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

Percentage Accurate: 69.9% → 99.8%

Time: 13.9s

Alternatives: 16

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: 69.9% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.2%

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

   1. +-commutative 61.2%

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

   2. +-commutative 61.2%

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

   3. +-commutative 61.2%

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

   4. *-commutative 61.2%

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

   5. distribute-rgt1-in 47.5%

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

   6. fma-define 61.2%

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

   7. +-commutative 61.2%

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

   8. +-commutative 61.2%

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

   9. cube-unmult 61.2%

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

   10. +-commutative 61.2%

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

3. Simplified 61.2%

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

   1. *-commutative 61.2%

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

   2. fma-define 47.5%

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

   3. cube-mult 47.5%

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

   4. distribute-rgt1-in 61.2%

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

   5. *-commutative 61.2%

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

   6. associate-*l* 61.2%

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

   7. times-frac 92.6%

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

   8. associate-+r+ 92.6%

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

6. Applied egg-rr 92.6%

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

   1. div-inv 92.5%

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

   2. +-commutative 92.5%

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

   3. +-commutative 92.5%

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

   4. associate-+l+ 92.5%

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

   5. +-commutative 92.5%

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

8. Applied egg-rr 92.5%

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

   1. associate-*r/ 92.6%

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

   2. *-rgt-identity 92.6%

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

   3. associate-/r* 99.8%

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

10. Simplified 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 64.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+85}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{y + \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-159}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.6e+85)
   (* (/ y (+ y x)) (/ 1.0 (+ y (+ x 1.0))))
   (if (<= x -1.1e-159)
     (* x (/ y (* (+ x 1.0) (* (+ y x) (+ y x)))))
     (/ (/ x y) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.6e+85) {
		tmp = (y / (y + x)) * (1.0 / (y + (x + 1.0)));
	} else if (x <= -1.1e-159) {
		tmp = x * (y / ((x + 1.0) * ((y + x) * (y + x))));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.6d+85)) then
        tmp = (y / (y + x)) * (1.0d0 / (y + (x + 1.0d0)))
    else if (x <= (-1.1d-159)) then
        tmp = x * (y / ((x + 1.0d0) * ((y + x) * (y + x))))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.6e+85) {
		tmp = (y / (y + x)) * (1.0 / (y + (x + 1.0)));
	} else if (x <= -1.1e-159) {
		tmp = x * (y / ((x + 1.0) * ((y + x) * (y + x))));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.6e+85:
		tmp = (y / (y + x)) * (1.0 / (y + (x + 1.0)))
	elif x <= -1.1e-159:
		tmp = x * (y / ((x + 1.0) * ((y + x) * (y + x))))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.6e+85)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / Float64(y + Float64(x + 1.0))));
	elseif (x <= -1.1e-159)
		tmp = Float64(x * Float64(y / Float64(Float64(x + 1.0) * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.6e+85)
		tmp = (y / (y + x)) * (1.0 / (y + (x + 1.0)));
	elseif (x <= -1.1e-159)
		tmp = x * (y / ((x + 1.0) * ((y + x) * (y + x))));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.6e+85], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.1e-159], N[(x * N[(y / N[(N[(x + 1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.5999999999999998e85

   1. Initial program 55.2%

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

      1. +-commutative 55.2%

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

      2. +-commutative 55.2%

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

      3. +-commutative 55.2%

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

      4. *-commutative 55.2%

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

      5. distribute-rgt1-in 22.9%

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

      6. fma-define 55.2%

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

      7. +-commutative 55.2%

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

      8. +-commutative 55.2%

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

      9. cube-unmult 55.2%

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

      10. +-commutative 55.2%

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

   3. Simplified 55.2%

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

      1. *-commutative 55.2%

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

      2. fma-define 22.9%

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

      3. cube-mult 22.9%

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

      4. distribute-rgt1-in 55.2%

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

      5. *-commutative 55.2%

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

      6. associate-*l* 55.2%

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

      7. times-frac 88.2%

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

      8. associate-+r+ 88.2%

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

   6. Applied egg-rr 88.2%

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

      1. div-inv 88.2%

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

      2. +-commutative 88.2%

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

      3. +-commutative 88.2%

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

      4. associate-+l+ 88.2%

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

      5. +-commutative 88.2%

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

   8. Applied egg-rr 88.2%

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

      1. associate-*r/ 88.2%

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

      2. *-rgt-identity 88.2%

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

      3. associate-/r* 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around inf 80.6%

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

   if -5.5999999999999998e85 < x < -1.1e-159

   1. Initial program 81.8%

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

      1. associate-/l* 90.5%

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

      2. associate-+l+ 90.5%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in y around 0 74.5%

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

      1. +-commutative 61.0%

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

   7. Simplified 74.5%

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

   if -1.1e-159 < x

   1. Initial program 56.3%

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

      1. associate-/l* 75.8%

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

      2. associate-+l+ 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in x around 0 58.3%

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

      1. associate-/r* 61.1%

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

      2. +-commutative 61.1%

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

   7. Simplified 61.1%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+85}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{y + \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-159}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{+167}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.8e+167)
   (* (/ y (+ y x)) (/ x (* (+ y x) (+ x (+ y 1.0)))))
   (/ (/ x (+ y x)) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.8e+167) {
		tmp = (y / (y + x)) * (x / ((y + x) * (x + (y + 1.0))));
	} else {
		tmp = (x / (y + x)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.8d+167) then
        tmp = (y / (y + x)) * (x / ((y + x) * (x + (y + 1.0d0))))
    else
        tmp = (x / (y + x)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.8e+167) {
		tmp = (y / (y + x)) * (x / ((y + x) * (x + (y + 1.0))));
	} else {
		tmp = (x / (y + x)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.8e+167:
		tmp = (y / (y + x)) * (x / ((y + x) * (x + (y + 1.0))))
	else:
		tmp = (x / (y + x)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.8e+167)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(Float64(y + x) * Float64(x + Float64(y + 1.0)))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.8e+167)
		tmp = (y / (y + x)) * (x / ((y + x) * (x + (y + 1.0))));
	else
		tmp = (x / (y + x)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.8e+167], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(y + x), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.80000000000000012e167

