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

Percentage Accurate: 69.2% → 99.8%

Time: 17.9s

Alternatives: 22

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: 69.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.0%

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

   1. times-frac 83.9%

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

   2. +-commutative 83.9%

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

   3. +-commutative 83.9%

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

   4. +-commutative 83.9%

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

   5. times-frac 66.0%

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

   6. associate-*l/ 77.7%

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

   7. *-commutative 77.7%

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

   8. *-commutative 77.7%

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

   9. distribute-rgt1-in 59.5%

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

   10. fma-def 77.8%

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

   11. +-commutative 77.8%

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

   12. +-commutative 77.8%

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

   13. cube-unmult 77.8%

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

   14. +-commutative 77.8%

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

3. Simplified 77.8%

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

   1. associate-*r/ 66.0%

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

   2. fma-udef 52.3%

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

   3. cube-mult 52.2%

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

   4. distribute-rgt1-in 66.0%

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

   5. associate-+r+ 66.0%

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

   6. *-commutative 66.0%

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

   7. frac-times 83.8%

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

   8. associate-/r* 99.7%

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

   9. associate-*l/ 99.8%

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

   10. associate-+r+ 99.8%

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

   11. +-commutative 99.8%

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

   12. associate-+l+ 99.8%

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

6. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. +-commutative 99.8%

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

8. Applied egg-rr 99.8%

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

   1. associate-*l/ 73.4%

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

   2. +-commutative 73.4%

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

   3. associate-/l* 99.8%

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

   4. +-commutative 99.8%

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

10. Applied egg-rr 99.8%

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

11. Taylor expanded in y around 0 99.8%

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

12. Final simplification 99.8%

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

Alternative 2: 67.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{y + x}\\ \mathbf{if}\;x \leq -2.55 \cdot 10^{-11}:\\ \;\;\;\;\frac{t\_0}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;x \leq -1.52 \cdot 10^{-167}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{y + 1}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-241}:\\ \;\;\;\;x \cdot \frac{t\_0}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y x))))
   (if (<= x -2.55e-11)
     (/ t_0 (+ x (+ y (+ y 1.0))))
     (if (<= x -1.52e-167)
       (* (/ y (* (+ y x) (+ y x))) (/ x (+ y 1.0)))
       (if (<= x 3.7e-241)
         (* x (/ t_0 (+ y x)))
         (/ (/ x (+ y (+ x 1.0))) (+ y x)))))))

C

double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (x <= -2.55e-11) {
		tmp = t_0 / (x + (y + (y + 1.0)));
	} else if (x <= -1.52e-167) {
		tmp = (y / ((y + x) * (y + x))) * (x / (y + 1.0));
	} else if (x <= 3.7e-241) {
		tmp = x * (t_0 / (y + x));
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (y + x)
    if (x <= (-2.55d-11)) then
        tmp = t_0 / (x + (y + (y + 1.0d0)))
    else if (x <= (-1.52d-167)) then
        tmp = (y / ((y + x) * (y + x))) * (x / (y + 1.0d0))
    else if (x <= 3.7d-241) then
        tmp = x * (t_0 / (y + x))
    else
        tmp = (x / (y + (x + 1.0d0))) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (x <= -2.55e-11) {
		tmp = t_0 / (x + (y + (y + 1.0)));
	} else if (x <= -1.52e-167) {
		tmp = (y / ((y + x) * (y + x))) * (x / (y + 1.0));
	} else if (x <= 3.7e-241) {
		tmp = x * (t_0 / (y + x));
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (y + x)
	tmp = 0
	if x <= -2.55e-11:
		tmp = t_0 / (x + (y + (y + 1.0)))
	elif x <= -1.52e-167:
		tmp = (y / ((y + x) * (y + x))) * (x / (y + 1.0))
	elif x <= 3.7e-241:
		tmp = x * (t_0 / (y + x))
	else:
		tmp = (x / (y + (x + 1.0))) / (y + x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(y + x))
	tmp = 0.0
	if (x <= -2.55e-11)
		tmp = Float64(t_0 / Float64(x + Float64(y + Float64(y + 1.0))));
	elseif (x <= -1.52e-167)
		tmp = Float64(Float64(y / Float64(Float64(y + x) * Float64(y + x))) * Float64(x / Float64(y + 1.0)));
	elseif (x <= 3.7e-241)
		tmp = Float64(x * Float64(t_0 / Float64(y + x)));
	else
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (y + x);
	tmp = 0.0;
	if (x <= -2.55e-11)
		tmp = t_0 / (x + (y + (y + 1.0)));
	elseif (x <= -1.52e-167)
		tmp = (y / ((y + x) * (y + x))) * (x / (y + 1.0));
	elseif (x <= 3.7e-241)
		tmp = x * (t_0 / (y + x));
	else
		tmp = (x / (y + (x + 1.0))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.55e-11], N[(t$95$0 / N[(x + N[(y + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.52e-167], N[(N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.7e-241], N[(x * N[(t$95$0 / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.54999999999999992e-11

   1. Initial program 54.4%

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

      1. times-frac 88.0%

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

      2. +-commutative 88.0%

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

      3. +-commutative 88.0%

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

      4. +-commutative 88.0%

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

      5. times-frac 54.4%

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

      6. associate-*l/ 71.6%

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

      7. *-commutative 71.6%

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

      8. *-commutative 71.6%

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

      9. distribute-rgt1-in 26.9%

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

      10. fma-def 71.7%

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

      11. +-commutative 71.7%

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

      12. +-commutative 71.7%

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

      13. cube-unmult 71.6%

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

      14. +-commutative 71.6%

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

   3. Simplified 71.6%

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

      1. associate-*r/ 54.4%

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

      2. fma-udef 26.9%

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

      3. cube-mult 26.9%

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

      4. distribute-rgt1-in 54.4%

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

      5. associate-+r+ 54.4%

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

      6. *-commutative 54.4%

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

      7. frac-times 88.0%

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

      8. *-commutative 88.0%

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

      9. clear-num 87.9%

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

      10. associate-/r* 99.7%

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

      11. frac-times 99.1%

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

      12. *-un-lft-identity 99.1%

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

      13. associate-+r+ 99.1%

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

      14. +-commutative 99.1%

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

      15. associate-+l+ 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in x around -inf 80.5%

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

      1. mul-1-neg 80.5%

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

      2. unsub-neg 80.5%

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

      3. neg-mul-1 80.5%

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

      4. +-commutative 80.5%

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

      5. unsub-neg 80.5%

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

      6. distribute-lft-in 80.5%

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

      7. metadata-eval 80.5%

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

      8. neg-mul-1 80.5%

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

      9. unsub-neg 80.5%

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

   9. Simplified 80.5%

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

   if -2.54999999999999992e-11 < x < -1.52e-167

   1. Initial program 86.9%

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

      1. associate-/r* 87.3%

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

      2. *-commutative 87.3%

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

      3. +-commutative 87.3%

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

      4. +-commutative 87.3%

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

      5. associate-*l/ 99.5%

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

      6. +-commutative 99.5%

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

      7. associate-*r/ 99.4%

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

      8. remove-double-neg 99.4%

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

      9. +-commutative 99.4%

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

      10. +-commutative 99.4%

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

      11. remove-double-neg 99.4%

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

      12. +-commutative 99.4%

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

      13. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.4%

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

      1. +-commutative 99.4%

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

   7. Simplified 99.4%

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

   if -1.52e-167 < x < 3.6999999999999999e-241

   1. Initial program 53.7%

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

      1. associate-/r* 53.7%

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

      2. *-commutative 53.7%

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

      3. +-commutative 53.7%

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

      4. +-commutative 53.7%

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

      5. associate-*l/ 69.6%

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

      6. +-commutative 69.6%

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

      7. associate-*r/ 69.6%

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

      8. remove-double-neg 69.6%

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

      9. +-commutative 69.6%

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

      10. +-commutative 69.6%

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

      11. remove-double-neg 69.6%

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

      12. +-commutative 69.6%

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

      13. associate-+l+ 69.6%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in x around 0 69.6%

