp = 1.0 / 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 (x <= (-3800.0d0)) then
        tmp = 1.0d0 / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3800.0) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3800.0:
		tmp = 1.0 / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3800.0)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3800.0)
		tmp = 1.0 / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3800.0], N[(1.0 / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3800

   1. Initial program 61.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 61.4%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 61.4%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 61.4%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 61.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 24.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 61.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 61.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 61.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 61.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 61.4%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 61.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 61.4%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 24.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 24.2%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 61.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 61.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 61.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 87.5%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 87.5%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 87.5%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   7. Step-by-step derivation

      1. clear-num 87.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + y}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)} \]

      2. associate-/r* 99.8%

         \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + \left(y + 1\right)}} \]

      3. frac-times 99.1%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(x + \left(y + 1\right)\right)}} \]

      4. metadata-eval 99.1%

         \[\leadsto \frac{\color{blue}{\frac{1}{1}} \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(x + \left(y + 1\right)\right)} \]

      5. times-frac 99.1%

         \[\leadsto \frac{\color{blue}{\frac{1 \cdot x}{1 \cdot \left(x + y\right)}}}{\frac{x + y}{y} \cdot \left(x + \left(y + 1\right)\right)} \]

      6. *-un-lft-identity 99.1%

         \[\leadsto \frac{\frac{\color{blue}{x}}{1 \cdot \left(x + y\right)}}{\frac{x + y}{y} \cdot \left(x + \left(y + 1\right)\right)} \]

      7. *-un-lft-identity 99.1%

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{\frac{x + y}{y} \cdot \left(x + \left(y + 1\right)\right)} \]

      8. +-commutative 99.1%

         \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{\frac{x + y}{y} \cdot \left(x + \left(y + 1\right)\right)} \]

      9. +-commutative 99.1%

         \[\leadsto \frac{\frac{x}{y + x}}{\frac{\color{blue}{y + x}}{y} \cdot \left(x + \left(y + 1\right)\right)} \]

   8. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{y + x}{y} \cdot \left(x + \left(y + 1\right)\right)}} \]

   9. Taylor expanded in x around 0 55.6%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{\frac{y + x}{y} \cdot \left(x + \left(y + 1\right)\right)} \]

   10. Taylor expanded in x around inf 6.4%

       \[\leadsto \color{blue}{\frac{1}{x}} \]

   if -3800 < x

   1. Initial program 68.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 81.0%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 81.0%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 81.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 62.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   6. Taylor expanded in y around 0 36.6%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 47.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y \cdot \left(1 + y\right)} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ x (* y (+ 1.0 y))))

C

double code(double x, double y) {
	return x / (y * (1.0 + y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / (y * (1.0d0 + y))
end function

Java

public static double code(double x, double y) {
	return x / (y * (1.0 + y));
}

Python

def code(x, y):
	return x / (y * (1.0 + y))

Julia

function code(x, y)
	return Float64(x / Float64(y * Float64(1.0 + y)))
end

MATLAB

function tmp = code(x, y)
	tmp = x / (y * (1.0 + y));
end

Wolfram

code[x_, y_] := N[(x / N[(y * N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.7%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l* 81.5%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   2. associate-+l+ 81.5%

      \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

3. Simplified 81.5%

   \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 52.1%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
6. Add Preprocessing

Alternative 22: 4.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 x))

C

double code(double x, double y) {
	return 1.0 / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / x
end function

Java

public static double code(double x, double y) {
	return 1.0 / x;
}

Python

def code(x, y):
	return 1.0 / x

Julia

function code(x, y)
	return Float64(1.0 / x)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / x;
end

Wolfram

code[x_, y_] := N[(1.0 / x), $MachinePrecision]

Derivation

1. Initial program 66.7%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. +-commutative 66.7%

      \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. +-commutative 66.7%

      \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   3. +-commutative 66.7%

      \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

   4. *-commutative 66.7%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   5. distribute-rgt1-in 50.5%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   6. fma-define 66.7%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