   1. Initial program 63.3%

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

      1. +-commutative 63.3%

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

      2. +-commutative 63.3%

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

      3. +-commutative 63.3%

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

      4. *-commutative 63.3%

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

      5. distribute-rgt1-in 47.6%

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

      6. fma-define 63.4%

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

      7. +-commutative 63.4%

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

      8. +-commutative 63.4%

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

      9. cube-unmult 63.4%

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

      10. +-commutative 63.4%

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

   3. Simplified 63.4%

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

      1. *-commutative 63.4%

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

      2. fma-define 47.6%

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

      3. cube-mult 47.6%

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

      4. distribute-rgt1-in 63.3%

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

      5. *-commutative 63.3%

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

      6. associate-*l* 63.3%

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

      7. times-frac 95.6%

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

      8. associate-+r+ 95.6%

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

   6. Applied egg-rr 95.6%

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

   if 1.80000000000000012e167 < y

   1. Initial program 47.1%

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

      1. +-commutative 47.1%

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

      2. +-commutative 47.1%

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

      3. +-commutative 47.1%

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

      4. *-commutative 47.1%

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

      5. distribute-rgt1-in 47.1%

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

      6. fma-define 47.1%

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

      7. +-commutative 47.1%

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

      8. +-commutative 47.1%

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

      9. cube-unmult 47.1%

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

      10. +-commutative 47.1%

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

   3. Simplified 47.1%

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

      1. *-commutative 47.1%

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

      2. fma-define 47.1%

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

      3. cube-mult 47.1%

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

      4. distribute-rgt1-in 47.1%

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

      5. *-commutative 47.1%

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

      6. associate-*l* 47.1%

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

      7. times-frac 73.1%

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

      8. associate-+r+ 73.1%

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

   6. Applied egg-rr 73.1%

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

      1. div-inv 73.1%

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

      2. +-commutative 73.1%

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

      3. +-commutative 73.1%

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

      4. associate-+l+ 73.1%

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

      5. +-commutative 73.1%

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

   8. Applied egg-rr 73.1%

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

      1. associate-*r/ 73.1%

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

      2. *-rgt-identity 73.1%

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

      3. associate-/r* 99.7%

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

   10. Simplified 99.7%

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

       1. +-commutative 99.7%

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

       2. clear-num 99.8%

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

       3. frac-times 99.7%

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

       4. *-un-lft-identity 99.7%

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

       5. +-commutative 99.7%

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

       6. +-commutative 99.7%

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

       7. associate-+r+ 99.7%

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

       8. +-commutative 99.7%

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

       9. associate-+r+ 99.7%

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

   12. Applied egg-rr 99.7%

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

   13. Taylor expanded in y around inf 84.4%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{+167}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-230}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+167}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7e-230)
   (/ (/ y (+ y x)) (+ x 1.0))
   (if (<= y 1.3e+167) (/ x (* (+ y x) (+ y (+ x 1.0)))) (/ (/ x (+ y x)) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7e-230) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else if (y <= 1.3e+167) {
		tmp = x / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = (x / (y + x)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 7d-230) then
        tmp = (y / (y + x)) / (x + 1.0d0)
    else if (y <= 1.3d+167) then
        tmp = x / ((y + x) * (y + (x + 1.0d0)))
    else
        tmp = (x / (y + x)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7e-230) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else if (y <= 1.3e+167) {
		tmp = x / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = (x / (y + x)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7e-230:
		tmp = (y / (y + x)) / (x + 1.0)
	elif y <= 1.3e+167:
		tmp = x / ((y + x) * (y + (x + 1.0)))
	else:
		tmp = (x / (y + x)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7e-230)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + 1.0));
	elseif (y <= 1.3e+167)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7e-230)
		tmp = (y / (y + x)) / (x + 1.0);
	elseif (y <= 1.3e+167)
		tmp = x / ((y + x) * (y + (x + 1.0)));
	else
		tmp = (x / (y + x)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7e-230], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e+167], N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.99999999999999975e-230

   1. Initial program 62.4%

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

      1. +-commutative 62.4%

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

      2. +-commutative 62.4%

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

      3. +-commutative 62.4%

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

      4. *-commutative 62.4%

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

      5. distribute-rgt1-in 43.8%

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

      6. fma-define 62.4%

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

      7. +-commutative 62.4%

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

      8. +-commutative 62.4%

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

      9. cube-unmult 62.4%

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

      10. +-commutative 62.4%

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

   3. Simplified 62.4%

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

      1. *-commutative 62.4%

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

      2. fma-define 43.8%

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

      3. cube-mult 43.8%

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

      4. distribute-rgt1-in 62.4%

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

      5. *-commutative 62.4%

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

      6. associate-*l* 62.4%

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

      7. times-frac 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. div-inv 93.0%