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

      1. +-commutative 69.6%

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

   7. Simplified 69.6%

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

   8. Taylor expanded in y around 0 60.6%

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

      1. associate-/r* 90.5%

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

      2. div-inv 90.5%

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

   10. Applied egg-rr 90.5%

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

       1. associate-*r/ 90.5%

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

       2. *-rgt-identity 90.5%

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

   12. Simplified 90.5%

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

   if 3.6999999999999999e-241 < x

   1. Initial program 72.5%

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

      1. times-frac 83.4%

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

      2. +-commutative 83.4%

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

      3. +-commutative 83.4%

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

      4. +-commutative 83.4%

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

      5. times-frac 72.5%

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

      6. associate-*l/ 80.9%

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

      7. *-commutative 80.9%

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

      8. *-commutative 80.9%

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

      9. distribute-rgt1-in 73.3%

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

      10. fma-def 80.8%

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

      11. +-commutative 80.8%

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

      12. +-commutative 80.8%

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

      13. cube-unmult 80.9%

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

      14. +-commutative 80.9%

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

   3. Simplified 80.9%

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

      1. associate-*r/ 72.5%

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

      2. fma-udef 65.2%

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

      3. cube-mult 65.2%

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

      4. distribute-rgt1-in 72.5%

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

      5. associate-+r+ 72.5%

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

      6. *-commutative 72.5%

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

      7. frac-times 83.3%

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

      8. associate-/r* 99.7%

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

      9. associate-*l/ 99.8%

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

      10. associate-+r+ 99.8%

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

      11. +-commutative 99.8%

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

      12. associate-+l+ 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in y around inf 46.7%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;x \leq -1.52 \cdot 10^{-167}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{y + 1}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-241}:\\ \;\;\;\;x \cdot \frac{\frac{y}{y + x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{x + \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.3e-5)
   (/ (* (/ y (+ y x)) (/ x (+ x 1.0))) (+ y x))
   (if (<= y 1.4e+154)
     (* (/ y (* (+ y x) (+ y x))) (/ x (+ y 1.0)))
     (* (/ 1.0 y) (/ x (+ x (+ y 1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.3e-5) {
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x);
	} else if (y <= 1.4e+154) {
		tmp = (y / ((y + x) * (y + x))) * (x / (y + 1.0));
	} else {
		tmp = (1.0 / y) * (x / (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 <= 2.3d-5) then
        tmp = ((y / (y + x)) * (x / (x + 1.0d0))) / (y + x)
    else if (y <= 1.4d+154) then
        tmp = (y / ((y + x) * (y + x))) * (x / (y + 1.0d0))
    else
        tmp = (1.0d0 / y) * (x / (x + (y + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.3e-5) {
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x);
	} else if (y <= 1.4e+154) {
		tmp = (y / ((y + x) * (y + x))) * (x / (y + 1.0));
	} else {
		tmp = (1.0 / y) * (x / (x + (y + 1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.3e-5:
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x)
	elif y <= 1.4e+154:
		tmp = (y / ((y + x) * (y + x))) * (x / (y + 1.0))
	else:
		tmp = (1.0 / y) * (x / (x + (y + 1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.3e-5)
		tmp = Float64(Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(x + 1.0))) / Float64(y + x));
	elseif (y <= 1.4e+154)
		tmp = Float64(Float64(y / Float64(Float64(y + x) * Float64(y + x))) * Float64(x / Float64(y + 1.0)));
	else
		tmp = Float64(Float64(1.0 / y) * Float64(x / Float64(x + Float64(y + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.3e-5)
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x);
	elseif (y <= 1.4e+154)
		tmp = (y / ((y + x) * (y + x))) * (x / (y + 1.0));
	else
		tmp = (1.0 / y) * (x / (x + (y + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.3e-5], N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.4e+154], N[(N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(x / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.3e-5

   1. Initial program 64.8%

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

      1. times-frac 82.8%

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

      2. +-commutative 82.8%

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

      3. +-commutative 82.8%

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

      4. +-commutative 82.8%

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

      5. times-frac 64.8%

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

      6. associate-*l/ 76.5%

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

      7. *-commutative 76.5%

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

      8. *-commutative 76.5%

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

      9. distribute-rgt1-in 57.1%

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

      10. fma-def 76.5%

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

      11. +-commutative 76.5%

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

      12. +-commutative 76.5%

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

      13. cube-unmult 76.5%

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

      14. +-commutative 76.5%

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

   3. Simplified 76.5%

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

      1. associate-*r/ 64.9%

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

      2. fma-udef 48.8%

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

      3. cube-mult 48.7%

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

      4. distribute-rgt1-in 64.8%

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

      5. associate-+r+ 64.8%

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

      6. *-commutative 64.8%

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

      7. frac-times 82.7%

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

      8. associate-/r* 99.6%

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

      9. associate-*l/ 99.8%

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

      10. associate-+r+ 99.8%

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

      11. +-commutative 99.8%

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

      12. associate-+l+ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around 0 81.6%

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

      1. +-commutative 81.6%

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

   9. Simplified 81.6%

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

   if 2.3e-5 < y < 1.4e154

   1. Initial program 70.5%

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

      1. associate-/r* 77.1%

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

      2. *-commutative 77.1%

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

      3. +-commutative 77.1%

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

      4. +-commutative 77.1%

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

      5. associate-*l/ 90.5%

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

      6. +-commutative 90.5%

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

      7. associate-*r/ 90.5%

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

      8. remove-double-neg 90.5%

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

      9. +-commutative 90.5%

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

      10. +-commutative 90.5%

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

      11. remove-double-neg 90.5%

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

      12. +-commutative 90.5%

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

      13. associate-+l+ 90.5%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in x around 0 83.7%

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

      1. +-commutative 83.7%

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

   7. Simplified 83.7%

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

   if 1.4e154 < y

   1. Initial program 69.3%

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

      1. associate-/r* 69.3%

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

      2. *-commutative 69.3%

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

      3. +-commutative 69.3%

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

      4. +-commutative 69.3%

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

      5. associate-*l/ 84.5%

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

      6. +-commutative 84.5%

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

      7. associate-*r/ 84.5%

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

      8. remove-double-neg 84.5%

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

      9. +-commutative 84.5%

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

      10. +-commutative 84.5%

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

      11. remove-double-neg 84.5%

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

      12. +-commutative 84.5%

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

      13. associate-+l+ 84.5%

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

   3. Simplified 84.5%

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

   5. Taylor expanded in y around inf 82.5%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{x + \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-150}:\\ \;\;\;\;x \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.5e-15)
   (/ (/ y (+ y x)) (+ x 1.0))
   (if (<= x -3.6e-150)
     (* x (/ y (* (+ y x) (+ y x))))
     (/ (/ x (+ y 1.0)) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.5e-15) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else if (x <= -3.6e-150) {
		tmp = x * (y / ((y + x) * (y + x)));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.5d-15)) then
        tmp = (y / (y + x)) / (x + 1.0d0)
    else if (x <= (-3.6d-150)) then
        tmp = x * (y / ((y + x) * (y + x)))
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.5e-15) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else if (x <= -3.6e-150) {
		tmp = x * (y / ((y + x) * (y + x)));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.5e-15:
		tmp = (y / (y + x)) / (x + 1.0)
	elif x <= -3.6e-150:
		tmp = x * (y / ((y + x) * (y + x)))
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.5e-15)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + 1.0));
	elseif (x <= -3.6e-150)
		tmp = Float64(x * Float64(y / Float64(Float64(y + x) * Float64(y + x))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.5e-15)
		tmp = (y / (y + x)) / (x + 1.0);
	elseif (x <= -3.6e-150)
		tmp = x * (y / ((y + x) * (y + x)));
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.5e-15], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.6e-150], N[(x * N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5e-15

   1. Initial program 54.4%

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

      1. times-frac 88.0%

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

      2. +-commutative 88.0%

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

      3. +-commutative 88.0%

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

      4. +-commutative 88.0%

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

      5. times-frac 54.4%

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

      6. associate-*l/ 71.6%

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

      7. *-commutative 71.6%

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

      8. *-commutative 71.6%

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

      9. distribute-rgt1-in 26.9%

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

      10. fma-def 71.7%

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

      11. +-commutative 71.7%

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

      12. +-commutative 71.7%

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

      13. cube-unmult 71.6%

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

      14. +-commutative 71.6%

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

   3. Simplified 71.6%

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

      1. associate-*r/ 54.4%

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

      2. fma-udef 26.9%

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

      3. cube-mult 26.9%

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

      4. distribute-rgt1-in 54.4%

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

      5. associate-+r+ 54.4%

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

      6. *-commutative 54.4%

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

      7. frac-times 88.0%

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

      8. *-commutative 88.0%

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

      9. clear-num 87.9%

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

      10. associate-/r* 99.7%

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

      11. frac-times 99.1%

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

      12. *-un-lft-identity 99.1%

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

      13. associate-+r+ 99.1%

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

      14. +-commutative 99.1%

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

      15. associate-+l+ 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in y around 0 79.5%