   7. +-commutative 66.7%

      \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   8. +-commutative 66.7%

      \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   9. cube-unmult 66.7%

      \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

   10. +-commutative 66.7%

       \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

3. Simplified 66.7%

   \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative 66.7%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

   2. fma-define 50.6%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   3. cube-mult 50.5%

      \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   4. distribute-rgt1-in 66.7%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   5. *-commutative 66.7%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   6. associate-*l* 66.7%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

   7. times-frac 91.6%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   8. associate-+r+ 91.6%

      \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

6. Applied egg-rr 91.6%

   \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
7. Step-by-step derivation

   1. clear-num 91.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{x + y}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)} \]

   2. associate-/r* 99.8%

      \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + \left(y + 1\right)}} \]

   3. frac-times 99.0%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(x + \left(y + 1\right)\right)}} \]

   4. metadata-eval 99.0%

      \[\leadsto \frac{\color{blue}{\frac{1}{1}} \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(x + \left(y + 1\right)\right)} \]

   5. times-frac 99.0%

      \[\leadsto \frac{\color{blue}{\frac{1 \cdot x}{1 \cdot \left(x + y\right)}}}{\frac{x + y}{y} \cdot \left(x + \left(y + 1\right)\right)} \]

   6. *-un-lft-identity 99.0%

      \[\leadsto \frac{\frac{\color{blue}{x}}{1 \cdot \left(x + y\right)}}{\frac{x + y}{y} \cdot \left(x + \left(y + 1\right)\right)} \]

   7. *-un-lft-identity 99.0%

      \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{\frac{x + y}{y} \cdot \left(x + \left(y + 1\right)\right)} \]

   8. +-commutative 99.0%

      \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{\frac{x + y}{y} \cdot \left(x + \left(y + 1\right)\right)} \]

   9. +-commutative 99.0%

      \[\leadsto \frac{\frac{x}{y + x}}{\frac{\color{blue}{y + x}}{y} \cdot \left(x + \left(y + 1\right)\right)} \]

8. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{y + x}{y} \cdot \left(x + \left(y + 1\right)\right)}} \]

9. Taylor expanded in x around 0 68.8%

   \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{\frac{y + x}{y} \cdot \left(x + \left(y + 1\right)\right)} \]

10. Taylor expanded in x around inf 4.6%

    \[\leadsto \color{blue}{\frac{1}{x}} \]
11. Add Preprocessing

Alternative 23: 3.5% 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 66.7%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l* 81.5%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   2. associate-+l+ 81.5%

      \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

3. Simplified 81.5%

   \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 52.1%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
6. Step-by-step derivation

   1. associate-/r* 54.5%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

   2. +-commutative 54.5%

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y + 1}} \]

7. Simplified 54.5%

   \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]

8. Taylor expanded in y around 0 15.8%

   \[\leadsto \color{blue}{\frac{x + -1 \cdot \left(x \cdot y\right)}{y}} \]
9. Step-by-step derivation

   1. neg-mul-1 15.8%

      \[\leadsto \frac{x + \color{blue}{\left(-x \cdot y\right)}}{y} \]

   2. unsub-neg 15.8%

      \[\leadsto \frac{\color{blue}{x - x \cdot y}}{y} \]

10. Simplified 15.8%

    \[\leadsto \color{blue}{\frac{x - x \cdot y}{y}} \]

11. Taylor expanded in y around inf 3.4%

    \[\leadsto \color{blue}{-1 \cdot x} \]
12. Step-by-step derivation

    1. neg-mul-1 3.4%

       \[\leadsto \color{blue}{-x} \]

13. Simplified 3.4%

    \[\leadsto \color{blue}{-x} \]
14. Add Preprocessing

Developer Target 1: 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 2024144 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (/ (/ x (+ (+ y 1) x)) (+ y x)) (/ 1 (/ y (+ y x)))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))