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

      2. +-commutative 93.0%

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

      3. +-commutative 93.0%

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

      4. associate-+l+ 93.0%

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

      5. +-commutative 93.0%

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

   8. Applied egg-rr 93.0%

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

      1. associate-*r/ 93.1%

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

      2. *-rgt-identity 93.1%

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

      3. associate-/r* 99.9%

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

   10. Simplified 99.9%

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

       1. +-commutative 99.9%

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

       2. +-commutative 99.9%

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

       3. associate-+r+ 99.9%

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

       4. +-commutative 99.9%

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

       5. associate-+r+ 99.9%

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

       6. associate-/r* 93.1%

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

       7. clear-num 93.0%

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

       8. un-div-inv 93.1%

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

       9. +-commutative 93.1%

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

       10. +-commutative 93.1%

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

       11. associate-+r+ 93.1%

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

       12. +-commutative 93.1%

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

       13. associate-+r+ 93.1%

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

       14. clear-num 93.0%

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

       15. *-commutative 93.0%

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

       16. associate-/l/ 99.8%

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

   12. Applied egg-rr 99.8%

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

   13. Taylor expanded in y around 0 52.8%

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

       1. +-commutative 52.8%

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

   15. Simplified 52.8%

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

   if 6.99999999999999975e-230 < y < 1.3000000000000001e167

   1. Initial program 64.9%

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

      1. +-commutative 64.9%

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

      2. +-commutative 64.9%

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

      3. +-commutative 64.9%

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

      4. *-commutative 64.9%

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

      5. distribute-rgt1-in 54.0%

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

      6. fma-define 65.0%

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

      7. +-commutative 65.0%

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

      8. +-commutative 65.0%

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

      9. cube-unmult 65.0%

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

      10. +-commutative 65.0%

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

   3. Simplified 65.0%

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

      1. *-commutative 65.0%

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

      2. fma-define 54.0%

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

      3. cube-mult 54.0%

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

      4. distribute-rgt1-in 64.9%

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

      5. *-commutative 64.9%

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

      6. associate-*l* 64.9%

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

      7. times-frac 99.8%

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

      8. associate-+r+ 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+l+ 99.8%

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

      6. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in y around inf 74.6%

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

   if 1.3000000000000001e167 < y

   1. Initial program 47.1%

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

      1. +-commutative 47.1%

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

      2. +-commutative 47.1%

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

      3. +-commutative 47.1%

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

      4. *-commutative 47.1%

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

      5. distribute-rgt1-in 47.1%

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

      6. fma-define 47.1%

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

      7. +-commutative 47.1%

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

      8. +-commutative 47.1%

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

      9. cube-unmult 47.1%

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

      10. +-commutative 47.1%

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

   3. Simplified 47.1%

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

      1. *-commutative 47.1%

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

      2. fma-define 47.1%

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

      3. cube-mult 47.1%

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

      4. distribute-rgt1-in 47.1%

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

      5. *-commutative 47.1%

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

      6. associate-*l* 47.1%

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

      7. times-frac 73.1%

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

      8. associate-+r+ 73.1%

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

   6. Applied egg-rr 73.1%

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

      1. div-inv 73.1%

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

      2. +-commutative 73.1%

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

      3. +-commutative 73.1%

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

      4. associate-+l+ 73.1%

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

      5. +-commutative 73.1%

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

   8. Applied egg-rr 73.1%

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

      1. associate-*r/ 73.1%

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

      2. *-rgt-identity 73.1%

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

      3. associate-/r* 99.7%

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

   10. Simplified 99.7%

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

       1. +-commutative 99.7%

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

       2. clear-num 99.8%

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

       3. frac-times 99.7%

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

       4. *-un-lft-identity 99.7%

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

       5. +-commutative 99.7%

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

       6. +-commutative 99.7%

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

       7. associate-+r+ 99.7%

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

       8. +-commutative 99.7%

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

       9. associate-+r+ 99.7%

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

   12. Applied egg-rr 99.7%

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

   13. Taylor expanded in y around inf 84.4%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-230}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+167}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -850000:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-142} \lor \neg \left(x \leq 7.2 \cdot 10^{-102}\right):\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -850000.0)
   (* (/ y x) (/ 1.0 x))
   (if (or (<= x -2e-142) (not (<= x 7.2e-102))) (/ x (* y y)) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -850000.0) {
		tmp = (y / x) * (1.0 / x);
	} else if ((x <= -2e-142) || !(x <= 7.2e-102)) {
		tmp = x / (y * y);
	} 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 <= (-850000.0d0)) then
        tmp = (y / x) * (1.0d0 / x)
    else if ((x <= (-2d-142)) .or. (.not. (x <= 7.2d-102))) then
        tmp = x / (y * y)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -850000.0) {
		tmp = (y / x) * (1.0 / x);
	} else if ((x <= -2e-142) || !(x <= 7.2e-102)) {
		tmp = x / (y * y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -850000.0:
		tmp = (y / x) * (1.0 / x)
	elif (x <= -2e-142) or not (x <= 7.2e-102):
		tmp = x / (y * y)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -850000.0)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	elseif ((x <= -2e-142) || !(x <= 7.2e-102))
		tmp = Float64(x / Float64(y * y));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -850000.0)
		tmp = (y / x) * (1.0 / x);
	elseif ((x <= -2e-142) || ~((x <= 7.2e-102)))
		tmp = x / (y * y);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -850000.0], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -2e-142], N[Not[LessEqual[x, 7.2e-102]], $MachinePrecision]], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.5e5

   1. Initial program 61.5%

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

      1. +-commutative 61.5%

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

      2. +-commutative 61.5%

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

      3. +-commutative 61.5%

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

      4. *-commutative 61.5%

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

      5. distribute-rgt1-in 34.9%

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

      6. fma-define 61.5%

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

      7. +-commutative 61.5%

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

      8. +-commutative 61.5%

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

      9. cube-unmult 61.5%

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

      10. +-commutative 61.5%

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

   3. Simplified 61.5%

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

      1. *-commutative 61.5%

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

      2. fma-define 34.9%

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

      3. cube-mult 34.9%

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

      4. distribute-rgt1-in 61.5%

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

      5. *-commutative 61.5%

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

      6. associate-*l* 61.5%

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

      7. times-frac 91.2%

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

      8. associate-+r+ 91.2%

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

   6. Applied egg-rr 91.2%

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

      1. div-inv 91.1%

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

      2. +-commutative 91.1%

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

      3. +-commutative 91.1%

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

      4. associate-+l+ 91.1%

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

      5. +-commutative 91.1%

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

   8. Applied egg-rr 91.1%

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

      1. associate-*r/ 91.2%

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

      2. *-rgt-identity 91.2%

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

      3. associate-/r* 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around inf 73.6%