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

      1. +-commutative 79.5%

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

   9. Simplified 79.5%

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

   if -1.5e-15 < x < -3.6000000000000002e-150

   1. Initial program 84.5%

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

      1. associate-/r* 84.9%

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

      2. *-commutative 84.9%

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

      3. +-commutative 84.9%

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

      4. +-commutative 84.9%

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

      5. associate-*l/ 99.5%

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

      6. +-commutative 99.5%

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

      7. associate-*r/ 99.4%

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

      8. remove-double-neg 99.4%

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

      9. +-commutative 99.4%

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

      10. +-commutative 99.4%

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

      11. remove-double-neg 99.4%

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

      12. +-commutative 99.4%

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

      13. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.4%

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

      1. +-commutative 99.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in y around 0 74.5%

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

   if -3.6000000000000002e-150 < x

   1. Initial program 67.7%

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

      1. times-frac 79.8%

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

      2. +-commutative 79.8%

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

      3. +-commutative 79.8%

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

      4. +-commutative 79.8%

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

      5. times-frac 67.7%

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

      6. associate-*l/ 78.0%

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

      7. *-commutative 78.0%

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

      8. *-commutative 78.0%

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

      9. distribute-rgt1-in 68.6%

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

      10. fma-def 78.0%

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

      11. +-commutative 78.0%

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

      12. +-commutative 78.0%

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

      13. cube-unmult 78.0%

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

      14. +-commutative 78.0%

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

   3. Simplified 78.0%

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

      1. associate-*r/ 67.7%

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

      2. fma-udef 58.6%

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

      3. cube-mult 58.6%

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

      4. distribute-rgt1-in 67.7%

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

      5. associate-+r+ 67.7%

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

      6. *-commutative 67.7%

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

      7. frac-times 79.7%

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

      8. associate-/r* 99.6%

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

      9. associate-*l/ 99.8%

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

      10. associate-+r+ 99.8%

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

      11. +-commutative 99.8%

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

      12. associate-+l+ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 58.0%

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

      1. +-commutative 58.0%

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

   9. Simplified 58.0%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-150}:\\ \;\;\;\;x \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{y + x}\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{-15}:\\ \;\;\;\;\frac{t\_0}{x + 1}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-241}:\\ \;\;\;\;x \cdot \frac{t\_0}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y x))))
   (if (<= x -1.45e-15)
     (/ t_0 (+ x 1.0))
     (if (<= x 3.2e-241) (* x (/ t_0 (+ y x))) (/ (/ x (+ y 1.0)) (+ y x))))))

C

double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (x <= -1.45e-15) {
		tmp = t_0 / (x + 1.0);
	} else if (x <= 3.2e-241) {
		tmp = x * (t_0 / (y + x));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (y + x)
    if (x <= (-1.45d-15)) then
        tmp = t_0 / (x + 1.0d0)
    else if (x <= 3.2d-241) then
        tmp = x * (t_0 / (y + x))
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (x <= -1.45e-15) {
		tmp = t_0 / (x + 1.0);
	} else if (x <= 3.2e-241) {
		tmp = x * (t_0 / (y + x));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (y + x)
	tmp = 0
	if x <= -1.45e-15:
		tmp = t_0 / (x + 1.0)
	elif x <= 3.2e-241:
		tmp = x * (t_0 / (y + x))
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(y + x))
	tmp = 0.0
	if (x <= -1.45e-15)
		tmp = Float64(t_0 / Float64(x + 1.0));
	elseif (x <= 3.2e-241)
		tmp = Float64(x * Float64(t_0 / Float64(y + x)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (y + x);
	tmp = 0.0;
	if (x <= -1.45e-15)
		tmp = t_0 / (x + 1.0);
	elseif (x <= 3.2e-241)
		tmp = x * (t_0 / (y + x));
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.45e-15], N[(t$95$0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.2e-241], N[(x * N[(t$95$0 / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.45000000000000009e-15

   1. Initial program 54.4%

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

      1. times-frac 88.0%

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

      2. +-commutative 88.0%

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

      3. +-commutative 88.0%

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

      4. +-commutative 88.0%

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

      5. times-frac 54.4%

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

      6. associate-*l/ 71.6%

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

      7. *-commutative 71.6%

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

      8. *-commutative 71.6%

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

      9. distribute-rgt1-in 26.9%

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

      10. fma-def 71.7%

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

      11. +-commutative 71.7%

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

      12. +-commutative 71.7%

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

      13. cube-unmult 71.6%

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

      14. +-commutative 71.6%

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

   3. Simplified 71.6%

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

      1. associate-*r/ 54.4%

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

      2. fma-udef 26.9%

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

      3. cube-mult 26.9%

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

      4. distribute-rgt1-in 54.4%

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

      5. associate-+r+ 54.4%

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

      6. *-commutative 54.4%

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

      7. frac-times 88.0%

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

      8. *-commutative 88.0%

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

      9. clear-num 87.9%

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

      10. associate-/r* 99.7%

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

      11. frac-times 99.1%

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

      12. *-un-lft-identity 99.1%

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

      13. associate-+r+ 99.1%

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

      14. +-commutative 99.1%

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

      15. associate-+l+ 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in y around 0 79.5%

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

      1. +-commutative 79.5%

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

   9. Simplified 79.5%

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

   if -1.45000000000000009e-15 < x < 3.2e-241

   1. Initial program 66.6%

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

      1. associate-/r* 66.7%

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

      2. *-commutative 66.7%

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

      3. +-commutative 66.7%

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

      4. +-commutative 66.7%

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

      5. associate-*l/ 81.2%

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

      6. +-commutative 81.2%

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

      7. associate-*r/ 81.1%

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

      8. remove-double-neg 81.1%

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

      9. +-commutative 81.1%

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

      10. +-commutative 81.1%

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

      11. remove-double-neg 81.1%

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

      12. +-commutative 81.1%

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

      13. associate-+l+ 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in x around 0 81.1%

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

      1. +-commutative 81.1%

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

   7. Simplified 81.1%

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

   8. Taylor expanded in y around 0 65.3%

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

      1. associate-/r* 79.4%

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

      2. div-inv 79.4%

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

   10. Applied egg-rr 79.4%

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

       1. associate-*r/ 79.4%

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

       2. *-rgt-identity 79.4%

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

   12. Simplified 79.4%

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

   if 3.2e-241 < x

   1. Initial program 72.5%

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

      1. times-frac 83.4%

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

      2. +-commutative 83.4%

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

      3. +-commutative 83.4%

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

      4. +-commutative 83.4%

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

      5. times-frac 72.5%

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

      6. associate-*l/ 80.9%

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

      7. *-commutative 80.9%

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

      8. *-commutative 80.9%

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

      9. distribute-rgt1-in 73.3%

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

      10. fma-def 80.8%

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

      11. +-commutative 80.8%

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

      12. +-commutative 80.8%

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

      13. cube-unmult 80.9%

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

      14. +-commutative 80.9%

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

   3. Simplified 80.9%

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

      1. associate-*r/ 72.5%

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

      2. fma-udef 65.2%

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

      3. cube-mult 65.2%

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

      4. distribute-rgt1-in 72.5%

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

      5. associate-+r+ 72.5%

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

      6. *-commutative 72.5%

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

      7. frac-times 83.3%

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

      8. associate-/r* 99.7%

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

      9. associate-*l/ 99.8%

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

      10. associate-+r+ 99.8%

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

      11. +-commutative 99.8%

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

      12. associate-+l+ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 45.5%

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

      1. +-commutative 45.5%

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

   9. Simplified 45.5%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-241}:\\ \;\;\;\;x \cdot \frac{\frac{y}{y + x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{y + x}\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{t\_0}{x + 1}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-241}:\\ \;\;\;\;x \cdot \frac{t\_0}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y x))))
   (if (<= x -1.5e-15)
     (/ t_0 (+ x 1.0))
     (if (<= x 4.8e-241)
       (* x (/ t_0 (+ y x)))
       (/ (/ x (+ y (+ x 1.0))) (+ y x))))))

C

double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (x <= -1.5e-15) {
		tmp = t_0 / (x + 1.0);
	} else if (x <= 4.8e-241) {
		tmp = x * (t_0 / (y + x));
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (y + x)
    if (x <= (-1.5d-15)) then
        tmp = t_0 / (x + 1.0d0)
    else if (x <= 4.8d-241) then
        tmp = x * (t_0 / (y + x))
    else
        tmp = (x / (y + (x + 1.0d0))) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (x <= -1.5e-15) {
		tmp = t_0 / (x + 1.0);
	} else if (x <= 4.8e-241) {
		tmp = x * (t_0 / (y + x));
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (y + x)
	tmp = 0
	if x <= -1.5e-15:
		tmp = t_0 / (x + 1.0)
	elif x <= 4.8e-241:
		tmp = x * (t_0 / (y + x))
	else:
		tmp = (x / (y + (x + 1.0))) / (y + x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(y + x))
	tmp = 0.0
	if (x <= -1.5e-15)
		tmp = Float64(t_0 / Float64(x + 1.0));
	elseif (x <= 4.8e-241)
		tmp = Float64(x * Float64(t_0 / Float64(y + x)));
	else
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (y + x);
	tmp = 0.0;
	if (x <= -1.5e-15)
		tmp = t_0 / (x + 1.0);
	elseif (x <= 4.8e-241)
		tmp = x * (t_0 / (y + x));
	else
		tmp = (x / (y + (x + 1.0))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.5e-15], N[(t$95$0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.8e-241], N[(x * N[(t$95$0 / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5e-15

   1. Initial program 54.4%

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

      1. times-frac 88.0%

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

      2. +-commutative 88.0%

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

      3. +-commutative 88.0%

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

      4. +-commutative 88.0%

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

      5. times-frac 54.4%

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

      6. associate-*l/ 71.6%

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

      7. *-commutative 71.6%

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

      8. *-commutative 71.6%

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

      9. distribute-rgt1-in 26.9%

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

      10. fma-def 71.7%

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

      11. +-commutative 71.7%

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

      12. +-commutative 71.7%

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

      13. cube-unmult 71.6%

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

      14. +-commutative 71.6%

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

   3. Simplified 71.6%

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

      1. associate-*r/ 54.4%

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

      2. fma-udef 26.9%

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

      3. cube-mult 26.9%

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

      4. distribute-rgt1-in 54.4%

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

      5. associate-+r+ 54.4%

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

      6. *-commutative 54.4%

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

      7. frac-times 88.0%

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

      8. *-commutative 88.0%

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

      9. clear-num 87.9%

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

      10. associate-/r* 99.7%

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

      11. frac-times 99.1%

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

      12. *-un-lft-identity 99.1%

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

      13. associate-+r+ 99.1%

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

      14. +-commutative 99.1%

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

      15. associate-+l+ 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in y around 0 79.5%

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

      1. +-commutative 79.5%

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

   9. Simplified 79.5%

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

   if -1.5e-15 < x < 4.8e-241

   1. Initial program 66.6%

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

      1. associate-/r* 66.7%

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

      2. *-commutative 66.7%

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

      3. +-commutative 66.7%

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

      4. +-commutative 66.7%

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

      5. associate-*l/ 81.2%

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

      6. +-commutative 81.2%

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

      7. associate-*r/ 81.1%

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

      8. remove-double-neg 81.1%

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

      9. +-commutative 81.1%

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

      10. +-commutative 81.1%

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

      11. remove-double-neg 81.1%

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

      12. +-commutative 81.1%

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

      13. associate-+l+ 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in x around 0 81.1%

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

      1. +-commutative 81.1%

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

   7. Simplified 81.1%

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

   8. Taylor expanded in y around 0 65.3%

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

      1. associate-/r* 79.4%

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

      2. div-inv 79.4%

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

   10. Applied egg-rr 79.4%

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

       1. associate-*r/ 79.4%

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

       2. *-rgt-identity 79.4%

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

   12. Simplified 79.4%

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

   if 4.8e-241 < x

   1. Initial program 72.5%

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

      1. times-frac 83.4%

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

      2. +-commutative 83.4%

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

      3. +-commutative 83.4%

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

      4. +-commutative 83.4%

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

      5. times-frac 72.5%

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

      6. associate-*l/ 80.9%

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

      7. *-commutative 80.9%

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

      8. *-commutative 80.9%

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

      9. distribute-rgt1-in 73.3%

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

      10. fma-def 80.8%

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

      11. +-commutative 80.8%

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

      12. +-commutative 80.8%

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

      13. cube-unmult 80.9%

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

      14. +-commutative 80.9%

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

   3. Simplified 80.9%

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

      1. associate-*r/ 72.5%

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

      2. fma-udef 65.2%

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

      3. cube-mult 65.2%

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

      4. distribute-rgt1-in 72.5%

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

      5. associate-+r+ 72.5%

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

      6. *-commutative 72.5%

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

      7. frac-times 83.3%

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

      8. associate-/r* 99.7%

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

      9. associate-*l/ 99.8%

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

      10. associate-+r+ 99.8%

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

      11. +-commutative 99.8%

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

      12. associate-+l+ 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in y around inf 46.7%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-241}:\\ \;\;\;\;x \cdot \frac{\frac{y}{y + x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{y + x}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-241}:\\ \;\;\;\;x \cdot \frac{\frac{y}{y + x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t\_0}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -7.2e-16)
     (/ (/ y t_0) (+ y x))
     (if (<= x 3.3e-241)
       (* x (/ (/ y (+ y x)) (+ y x)))
       (/ (/ x t_0) (+ y x))))))

C

double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -7.2e-16) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= 3.3e-241) {
		tmp = x * ((y / (y + x)) / (y + x));
	} else {
		tmp = (x / t_0) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (x <= (-7.2d-16)) then
        tmp = (y / t_0) / (y + x)
    else if (x <= 3.3d-241) then
        tmp = x * ((y / (y + x)) / (y + x))
    else
        tmp = (x / t_0) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -7.2e-16) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= 3.3e-241) {
		tmp = x * ((y / (y + x)) / (y + x));
	} else {
		tmp = (x / t_0) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -7.2e-16:
		tmp = (y / t_0) / (y + x)
	elif x <= 3.3e-241:
		tmp = x * ((y / (y + x)) / (y + x))
	else:
		tmp = (x / t_0) / (y + x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -7.2e-16)
		tmp = Float64(Float64(y / t_0) / Float64(y + x));
	elseif (x <= 3.3e-241)
		tmp = Float64(x * Float64(Float64(y / Float64(y + x)) / Float64(y + x)));
	else
		tmp = Float64(Float64(x / t_0) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -7.2e-16)
		tmp = (y / t_0) / (y + x);
	elseif (x <= 3.3e-241)
		tmp = x * ((y / (y + x)) / (y + x));
	else
		tmp = (x / t_0) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.2e-16], N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.3e-241], N[(x * N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.19999999999999965e-16

   1. Initial program 54.4%

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

      1. times-frac 88.0%

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

      2. +-commutative 88.0%

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

      3. +-commutative 88.0%

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

      4. +-commutative 88.0%

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

      5. times-frac 54.4%

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

      6. associate-*l/ 71.6%

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

      7. *-commutative 71.6%

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

      8. *-commutative 71.6%

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

      9. distribute-rgt1-in 26.9%

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

      10. fma-def 71.7%

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

      11. +-commutative 71.7%

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

      12. +-commutative 71.7%

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

      13. cube-unmult 71.6%

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

      14. +-commutative 71.6%

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

   3. Simplified 71.6%

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

      1. associate-*r/ 54.4%

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

      2. fma-udef 26.9%

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

      3. cube-mult 26.9%

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

      4. distribute-rgt1-in 54.4%

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

      5. associate-+r+ 54.4%

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

      6. *-commutative 54.4%

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

      7. frac-times 88.0%

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

      8. associate-/r* 99.7%

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

      9. associate-*l/ 99.7%

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

      10. associate-+r+ 99.7%

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

      11. +-commutative 99.7%

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

      12. associate-+l+ 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.7%

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

      2. +-commutative 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in y around 0 79.9%

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

   if -7.19999999999999965e-16 < x < 3.2999999999999999e-241

   1. Initial program 66.6%

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

      1. associate-/r* 66.7%

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

      2. *-commutative 66.7%

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

      3. +-commutative 66.7%

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

      4. +-commutative 66.7%

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

      5. associate-*l/ 81.2%

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

      6. +-commutative 81.2%

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

      7. associate-*r/ 81.1%

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

      8. remove-double-neg 81.1%

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

      9. +-commutative 81.1%

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

      10. +-commutative 81.1%

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

      11. remove-double-neg 81.1%

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

      12. +-commutative 81.1%

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

      13. associate-+l+ 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in x around 0 81.1%