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

   12. Taylor expanded in x around inf 73.2%

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

   if -8.5e5 < x < -2.0000000000000001e-142 or 7.2e-102 < x

   1. Initial program 62.0%

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

      1. associate-/l* 81.1%

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

      2. associate-+l+ 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in x around 0 43.7%

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

   6. Taylor expanded in y around inf 36.0%

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

   if -2.0000000000000001e-142 < x < 7.2e-102

   1. Initial program 59.9%

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

      1. associate-/l* 76.7%

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

      2. associate-+l+ 76.7%

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

   3. Simplified 76.7%

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

   5. Taylor expanded in x around 0 84.5%

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

   6. Taylor expanded in y around 0 74.7%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -850000:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-142} \lor \neg \left(x \leq 7.2 \cdot 10^{-102}\right):\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.65 \cdot 10^{+55}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + \left(y + 1\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.65e+55)
   (* (/ y (+ y x)) (/ x (* (+ y x) (+ x 1.0))))
   (/ (/ x (+ x (+ y 1.0))) (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.65e+55) {
		tmp = (y / (y + x)) * (x / ((y + x) * (x + 1.0)));
	} else {
		tmp = (x / (x + (y + 1.0))) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.65d+55) then
        tmp = (y / (y + x)) * (x / ((y + x) * (x + 1.0d0)))
    else
        tmp = (x / (x + (y + 1.0d0))) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.65e+55) {
		tmp = (y / (y + x)) * (x / ((y + x) * (x + 1.0)));
	} else {
		tmp = (x / (x + (y + 1.0))) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.65e+55:
		tmp = (y / (y + x)) * (x / ((y + x) * (x + 1.0)))
	else:
		tmp = (x / (x + (y + 1.0))) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.65e+55)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(Float64(y + x) * Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(x + Float64(y + 1.0))) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.65e+55)
		tmp = (y / (y + x)) * (x / ((y + x) * (x + 1.0)));
	else
		tmp = (x / (x + (y + 1.0))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.65e+55], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(y + x), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.6499999999999998e55

   1. Initial program 64.6%

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

      1. +-commutative 64.6%

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

      2. +-commutative 64.6%

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

      3. +-commutative 64.6%

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

      4. *-commutative 64.6%

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

      5. distribute-rgt1-in 47.5%

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

      6. fma-define 64.6%

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

      7. +-commutative 64.6%

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

      8. +-commutative 64.6%

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

      9. cube-unmult 64.7%

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

      10. +-commutative 64.7%

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

   3. Simplified 64.7%

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

      1. *-commutative 64.7%

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

      2. fma-define 47.6%

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

      3. cube-mult 47.5%

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

      4. distribute-rgt1-in 64.6%

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

      5. *-commutative 64.6%

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

      6. associate-*l* 64.6%

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

      7. times-frac 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. Taylor expanded in y around 0 83.3%

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

      1. +-commutative 83.3%

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

   9. Simplified 83.3%

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

   if 3.6499999999999998e55 < y

   1. Initial program 47.3%

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

      1. +-commutative 47.3%

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

      2. +-commutative 47.3%

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

      3. +-commutative 47.3%

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

      4. *-commutative 47.3%

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

      5. distribute-rgt1-in 47.3%

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

      6. fma-define 47.3%

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

      7. +-commutative 47.3%

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

      8. +-commutative 47.3%

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

      9. cube-unmult 47.3%

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

      10. +-commutative 47.3%

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

   3. Simplified 47.3%

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

      1. *-commutative 47.3%

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

      2. fma-define 47.3%

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

      3. cube-mult 47.3%

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

      4. distribute-rgt1-in 47.3%

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

      5. *-commutative 47.3%

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

      6. associate-*l* 47.3%

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

      7. times-frac 82.0%

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

      8. associate-+r+ 82.0%

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

   6. Applied egg-rr 82.0%

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

   7. Taylor expanded in y around inf 81.3%

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

      1. *-un-lft-identity 81.3%

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

      2. associate-/r* 78.3%

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

      3. +-commutative 78.3%

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

      4. associate-+r+ 78.3%

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

      5. +-commutative 78.3%

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

      6. associate-+r+ 78.3%

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

      7. div-inv 78.3%

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

      8. associate-*l/ 78.3%

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

      9. div-inv 78.3%

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

      10. associate-+r+ 78.3%

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

      11. +-commutative 78.3%

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

      12. associate-+r+ 78.3%

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

      13. +-commutative 78.3%

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

   9. Applied egg-rr 78.3%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.65 \cdot 10^{+55}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + \left(y + 1\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-230}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7e-230)
   (/ (/ y (+ y x)) (+ x 1.0))
   (if (<= y 1.02e+41)
     (/ x (* (+ y x) (+ x 1.0)))
     (/ (/ x (+ y x)) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7e-230) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else if (y <= 1.02e+41) {
		tmp = x / ((y + x) * (x + 1.0));
	} else {
		tmp = (x / (y + x)) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 7d-230) then
        tmp = (y / (y + x)) / (x + 1.0d0)
    else if (y <= 1.02d+41) then
        tmp = x / ((y + x) * (x + 1.0d0))
    else
        tmp = (x / (y + x)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7e-230) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else if (y <= 1.02e+41) {
		tmp = x / ((y + x) * (x + 1.0));
	} else {
		tmp = (x / (y + x)) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7e-230:
		tmp = (y / (y + x)) / (x + 1.0)
	elif y <= 1.02e+41:
		tmp = x / ((y + x) * (x + 1.0))
	else:
		tmp = (x / (y + x)) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7e-230)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + 1.0));
	elseif (y <= 1.02e+41)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7e-230)
		tmp = (y / (y + x)) / (x + 1.0);
	elseif (y <= 1.02e+41)
		tmp = x / ((y + x) * (x + 1.0));
	else
		tmp = (x / (y + x)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7e-230], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.02e+41], N[(x / N[(N[(y + x), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.99999999999999975e-230