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

      1. +-commutative 81.1%

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

   7. Simplified 81.1%

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

   8. Taylor expanded in y around 0 65.3%

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

      1. associate-/r* 79.4%

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

      2. div-inv 79.4%

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

   10. Applied egg-rr 79.4%

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

       1. associate-*r/ 79.4%

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

       2. *-rgt-identity 79.4%

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

   12. Simplified 79.4%

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

   if 3.2999999999999999e-241 < x

   1. Initial program 72.5%

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

      1. times-frac 83.4%

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

      2. +-commutative 83.4%

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

      3. +-commutative 83.4%

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

      4. +-commutative 83.4%

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

      5. times-frac 72.5%

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

      6. associate-*l/ 80.9%

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

      7. *-commutative 80.9%

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

      8. *-commutative 80.9%

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

      9. distribute-rgt1-in 73.3%

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

      10. fma-def 80.8%

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

      11. +-commutative 80.8%

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

      12. +-commutative 80.8%

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

      13. cube-unmult 80.9%

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

      14. +-commutative 80.9%

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

   3. Simplified 80.9%

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

      1. associate-*r/ 72.5%

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

      2. fma-udef 65.2%

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

      3. cube-mult 65.2%

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

      4. distribute-rgt1-in 72.5%

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

      5. associate-+r+ 72.5%

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

      6. *-commutative 72.5%

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

      7. frac-times 83.3%

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

      8. associate-/r* 99.7%

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

      9. associate-*l/ 99.8%

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

      10. associate-+r+ 99.8%

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

      11. +-commutative 99.8%

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

      12. associate-+l+ 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in y around inf 46.7%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{y + x}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-241}:\\ \;\;\;\;x \cdot \frac{\frac{y}{y + x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{y + x}\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{t\_0}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-241}:\\ \;\;\;\;x \cdot \frac{t\_0}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y x))))
   (if (<= x -1.6e-17)
     (/ t_0 (+ x (+ y (+ y 1.0))))
     (if (<= x 2.5e-241)
       (* x (/ t_0 (+ y x)))
       (/ (/ x (+ y (+ x 1.0))) (+ y x))))))

C

double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (x <= -1.6e-17) {
		tmp = t_0 / (x + (y + (y + 1.0)));
	} else if (x <= 2.5e-241) {
		tmp = x * (t_0 / (y + x));
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (y + x)
    if (x <= (-1.6d-17)) then
        tmp = t_0 / (x + (y + (y + 1.0d0)))
    else if (x <= 2.5d-241) then
        tmp = x * (t_0 / (y + x))
    else
        tmp = (x / (y + (x + 1.0d0))) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (x <= -1.6e-17) {
		tmp = t_0 / (x + (y + (y + 1.0)));
	} else if (x <= 2.5e-241) {
		tmp = x * (t_0 / (y + x));
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (y + x)
	tmp = 0
	if x <= -1.6e-17:
		tmp = t_0 / (x + (y + (y + 1.0)))
	elif x <= 2.5e-241:
		tmp = x * (t_0 / (y + x))
	else:
		tmp = (x / (y + (x + 1.0))) / (y + x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(y + x))
	tmp = 0.0
	if (x <= -1.6e-17)
		tmp = Float64(t_0 / Float64(x + Float64(y + Float64(y + 1.0))));
	elseif (x <= 2.5e-241)
		tmp = Float64(x * Float64(t_0 / Float64(y + x)));
	else
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (y + x);
	tmp = 0.0;
	if (x <= -1.6e-17)
		tmp = t_0 / (x + (y + (y + 1.0)));
	elseif (x <= 2.5e-241)
		tmp = x * (t_0 / (y + x));
	else
		tmp = (x / (y + (x + 1.0))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.6e-17], N[(t$95$0 / N[(x + N[(y + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.5e-241], N[(x * N[(t$95$0 / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6000000000000001e-17

   1. Initial program 55.1%

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

      1. times-frac 88.2%

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

      2. +-commutative 88.2%

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

      3. +-commutative 88.2%

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

      4. +-commutative 88.2%

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

      5. times-frac 55.1%

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

      6. associate-*l/ 72.1%

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

      7. *-commutative 72.1%

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

      8. *-commutative 72.1%

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

      9. distribute-rgt1-in 28.0%

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

      10. fma-def 72.1%

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

      11. +-commutative 72.1%

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

      12. +-commutative 72.1%

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

      13. cube-unmult 72.1%

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

      14. +-commutative 72.1%

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

   3. Simplified 72.1%

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

      1. associate-*r/ 55.1%

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

      2. fma-udef 28.0%

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

      3. cube-mult 28.0%

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

      4. distribute-rgt1-in 55.1%

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

      5. associate-+r+ 55.1%

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

      6. *-commutative 55.1%

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

      7. frac-times 88.1%

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

      8. *-commutative 88.1%

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

      9. clear-num 88.1%

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

      10. associate-/r* 99.7%

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

      11. frac-times 99.2%

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

      12. *-un-lft-identity 99.2%

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

      13. associate-+r+ 99.2%

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

      14. +-commutative 99.2%

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

      15. associate-+l+ 99.2%

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

   6. Applied egg-rr 99.2%

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

   7. Taylor expanded in x around -inf 79.4%

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

      1. mul-1-neg 79.4%

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

      2. unsub-neg 79.4%

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

      3. neg-mul-1 79.4%

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

      4. +-commutative 79.4%

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

      5. unsub-neg 79.4%

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

      6. distribute-lft-in 79.4%

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

      7. metadata-eval 79.4%

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

      8. neg-mul-1 79.4%

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

      9. unsub-neg 79.4%

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

   9. Simplified 79.4%

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

   if -1.6000000000000001e-17 < x < 2.4999999999999999e-241

   1. Initial program 66.2%

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

      1. associate-/r* 66.3%

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

      2. *-commutative 66.3%

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

      3. +-commutative 66.3%

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

      4. +-commutative 66.3%

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

      5. associate-*l/ 81.0%

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

      6. +-commutative 81.0%

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

      7. associate-*r/ 80.9%

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

      8. remove-double-neg 80.9%

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

      9. +-commutative 80.9%

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

      10. +-commutative 80.9%

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

      11. remove-double-neg 80.9%

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

      12. +-commutative 80.9%

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

      13. associate-+l+ 80.9%

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

   3. Simplified 80.9%

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

   5. Taylor expanded in x around 0 80.9%

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

      1. +-commutative 80.9%

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

   7. Simplified 80.9%

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

   8. Taylor expanded in y around 0 66.1%

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

      1. associate-/r* 80.4%

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

      2. div-inv 80.3%

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

   10. Applied egg-rr 80.3%

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

       1. associate-*r/ 80.4%

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

       2. *-rgt-identity 80.4%

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

   12. Simplified 80.4%

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

   if 2.4999999999999999e-241 < x

   1. Initial program 72.5%

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

      1. times-frac 83.4%

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

      2. +-commutative 83.4%

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

      3. +-commutative 83.4%

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

      4. +-commutative 83.4%

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

      5. times-frac 72.5%

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

      6. associate-*l/ 80.9%

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

      7. *-commutative 80.9%

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

      8. *-commutative 80.9%

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

      9. distribute-rgt1-in 73.3%

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

      10. fma-def 80.8%

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

      11. +-commutative 80.8%

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

      12. +-commutative 80.8%

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

      13. cube-unmult 80.9%

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

      14. +-commutative 80.9%

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

   3. Simplified 80.9%

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

      1. associate-*r/ 72.5%

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

      2. fma-udef 65.2%

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

      3. cube-mult 65.2%

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

      4. distribute-rgt1-in 72.5%

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

      5. associate-+r+ 72.5%

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

      6. *-commutative 72.5%

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

      7. frac-times 83.3%

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

      8. associate-/r* 99.7%

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

      9. associate-*l/ 99.8%

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

      10. associate-+r+ 99.8%

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

      11. +-commutative 99.8%

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

      12. associate-+l+ 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in y around inf 46.7%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-241}:\\ \;\;\;\;x \cdot \frac{\frac{y}{y + x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{y + x}\\ \mathbf{if}\;x \leq -2.55 \cdot 10^{-11}:\\ \;\;\;\;\frac{t\_0}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(y + x\right) \cdot \frac{y + 1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y x))))
   (if (<= x -2.55e-11)
     (/ t_0 (+ x (+ y (+ y 1.0))))
     (/ t_0 (* (+ y x) (/ (+ y 1.0) x))))))

C

double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (x <= -2.55e-11) {
		tmp = t_0 / (x + (y + (y + 1.0)));
	} else {
		tmp = t_0 / ((y + x) * ((y + 1.0) / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (y + x)
    if (x <= (-2.55d-11)) then
        tmp = t_0 / (x + (y + (y + 1.0d0)))
    else
        tmp = t_0 / ((y + x) * ((y + 1.0d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (x <= -2.55e-11) {
		tmp = t_0 / (x + (y + (y + 1.0)));
	} else {
		tmp = t_0 / ((y + x) * ((y + 1.0) / x));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (y + x)
	tmp = 0
	if x <= -2.55e-11:
		tmp = t_0 / (x + (y + (y + 1.0)))
	else:
		tmp = t_0 / ((y + x) * ((y + 1.0) / x))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(y + x))
	tmp = 0.0
	if (x <= -2.55e-11)
		tmp = Float64(t_0 / Float64(x + Float64(y + Float64(y + 1.0))));
	else
		tmp = Float64(t_0 / Float64(Float64(y + x) * Float64(Float64(y + 1.0) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (y + x);
	tmp = 0.0;
	if (x <= -2.55e-11)
		tmp = t_0 / (x + (y + (y + 1.0)));
	else
		tmp = t_0 / ((y + x) * ((y + 1.0) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.55e-11], N[(t$95$0 / N[(x + N[(y + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(y + x), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.54999999999999992e-11

   1. Initial program 54.4%

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

      1. times-frac 88.0%

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

      2. +-commutative 88.0%

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

      3. +-commutative 88.0%

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

      4. +-commutative 88.0%

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

      5. times-frac 54.4%

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

      6. associate-*l/ 71.6%

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

      7. *-commutative 71.6%

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

      8. *-commutative 71.6%

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

      9. distribute-rgt1-in 26.9%

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

      10. fma-def 71.7%

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

      11. +-commutative 71.7%

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

      12. +-commutative 71.7%

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

      13. cube-unmult 71.6%

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

      14. +-commutative 71.6%

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

   3. Simplified 71.6%

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

      1. associate-*r/ 54.4%

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

      2. fma-udef 26.9%

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

      3. cube-mult 26.9%

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

      4. distribute-rgt1-in 54.4%

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

      5. associate-+r+ 54.4%

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

      6. *-commutative 54.4%

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

      7. frac-times 88.0%

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

      8. *-commutative 88.0%

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

      9. clear-num 87.9%

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

      10. associate-/r* 99.7%

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

      11. frac-times 99.1%

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

      12. *-un-lft-identity 99.1%

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

      13. associate-+r+ 99.1%

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

      14. +-commutative 99.1%

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

      15. associate-+l+ 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in x around -inf 80.5%

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

      1. mul-1-neg 80.5%

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

      2. unsub-neg 80.5%

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

      3. neg-mul-1 80.5%

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

      4. +-commutative 80.5%

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

      5. unsub-neg 80.5%

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

      6. distribute-lft-in 80.5%

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

      7. metadata-eval 80.5%

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

      8. neg-mul-1 80.5%

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

      9. unsub-neg 80.5%

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

   9. Simplified 80.5%

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

   if -2.54999999999999992e-11 < x

   1. Initial program 70.0%

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

      1. associate-/r* 71.0%

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

      2. *-commutative 71.0%

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

      3. +-commutative 71.0%

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

      4. +-commutative 71.0%

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

      5. associate-*l/ 82.4%

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

      6. +-commutative 82.4%

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

      7. associate-*r/ 82.4%

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

      8. remove-double-neg 82.4%

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

      9. +-commutative 82.4%

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

      10. +-commutative 82.4%

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

      11. remove-double-neg 82.4%

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

      12. +-commutative 82.4%

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

      13. associate-+l+ 82.4%

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

   3. Simplified 82.4%

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

   5. Taylor expanded in x around 0 79.2%

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

      1. +-commutative 79.2%

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

   7. Simplified 79.2%

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

      1. *-commutative 79.2%

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

      2. clear-num 79.2%

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

      3. associate-/r* 92.2%

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

      4. frac-times 86.1%

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

      5. *-un-lft-identity 86.1%

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

      6. +-commutative 86.1%

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

      7. +-commutative 86.1%

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

   9. Applied egg-rr 86.1%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{\left(y + x\right) \cdot \frac{y + 1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{-129}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-58}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.65e-129)
   (/ y (* x (+ x 1.0)))
   (if (<= x 3.8e-58) (/ x (* y (+ y 1.0))) (/ (/ x y) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.65e-129) {
		tmp = y / (x * (x + 1.0));
	} else if (x <= 3.8e-58) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / (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 <= (-2.65d-129)) then
        tmp = y / (x * (x + 1.0d0))
    else if (x <= 3.8d-58) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.65e-129) {
		tmp = y / (x * (x + 1.0));
	} else if (x <= 3.8e-58) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.65e-129:
		tmp = y / (x * (x + 1.0))
	elif x <= 3.8e-58:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.65e-129)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (x <= 3.8e-58)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.65e-129)
		tmp = y / (x * (x + 1.0));
	elseif (x <= 3.8e-58)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.65e-129], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.8e-58], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.64999999999999987e-129

   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. associate-/r* 73.5%

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

      2. *-commutative 73.5%

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

      3. +-commutative 73.5%

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

      4. +-commutative 73.5%

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

      5. associate-*l/ 90.8%

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

      6. +-commutative 90.8%

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

      7. associate-*r/ 90.8%

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

      8. remove-double-neg 90.8%

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

      9. +-commutative 90.8%

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

      10. +-commutative 90.8%

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

      11. remove-double-neg 90.8%

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

      12. +-commutative 90.8%

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

      13. associate-+l+ 90.8%

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

   3. Simplified 90.8%

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

   5. Taylor expanded in y around 0 66.7%

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

   if -2.64999999999999987e-129 < x < 3.7999999999999997e-58

   1. Initial program 64.7%

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

      1. associate-/r* 64.6%

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

      2. *-commutative 64.6%

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

      3. +-commutative 64.6%

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

      4. +-commutative 64.6%

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

      5. associate-*l/ 78.1%

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

      6. +-commutative 78.1%

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

      7. associate-*r/ 78.1%

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

      8. remove-double-neg 78.1%

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

      9. +-commutative 78.1%

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

      10. +-commutative 78.1%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 78.1%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 78.1%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 78.1%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 78.1%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 75.6%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 75.6%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   7. Simplified 75.6%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   if 3.7999999999999997e-58 < x

   1. Initial program 74.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 83.3%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 83.3%

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1} \]

      3. +-commutative 83.3%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      4. +-commutative 83.3%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      5. times-frac 74.0%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. associate-*l/ 80.6%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

      7. *-commutative 80.6%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 80.6%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 77.2%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 80.5%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 80.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 80.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 80.6%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 80.6%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 80.6%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 74.1%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 71.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 71.1%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 74.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. associate-+r+ 74.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 74.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. frac-times 83.2%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

      8. associate-/r* 99.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \cdot \frac{x}{x + \left(y + 1\right)} \]

      9. associate-*l/ 99.8%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot \frac{x}{x + \left(y + 1\right)}}{x + y}} \]

      10. associate-+r+ 99.8%

          \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      11. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{\left(y + x\right)} + 1}}{x + y} \]

      12. associate-+l+ 99.8%

          \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{y + \left(x + 1\right)}}}{x + y} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot \frac{x}{y + \left(x + 1\right)}}{x + y}} \]

   7. Taylor expanded in y around inf 34.2%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{x + y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{-129}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-58}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.11 \cdot 10^{-128}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-58}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.11e-128)
   (/ (/ y x) (+ x 1.0))
   (if (<= x 3.8e-58) (/ x (* y (+ y 1.0))) (/ (/ x y) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.11e-128) {
		tmp = (y / x) / (x + 1.0);
	} else if (x <= 3.8e-58) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.11d-128)) then
        tmp = (y / x) / (x + 1.0d0)
    else if (x <= 3.8d-58) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.11e-128) {
		tmp = (y / x) / (x + 1.0);
	} else if (x <= 3.8e-58) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.11e-128:
		tmp = (y / x) / (x + 1.0)
	elif x <= 3.8e-58:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.11e-128)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (x <= 3.8e-58)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.11e-128)
		tmp = (y / x) / (x + 1.0);
	elseif (x <= 3.8e-58)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.11e-128], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.8e-58], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1099999999999999e-128

   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. associate-/r* 73.5%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. *-commutative 73.5%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 73.5%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      4. +-commutative 73.5%

         \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      5. associate-*l/ 90.8%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 90.8%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 90.8%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 90.8%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 90.8%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 90.8%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 90.8%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 90.8%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 90.8%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 90.8%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 66.7%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 70.6%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 70.6%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   7. Simplified 70.6%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if -1.1099999999999999e-128 < x < 3.7999999999999997e-58

   1. Initial program 64.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 64.6%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. *-commutative 64.6%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 64.6%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      4. +-commutative 64.6%

         \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      5. associate-*l/ 78.1%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 78.1%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 78.1%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 78.1%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 78.1%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 78.1%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 78.1%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 78.1%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 78.1%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 78.1%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 75.6%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 75.6%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   7. Simplified 75.6%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   if 3.7999999999999997e-58 < x

   1. Initial program 74.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 83.3%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 83.3%

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1} \]

      3. +-commutative 83.3%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      4. +-commutative 83.3%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      5. times-frac 74.0%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. associate-*l/ 80.6%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

      7. *-commutative 80.6%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 80.6%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 77.2%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 80.5%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 80.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 80.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 80.6%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 80.6%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 80.6%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 74.1%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 71.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 71.1%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 74.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. associate-+r+ 74.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 74.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. frac-times 83.2%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