   1. Initial program 62.4%

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

      1. +-commutative 62.4%

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

      2. +-commutative 62.4%

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

      3. +-commutative 62.4%

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

      4. *-commutative 62.4%

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

      5. distribute-rgt1-in 43.8%

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

      6. fma-define 62.4%

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

      7. +-commutative 62.4%

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

      8. +-commutative 62.4%

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

      9. cube-unmult 62.4%

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

      10. +-commutative 62.4%

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

   3. Simplified 62.4%

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

      1. *-commutative 62.4%

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

      2. fma-define 43.8%

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

      3. cube-mult 43.8%

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

      4. distribute-rgt1-in 62.4%

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

      5. *-commutative 62.4%

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

      6. associate-*l* 62.4%

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

      7. times-frac 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. div-inv 93.0%

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

      2. +-commutative 93.0%

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

      3. +-commutative 93.0%

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

      4. associate-+l+ 93.0%

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

      5. +-commutative 93.0%

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

   8. Applied egg-rr 93.0%

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

      1. associate-*r/ 93.1%

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

      2. *-rgt-identity 93.1%

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

      3. associate-/r* 99.9%

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

   10. Simplified 99.9%

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

       1. +-commutative 99.9%

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

       2. +-commutative 99.9%

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

       3. associate-+r+ 99.9%

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

       4. +-commutative 99.9%

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

       5. associate-+r+ 99.9%

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

       6. associate-/r* 93.1%

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

       7. clear-num 93.0%

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

       8. un-div-inv 93.1%

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

       9. +-commutative 93.1%

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

       10. +-commutative 93.1%

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

       11. associate-+r+ 93.1%

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

       12. +-commutative 93.1%

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

       13. associate-+r+ 93.1%

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

       14. clear-num 93.0%

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

       15. *-commutative 93.0%

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

       16. associate-/l/ 99.8%

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

   12. Applied egg-rr 99.8%

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

   13. Taylor expanded in y around 0 52.8%

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

       1. +-commutative 52.8%

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

   15. Simplified 52.8%

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

   if 6.99999999999999975e-230 < y < 1.01999999999999992e41

   1. Initial program 71.2%

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

      1. +-commutative 71.2%

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

      2. +-commutative 71.2%

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

      3. +-commutative 71.2%

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

      4. *-commutative 71.2%

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

      5. distribute-rgt1-in 56.0%

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

      6. fma-define 71.3%

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

      7. +-commutative 71.3%

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

      8. +-commutative 71.3%

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

      9. cube-unmult 71.3%

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

      10. +-commutative 71.3%

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

   3. Simplified 71.3%

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

      1. *-commutative 71.3%

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

      2. fma-define 56.0%

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

      3. cube-mult 56.0%

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

      4. distribute-rgt1-in 71.2%

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

      5. *-commutative 71.2%

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

      6. associate-*l* 71.2%

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

      7. times-frac 99.8%

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

      8. associate-+r+ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around inf 70.1%

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

   8. Taylor expanded in y around 0 69.1%

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

   if 1.01999999999999992e41 < y

   1. Initial program 47.8%

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

      1. +-commutative 47.8%

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

      2. +-commutative 47.8%

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

      3. +-commutative 47.8%

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

      4. *-commutative 47.8%

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

      5. distribute-rgt1-in 47.8%

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

      6. fma-define 47.8%

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

      7. +-commutative 47.8%

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

      8. +-commutative 47.8%

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

      9. cube-unmult 47.8%

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

      10. +-commutative 47.8%

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

   3. Simplified 47.8%

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

      1. *-commutative 47.8%

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

      2. fma-define 47.8%

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

      3. cube-mult 47.8%

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

      4. distribute-rgt1-in 47.8%

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

      5. *-commutative 47.8%

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

      6. associate-*l* 47.8%

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

      7. times-frac 83.8%

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

      8. associate-+r+ 83.8%

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

   6. Applied egg-rr 83.8%

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

      1. div-inv 83.7%

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

      2. +-commutative 83.7%

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

      3. +-commutative 83.7%

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

      4. associate-+l+ 83.7%

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

      5. +-commutative 83.7%

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

   8. Applied egg-rr 83.7%

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

      1. associate-*r/ 83.8%

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

      2. *-rgt-identity 83.8%

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

      3. associate-/r* 99.7%

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

   10. Simplified 99.7%

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

       1. +-commutative 99.7%

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

       2. clear-num 99.8%

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

       3. frac-times 99.8%

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

       4. *-un-lft-identity 99.8%

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

       5. +-commutative 99.8%

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

       6. +-commutative 99.8%

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

       7. associate-+r+ 99.8%

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

       8. +-commutative 99.8%

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

       9. associate-+r+ 99.8%

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

   12. Applied egg-rr 99.8%

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

   13. Taylor expanded in x around 0 73.2%

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

       1. +-commutative 73.2%

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

   15. Simplified 73.2%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-230}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-171}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.1e-171) (/ y (* (+ y x) (+ y (+ x 1.0)))) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.1e-171) {
		tmp = y / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.1d-171)) then
        tmp = y / ((y + x) * (y + (x + 1.0d0)))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.1e-171) {
		tmp = y / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.1e-171:
		tmp = y / ((y + x) * (y + (x + 1.0)))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.1e-171)
		tmp = Float64(y / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.1e-171)
		tmp = y / ((y + x) * (y + (x + 1.0)));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.1e-171], N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1000000000000001e-171

   1. Initial program 68.8%

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

      1. +-commutative 68.8%

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

      2. +-commutative 68.8%

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

      3. +-commutative 68.8%

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

      4. *-commutative 68.8%

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

      5. distribute-rgt1-in 49.0%

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

      6. fma-define 68.8%

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

      7. +-commutative 68.8%

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

      8. +-commutative 68.8%

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

      9. cube-unmult 68.9%

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

      10. +-commutative 68.9%

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

   3. Simplified 68.9%

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

      1. *-commutative 68.9%

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

      2. fma-define 49.0%

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

      3. cube-mult 49.0%

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

      4. distribute-rgt1-in 68.8%

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

      5. *-commutative 68.8%

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

      6. associate-*l* 68.8%

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

      7. times-frac 94.0%

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

      8. associate-+r+ 94.0%

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

   6. Applied egg-rr 94.0%

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

      1. associate-*r/ 94.1%

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

      2. +-commutative 94.1%

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

      3. +-commutative 94.1%

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

      4. +-commutative 94.1%

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

      5. associate-+l+ 94.1%

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

      6. +-commutative 94.1%

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

   8. Applied egg-rr 94.1%

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

   9. Taylor expanded in y around 0 73.8%

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

   if -1.1000000000000001e-171 < x

   1. Initial program 56.3%

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

      1. associate-/l* 75.8%

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

      2. associate-+l+ 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in x around 0 58.3%

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

      1. associate-/r* 61.1%

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

      2. +-commutative 61.1%

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

   7. Simplified 61.1%

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

Alternative 9: 60.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-111}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.55e-111) (/ (/ y (+ y x)) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.55e-111) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.55d-111)) then
        tmp = (y / (y + x)) / (x + 1.0d0)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.55e-111) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.55e-111:
		tmp = (y / (y + x)) / (x + 1.0)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.55e-111)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.55e-111)
		tmp = (y / (y + x)) / (x + 1.0);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.55e-111], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000007e-111

   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 44.2%

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

      6. fma-define 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 44.2%

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

      3. cube-mult 44.2%

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

      4. distribute-rgt1-in 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 93.2%

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

      8. associate-+r+ 93.2%

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

   6. Applied egg-rr 93.2%

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

      1. div-inv 93.2%

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

      2. +-commutative 93.2%

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

      3. +-commutative 93.2%

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

      4. associate-+l+ 93.2%

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

      5. +-commutative 93.2%

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

   8. Applied egg-rr 93.2%

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

      1. associate-*r/ 93.2%

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

      2. *-rgt-identity 93.2%

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

      3. associate-/r* 99.8%

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

   10. Simplified 99.8%

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

       1. +-commutative 99.8%

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

       2. +-commutative 99.8%

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

       3. associate-+r+ 99.8%

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

       4. +-commutative 99.8%

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

       5. associate-+r+ 99.8%

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

       6. associate-/r* 93.2%

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

       7. clear-num 93.2%

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

       8. un-div-inv 93.3%

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

       9. +-commutative 93.3%

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

       10. +-commutative 93.3%

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

       11. associate-+r+ 93.3%

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

       12. +-commutative 93.3%

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

       13. associate-+r+ 93.3%

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

       14. clear-num 93.2%

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

       15. *-commutative 93.2%

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

       16. associate-/l/ 99.8%

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

   12. Applied egg-rr 99.8%

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

   13. Taylor expanded in y around 0 64.8%

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

       1. +-commutative 64.8%

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

   15. Simplified 64.8%

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

   if -1.55000000000000007e-111 < x

   1. Initial program 58.3%

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

      1. associate-/l* 77.2%

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

      2. associate-+l+ 77.2%

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

   3. Simplified 77.2%

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

   5. Taylor expanded in x around 0 60.2%

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

      1. associate-/r* 62.7%

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

      2. +-commutative 62.7%

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

   7. Simplified 62.7%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-111}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-111}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.5e-111) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.5e-111) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.5d-111)) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.5e-111) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.5e-111:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.5e-111)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.5e-111)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.5e-111], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.50000000000000004e-111

   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 44.2%

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

      6. fma-define 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 44.2%

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

      3. cube-mult 44.2%

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

      4. distribute-rgt1-in 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 93.2%

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

      8. associate-+r+ 93.2%

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

   6. Applied egg-rr 93.2%

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

      1. div-inv 93.2%

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

      2. +-commutative 93.2%

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

      3. +-commutative 93.2%

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

      4. associate-+l+ 93.2%

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

      5. +-commutative 93.2%

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

   8. Applied egg-rr 93.2%

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

      1. associate-*r/ 93.2%

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

      2. *-rgt-identity 93.2%

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

      3. associate-/r* 99.8%

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

   10. Simplified 99.8%

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

       1. +-commutative 99.8%

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

       2. clear-num 99.8%

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

       3. frac-times 98.9%

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

       4. *-un-lft-identity 98.9%

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

       5. +-commutative 98.9%

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

       6. +-commutative 98.9%

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

       7. associate-+r+ 98.9%

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

       8. +-commutative 98.9%

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

       9. associate-+r+ 98.9%

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

   12. Applied egg-rr 98.9%

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

   13. Taylor expanded in y around 0 65.4%

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

       1. associate-/r* 64.5%

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

       2. +-commutative 64.5%

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

   15. Simplified 64.5%

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

   if -1.50000000000000004e-111 < x

   1. Initial program 58.3%

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

      1. associate-/l* 77.2%

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

      2. associate-+l+ 77.2%

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

   3. Simplified 77.2%

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

   5. Taylor expanded in x around 0 60.2%

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

      1. associate-/r* 62.7%

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

      2. +-commutative 62.7%

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

   7. Simplified 62.7%

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

Alternative 11: 62.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -850000:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -850000.0) (* (/ y x) (/ 1.0 x)) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -850000.0) {
		tmp = (y / x) * (1.0 / x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-850000.0d0)) then
        tmp = (y / x) * (1.0d0 / x)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -850000.0) {
		tmp = (y / x) * (1.0 / x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -850000.0:
		tmp = (y / x) * (1.0 / x)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -850000.0)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -850000.0)
		tmp = (y / x) * (1.0 / x);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -850000.0], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.5e5