      8. associate-/r* 99.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \cdot \frac{x}{x + \left(y + 1\right)} \]

      9. associate-*l/ 99.8%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot \frac{x}{x + \left(y + 1\right)}}{x + y}} \]

      10. associate-+r+ 99.8%

          \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      11. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{\left(y + x\right)} + 1}}{x + y} \]

      12. associate-+l+ 99.8%

          \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{y + \left(x + 1\right)}}}{x + y} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot \frac{x}{y + \left(x + 1\right)}}{x + y}} \]

   7. Taylor expanded in y around inf 34.2%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{x + y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.11 \cdot 10^{-128}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-58}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{y}{y + x} \cdot \frac{x}{y + \left(x + 1\right)}}{y + x} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (* (/ y (+ y x)) (/ x (+ y (+ x 1.0)))) (+ y x)))

C

double code(double x, double y) {
	return ((y / (y + x)) * (x / (y + (x + 1.0)))) / (y + x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((y / (y + x)) * (x / (y + (x + 1.0d0)))) / (y + x)
end function

Java

public static double code(double x, double y) {
	return ((y / (y + x)) * (x / (y + (x + 1.0)))) / (y + x);
}

Python

def code(x, y):
	return ((y / (y + x)) * (x / (y + (x + 1.0)))) / (y + x)

Julia

function code(x, y)
	return Float64(Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(y + Float64(x + 1.0)))) / Float64(y + x))
end

MATLAB

function tmp = code(x, y)
	tmp = ((y / (y + x)) * (x / (y + (x + 1.0)))) / (y + x);
end

Wolfram

code[x_, y_] := N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.0%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. times-frac 83.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. +-commutative 83.9%

      \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1} \]

   3. +-commutative 83.9%

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

   4. +-commutative 83.9%

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   5. times-frac 66.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

   6. associate-*l/ 77.7%

      \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

   7. *-commutative 77.7%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

   8. *-commutative 77.7%

      \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   9. distribute-rgt1-in 59.5%

      \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   10. fma-def 77.8%

       \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

   11. +-commutative 77.8%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   12. +-commutative 77.8%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   13. cube-unmult 77.8%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

   14. +-commutative 77.8%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

3. Simplified 77.8%

   \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r/ 66.0%

      \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   2. fma-udef 52.3%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   3. cube-mult 52.2%

      \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   4. distribute-rgt1-in 66.0%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   5. associate-+r+ 66.0%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

   6. *-commutative 66.0%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

   7. frac-times 83.8%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   8. associate-/r* 99.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \cdot \frac{x}{x + \left(y + 1\right)} \]

   9. associate-*l/ 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot \frac{x}{x + \left(y + 1\right)}}{x + y}} \]

   10. associate-+r+ 99.8%

       \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

   11. +-commutative 99.8%

       \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{\left(y + x\right)} + 1}}{x + y} \]

   12. associate-+l+ 99.8%

       \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{y + \left(x + 1\right)}}}{x + y} \]

6. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot \frac{x}{y + \left(x + 1\right)}}{x + y}} \]

7. Final simplification 99.8%

   \[\leadsto \frac{\frac{y}{y + x} \cdot \frac{x}{y + \left(x + 1\right)}}{y + x} \]
8. Add Preprocessing

Alternative 13: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{x \cdot \frac{y}{y + x}}{y + \left(x + 1\right)}}{y + x} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (/ (* x (/ y (+ y x))) (+ y (+ x 1.0))) (+ y x)))

C

double code(double x, double y) {
	return ((x * (y / (y + x))) / (y + (x + 1.0))) / (y + x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * (y / (y + x))) / (y + (x + 1.0d0))) / (y + x)
end function

Java

public static double code(double x, double y) {
	return ((x * (y / (y + x))) / (y + (x + 1.0))) / (y + x);
}

Python

def code(x, y):
	return ((x * (y / (y + x))) / (y + (x + 1.0))) / (y + x)

Julia

function code(x, y)
	return Float64(Float64(Float64(x * Float64(y / Float64(y + x))) / Float64(y + Float64(x + 1.0))) / Float64(y + x))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * (y / (y + x))) / (y + (x + 1.0))) / (y + x);
end

Wolfram

code[x_, y_] := N[(N[(N[(x * N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.0%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. times-frac 83.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. +-commutative 83.9%

      \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1} \]

   3. +-commutative 83.9%

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

   4. +-commutative 83.9%

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   5. times-frac 66.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

   6. associate-*l/ 77.7%

      \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

   7. *-commutative 77.7%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

   8. *-commutative 77.7%

      \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   9. distribute-rgt1-in 59.5%

      \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   10. fma-def 77.8%

       \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

   11. +-commutative 77.8%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   12. +-commutative 77.8%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   13. cube-unmult 77.8%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

   14. +-commutative 77.8%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

3. Simplified 77.8%

   \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r/ 66.0%

      \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   2. fma-udef 52.3%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   3. cube-mult 52.2%

      \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   4. distribute-rgt1-in 66.0%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   5. associate-+r+ 66.0%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

   6. *-commutative 66.0%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

   7. frac-times 83.8%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   8. associate-/r* 99.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \cdot \frac{x}{x + \left(y + 1\right)} \]

   9. associate-*l/ 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot \frac{x}{x + \left(y + 1\right)}}{x + y}} \]

   10. associate-+r+ 99.8%

       \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

   11. +-commutative 99.8%

       \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{\left(y + x\right)} + 1}}{x + y} \]

   12. associate-+l+ 99.8%

       \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{y + \left(x + 1\right)}}}{x + y} \]

6. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot \frac{x}{y + \left(x + 1\right)}}{x + y}} \]
7. Step-by-step derivation

   1. associate-*r/ 99.8%

      \[\leadsto \frac{\color{blue}{\frac{\frac{y}{x + y} \cdot x}{y + \left(x + 1\right)}}}{x + y} \]

   2. +-commutative 99.8%

      \[\leadsto \frac{\frac{\frac{y}{\color{blue}{y + x}} \cdot x}{y + \left(x + 1\right)}}{x + y} \]

8. Applied egg-rr 99.8%

   \[\leadsto \frac{\color{blue}{\frac{\frac{y}{y + x} \cdot x}{y + \left(x + 1\right)}}}{x + y} \]

9. Final simplification 99.8%

   \[\leadsto \frac{\frac{x \cdot \frac{y}{y + x}}{y + \left(x + 1\right)}}{y + x} \]
10. Add Preprocessing

Alternative 14: 59.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.11 \cdot 10^{-128}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.11e-128) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ y 1.0)) (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.11e-128) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.11d-128)) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.11e-128) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.11e-128:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.11e-128)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.11e-128)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.11e-128], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1099999999999999e-128

   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. associate-/r* 73.5%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. *-commutative 73.5%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 73.5%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      4. +-commutative 73.5%

         \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      5. associate-*l/ 90.8%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 90.8%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 90.8%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 90.8%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 90.8%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 90.8%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 90.8%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 90.8%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 90.8%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 90.8%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 66.7%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 70.6%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 70.6%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   7. Simplified 70.6%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if -1.1099999999999999e-128 < x

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 80.2%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 80.2%

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1} \]

      3. +-commutative 80.2%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      4. +-commutative 80.2%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      5. times-frac 68.5%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. associate-*l/ 78.5%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

      7. *-commutative 78.5%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 78.5%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 69.4%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 78.5%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 78.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 78.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 78.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 78.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 78.5%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 68.5%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 59.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 59.5%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 68.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. associate-+r+ 68.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 68.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. frac-times 80.1%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

      8. associate-/r* 99.6%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \cdot \frac{x}{x + \left(y + 1\right)} \]

      9. associate-*l/ 99.8%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot \frac{x}{x + \left(y + 1\right)}}{x + y}} \]

      10. associate-+r+ 99.8%

          \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      11. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{\left(y + x\right)} + 1}}{x + y} \]

      12. associate-+l+ 99.8%

          \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{y + \left(x + 1\right)}}}{x + y} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot \frac{x}{y + \left(x + 1\right)}}{x + y}} \]

   7. Taylor expanded in x around 0 59.0%

      \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{x + y} \]
   8. Step-by-step derivation

      1. +-commutative 59.0%

         \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{x + y} \]

   9. Simplified 59.0%

      \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{x + y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.11 \cdot 10^{-128}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 59.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.11 \cdot 10^{-128}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.11e-128)
   (/ (/ y (+ x 1.0)) (+ y x))
   (/ (/ x (+ y 1.0)) (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.11e-128) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.11d-128)) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.11e-128) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.11e-128:
		tmp = (y / (x + 1.0)) / (y + x)
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.11e-128)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.11e-128)
		tmp = (y / (x + 1.0)) / (y + x);
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.11e-128], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1099999999999999e-128