   1. Initial program 61.5%

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

      1. +-commutative 61.5%

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

      2. +-commutative 61.5%

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

      3. +-commutative 61.5%

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

      4. *-commutative 61.5%

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

      5. distribute-rgt1-in 34.9%

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

      6. fma-define 61.5%

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

      7. +-commutative 61.5%

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

      8. +-commutative 61.5%

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

      9. cube-unmult 61.5%

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

      10. +-commutative 61.5%

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

   3. Simplified 61.5%

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

      1. *-commutative 61.5%

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

      2. fma-define 34.9%

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

      3. cube-mult 34.9%

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

      4. distribute-rgt1-in 61.5%

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

      5. *-commutative 61.5%

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

      6. associate-*l* 61.5%

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

      7. times-frac 91.2%

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

      8. associate-+r+ 91.2%

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

   6. Applied egg-rr 91.2%

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

      1. div-inv 91.1%

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

      2. +-commutative 91.1%

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

      3. +-commutative 91.1%

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

      4. associate-+l+ 91.1%

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

      5. +-commutative 91.1%

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

   8. Applied egg-rr 91.1%

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

      1. associate-*r/ 91.2%

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

      2. *-rgt-identity 91.2%

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

      3. associate-/r* 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around inf 73.6%

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

   12. Taylor expanded in x around inf 73.2%

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

   if -8.5e5 < x

   1. Initial program 61.1%

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

      1. associate-/l* 79.2%

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

      2. associate-+l+ 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in x around 0 61.2%

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

      1. associate-/r* 63.5%

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

      2. +-commutative 63.5%

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

   7. Simplified 63.5%

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

Alternative 12: 61.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -620000:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -620000.0) (* (/ y x) (/ 1.0 x)) (/ x (* y (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -620000.0) {
		tmp = (y / x) * (1.0 / x);
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-620000.0d0)) then
        tmp = (y / x) * (1.0d0 / x)
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -620000.0) {
		tmp = (y / x) * (1.0 / x);
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -620000.0:
		tmp = (y / x) * (1.0 / x)
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -620000.0)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -620000.0)
		tmp = (y / x) * (1.0 / x);
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -620000.0], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.2e5

   1. Initial program 61.5%

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

      1. +-commutative 61.5%

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

      2. +-commutative 61.5%

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

      3. +-commutative 61.5%

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

      4. *-commutative 61.5%

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

      5. distribute-rgt1-in 34.9%

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

      6. fma-define 61.5%

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

      7. +-commutative 61.5%

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

      8. +-commutative 61.5%

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

      9. cube-unmult 61.5%

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

      10. +-commutative 61.5%

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

   3. Simplified 61.5%

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

      1. *-commutative 61.5%

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

      2. fma-define 34.9%

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

      3. cube-mult 34.9%

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

      4. distribute-rgt1-in 61.5%

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

      5. *-commutative 61.5%

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

      6. associate-*l* 61.5%

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

      7. times-frac 91.2%

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

      8. associate-+r+ 91.2%

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

   6. Applied egg-rr 91.2%

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

      1. div-inv 91.1%

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

      2. +-commutative 91.1%

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

      3. +-commutative 91.1%

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

      4. associate-+l+ 91.1%

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

      5. +-commutative 91.1%

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

   8. Applied egg-rr 91.1%

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

      1. associate-*r/ 91.2%

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

      2. *-rgt-identity 91.2%

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

      3. associate-/r* 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around inf 73.6%

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

   12. Taylor expanded in x around inf 73.2%

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

   if -6.2e5 < x

   1. Initial program 61.1%

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

      1. associate-/l* 79.2%

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

      2. associate-+l+ 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in x around 0 61.2%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -620000:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 36.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.0) {
		tmp = x / y;
	} 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 <= 1.0d0) then
        tmp = x / y
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 65.1%

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

      1. associate-/l* 80.3%

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

      2. associate-+l+ 80.3%

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

   3. Simplified 80.3%

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

   5. Taylor expanded in x around 0 47.6%

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

   6. Taylor expanded in y around 0 31.4%

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

   if 1 < y

   1. Initial program 48.9%

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

      1. associate-/l* 76.0%

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

      2. associate-+l+ 76.0%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in x around 0 64.1%

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

   6. Taylor expanded in y around inf 64.1%

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

Alternative 14: 27.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -118000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -118000.0) (/ 1.0 x) (/ 1.0 (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -118000.0) {
		tmp = 1.0 / x;
	} 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 <= (-118000.0d0)) then
        tmp = 1.0d0 / x
    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 <= -118000.0) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -118000.0:
		tmp = 1.0 / x
	else:
		tmp = 1.0 / (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -118000.0)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(1.0 / Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -118000.0)
		tmp = 1.0 / x;
	else
		tmp = 1.0 / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -118000.0], N[(1.0 / x), $MachinePrecision], N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -118000

   1. Initial program 61.5%

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

      1. +-commutative 61.5%

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

      2. +-commutative 61.5%

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

      3. +-commutative 61.5%

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

      4. *-commutative 61.5%

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

      5. distribute-rgt1-in 34.9%

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

      6. fma-define 61.5%

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

      7. +-commutative 61.5%

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

      8. +-commutative 61.5%

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

      9. cube-unmult 61.5%

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

      10. +-commutative 61.5%

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

   3. Simplified 61.5%

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

      1. *-commutative 61.5%

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

      2. fma-define 34.9%

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

      3. cube-mult 34.9%

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

      4. distribute-rgt1-in 61.5%

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

      5. *-commutative 61.5%

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

      6. associate-*l* 61.5%

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

      7. times-frac 91.2%

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

      8. associate-+r+ 91.2%

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

   6. Applied egg-rr 91.2%

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

      1. div-inv 91.1%

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

      2. +-commutative 91.1%

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

      3. +-commutative 91.1%

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

      4. associate-+l+ 91.1%

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

      5. +-commutative 91.1%

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

   8. Applied egg-rr 91.1%

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

      1. associate-*r/ 91.2%

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

      2. *-rgt-identity 91.2%

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

      3. associate-/r* 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around inf 73.6%

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

   12. Taylor expanded in y around inf 5.5%

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

   if -118000 < x

   1. Initial program 61.1%

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

      1. associate-/l* 79.2%

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

      2. associate-+l+ 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in x around 0 61.2%

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

   6. Taylor expanded in y around 0 39.0%

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

      1. clear-num 40.3%

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

      2. inv-pow 40.3%

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

   8. Applied egg-rr 40.3%

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

      1. unpow-1 40.3%

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

   10. Simplified 40.3%

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

Alternative 15: 26.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4600:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -4600.0) (/ 1.0 x) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4600.0) {
		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 <= (-4600.0d0)) 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 <= -4600.0) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4600.0:
		tmp = 1.0 / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4600.0)
		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 <= -4600.0)
		tmp = 1.0 / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4600.0], N[(1.0 / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4600

   1. Initial program 61.5%

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

      1. +-commutative 61.5%

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

      2. +-commutative 61.5%

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

      3. +-commutative 61.5%

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

      4. *-commutative 61.5%

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

      5. distribute-rgt1-in 34.9%

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

      6. fma-define 61.5%

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

      7. +-commutative 61.5%

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

      8. +-commutative 61.5%

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

      9. cube-unmult 61.5%

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

      10. +-commutative 61.5%

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

   3. Simplified 61.5%

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

      1. *-commutative 61.5%

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

      2. fma-define 34.9%

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

      3. cube-mult 34.9%

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

      4. distribute-rgt1-in 61.5%

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

      5. *-commutative 61.5%

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

      6. associate-*l* 61.5%

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

      7. times-frac 91.2%

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

      8. associate-+r+ 91.2%

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

   6. Applied egg-rr 91.2%

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

      1. div-inv 91.1%

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

      2. +-commutative 91.1%

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

      3. +-commutative 91.1%

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

      4. associate-+l+ 91.1%

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

      5. +-commutative 91.1%

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

   8. Applied egg-rr 91.1%

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

      1. associate-*r/ 91.2%

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

      2. *-rgt-identity 91.2%

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

      3. associate-/r* 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around inf 73.6%

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

   12. Taylor expanded in y around inf 5.5%

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

   if -4600 < x

   1. Initial program 61.1%

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

      1. associate-/l* 79.2%

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

      2. associate-+l+ 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in x around 0 61.2%

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

   6. Taylor expanded in y around 0 39.0%

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

Alternative 16: 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 61.2%

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

   1. +-commutative 61.2%

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

   2. +-commutative 61.2%

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

   3. +-commutative 61.2%

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

   4. *-commutative 61.2%

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

   5. distribute-rgt1-in 47.5%

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

   6. fma-define 61.2%

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

   7. +-commutative 61.2%

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

   8. +-commutative 61.2%

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

   9. cube-unmult 61.2%

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

   10. +-commutative 61.2%

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

3. Simplified 61.2%

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

   1. *-commutative 61.2%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

   2. fma-define 47.5%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   3. cube-mult 47.5%

      \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   4. distribute-rgt1-in 61.2%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   5. *-commutative 61.2%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   6. associate-*l* 61.2%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

   7. times-frac 92.6%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   8. associate-+r+ 92.6%

      \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

6. Applied egg-rr 92.6%

   \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
7. Step-by-step derivation

   1. div-inv 92.5%

      \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\left(x \cdot \frac{1}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}\right)} \]

   2. +-commutative 92.5%

      \[\leadsto \frac{y}{x + y} \cdot \left(x \cdot \frac{1}{\color{blue}{\left(y + x\right)} \cdot \left(x + \left(y + 1\right)\right)}\right) \]

   3. +-commutative 92.5%

      \[\leadsto \frac{y}{x + y} \cdot \left(x \cdot \frac{1}{\left(y + x\right) \cdot \left(x + \color{blue}{\left(1 + y\right)}\right)}\right) \]

   4. associate-+l+ 92.5%

      \[\leadsto \frac{y}{x + y} \cdot \left(x \cdot \frac{1}{\left(y + x\right) \cdot \color{blue}{\left(\left(x + 1\right) + y\right)}}\right) \]

   5. +-commutative 92.5%

      \[\leadsto \frac{y}{x + y} \cdot \left(x \cdot \frac{1}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}}\right) \]

8. Applied egg-rr 92.5%

   \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\left(x \cdot \frac{1}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\right)} \]
9. Step-by-step derivation

   1. associate-*r/ 92.6%

      \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{x \cdot 1}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}} \]

   2. *-rgt-identity 92.6%

      \[\leadsto \frac{y}{x + y} \cdot \frac{\color{blue}{x}}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)} \]

   3. associate-/r* 99.8%

      \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{\frac{x}{y + x}}{y + \left(x + 1\right)}} \]

10. Simplified 99.8%

    \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{\frac{x}{y + x}}{y + \left(x + 1\right)}} \]

11. Taylor expanded in x around inf 37.8%

    \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{x}} \]

12. Taylor expanded in y around inf 4.4%

    \[\leadsto \color{blue}{\frac{1}{x}} \]
13. 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 2024145 
(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))))