   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. times-frac 90.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 90.9%

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1} \]

      3. +-commutative 90.9%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      4. +-commutative 90.9%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      5. times-frac 61.2%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. associate-*l/ 76.4%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

      7. *-commutative 76.4%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 76.4%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 40.5%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 76.4%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 76.4%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 76.4%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 76.4%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 76.4%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 76.4%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 61.2%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 38.3%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 38.3%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 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. associate-+r+ 61.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 61.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. frac-times 90.8%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

      8. associate-/r* 99.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \cdot \frac{x}{x + \left(y + 1\right)} \]

      9. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot \frac{x}{x + \left(y + 1\right)}}{x + y}} \]

      10. associate-+r+ 99.7%

          \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      11. +-commutative 99.7%

          \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{\left(y + x\right)} + 1}}{x + y} \]

      12. associate-+l+ 99.7%

          \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{y + \left(x + 1\right)}}}{x + y} \]

   6. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot \frac{x}{y + \left(x + 1\right)}}{x + y}} \]

   7. Taylor expanded in y around 0 70.8%

      \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]
   8. Step-by-step derivation

      1. +-commutative 70.8%

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

   9. Simplified 70.8%

      \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{x + y} \]

   if -1.1099999999999999e-128 < x

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 80.2%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 80.2%

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1} \]

      3. +-commutative 80.2%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      4. +-commutative 80.2%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      5. times-frac 68.5%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. associate-*l/ 78.5%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

      7. *-commutative 78.5%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 78.5%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 69.4%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 78.5%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 78.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 78.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 78.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 78.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 78.5%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 68.5%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 59.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 59.5%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 68.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. associate-+r+ 68.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 68.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. frac-times 80.1%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

      8. associate-/r* 99.6%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \cdot \frac{x}{x + \left(y + 1\right)} \]

      9. associate-*l/ 99.8%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot \frac{x}{x + \left(y + 1\right)}}{x + y}} \]

      10. associate-+r+ 99.8%

          \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      11. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{\left(y + x\right)} + 1}}{x + y} \]

      12. associate-+l+ 99.8%

          \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{y + \left(x + 1\right)}}}{x + y} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot \frac{x}{y + \left(x + 1\right)}}{x + y}} \]

   7. Taylor expanded in x around 0 59.0%

      \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{x + y} \]
   8. Step-by-step derivation

      1. +-commutative 59.0%

         \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{x + y} \]

   9. Simplified 59.0%

      \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{x + y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.11 \cdot 10^{-128}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 59.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.25e-129)
   (/ (/ y (+ y x)) (+ x 1.0))
   (/ (/ x (+ y 1.0)) (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.25e-129) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.25d-129)) then
        tmp = (y / (y + x)) / (x + 1.0d0)
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.25e-129) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.25e-129:
		tmp = (y / (y + x)) / (x + 1.0)
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.25e-129)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.25e-129)
		tmp = (y / (y + x)) / (x + 1.0);
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.25e-129], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.25000000000000015e-129

   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. times-frac 90.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 90.9%

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1} \]

      3. +-commutative 90.9%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      4. +-commutative 90.9%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      5. times-frac 61.2%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. associate-*l/ 76.4%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

      7. *-commutative 76.4%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 76.4%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 40.5%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 76.4%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 76.4%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 76.4%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 76.4%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 76.4%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 76.4%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 61.2%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 38.3%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 38.3%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 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. associate-+r+ 61.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 61.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. frac-times 90.8%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

      8. *-commutative 90.8%

         \[\leadsto \color{blue}{\frac{x}{x + \left(y + 1\right)} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      9. clear-num 90.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + \left(y + 1\right)}{x}}} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \]

      10. associate-/r* 99.7%

          \[\leadsto \frac{1}{\frac{x + \left(y + 1\right)}{x}} \cdot \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \]

      11. frac-times 99.1%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{y}{x + y}}{\frac{x + \left(y + 1\right)}{x} \cdot \left(x + y\right)}} \]

      12. *-un-lft-identity 99.1%

          \[\leadsto \frac{\color{blue}{\frac{y}{x + y}}}{\frac{x + \left(y + 1\right)}{x} \cdot \left(x + y\right)} \]

      13. associate-+r+ 99.1%

          \[\leadsto \frac{\frac{y}{x + y}}{\frac{\color{blue}{\left(x + y\right) + 1}}{x} \cdot \left(x + y\right)} \]

      14. +-commutative 99.1%

          \[\leadsto \frac{\frac{y}{x + y}}{\frac{\color{blue}{\left(y + x\right)} + 1}{x} \cdot \left(x + y\right)} \]

      15. associate-+l+ 99.1%

          \[\leadsto \frac{\frac{y}{x + y}}{\frac{\color{blue}{y + \left(x + 1\right)}}{x} \cdot \left(x + y\right)} \]

   6. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)}} \]

   7. Taylor expanded in y around 0 70.8%

      \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{1 + x}} \]
   8. Step-by-step derivation

      1. +-commutative 70.8%

         \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{x + 1}} \]

   9. Simplified 70.8%

      \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{x + 1}} \]

   if -2.25000000000000015e-129 < x

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 80.2%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 80.2%

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1} \]

      3. +-commutative 80.2%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      4. +-commutative 80.2%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      5. times-frac 68.5%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. associate-*l/ 78.5%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

      7. *-commutative 78.5%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 78.5%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 69.4%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 78.5%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 78.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 78.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 78.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 78.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 78.5%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 68.5%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 59.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 59.5%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 68.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. associate-+r+ 68.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 68.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. frac-times 80.1%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

      8. associate-/r* 99.6%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \cdot \frac{x}{x + \left(y + 1\right)} \]

      9. associate-*l/ 99.8%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot \frac{x}{x + \left(y + 1\right)}}{x + y}} \]

      10. associate-+r+ 99.8%

          \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      11. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{\left(y + x\right)} + 1}}{x + y} \]

      12. associate-+l+ 99.8%

          \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{y + \left(x + 1\right)}}}{x + y} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot \frac{x}{y + \left(x + 1\right)}}{x + y}} \]

   7. Taylor expanded in x around 0 59.0%

      \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{x + y} \]
   8. Step-by-step derivation

      1. +-commutative 59.0%

         \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{x + y} \]

   9. Simplified 59.0%

      \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{x + y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 58.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.11 \cdot 10^{-128}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.11e-128) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.11e-128) {
		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 <= (-1.11d-128)) 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 <= -1.11e-128) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.11e-128:
		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 <= -1.11e-128)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.11e-128)
		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, -1.11e-128], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1099999999999999e-128

   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. associate-/r* 73.5%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. *-commutative 73.5%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 73.5%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      4. +-commutative 73.5%

         \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      5. associate-*l/ 90.8%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 90.8%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 90.8%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 90.8%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 90.8%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 90.8%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 90.8%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 90.8%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 90.8%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 90.8%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 66.7%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   if -1.1099999999999999e-128 < x

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 69.5%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. *-commutative 69.5%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 69.5%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      4. +-commutative 69.5%

         \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      5. associate-*l/ 80.2%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 80.2%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 80.1%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 80.1%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 80.1%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 80.1%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 80.1%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 80.1%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 80.1%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 80.1%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 56.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 56.7%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   7. Simplified 56.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.11 \cdot 10^{-128}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 26.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -2.55e-11) (/ 1.0 x) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.55e-11) {
		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 <= (-2.55d-11)) 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 <= -2.55e-11) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.55e-11:
		tmp = 1.0 / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.55e-11)
		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 <= -2.55e-11)
		tmp = 1.0 / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.55e-11], N[(1.0 / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.54999999999999992e-11

   1. Initial program 54.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 88.0%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 88.0%

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1} \]

      3. +-commutative 88.0%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      4. +-commutative 88.0%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      5. times-frac 54.4%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. associate-*l/ 71.6%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

      7. *-commutative 71.6%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 71.6%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 26.9%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 71.7%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 71.7%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 71.7%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 71.6%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 71.6%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 71.6%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 54.4%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 26.9%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 26.9%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 54.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. associate-+r+ 54.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 54.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. frac-times 88.0%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

      8. associate-/r* 99.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \cdot \frac{x}{x + \left(y + 1\right)} \]

      9. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot \frac{x}{x + \left(y + 1\right)}}{x + y}} \]

      10. associate-+r+ 99.7%

          \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      11. +-commutative 99.7%

          \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{\left(y + x\right)} + 1}}{x + y} \]

      12. associate-+l+ 99.7%

          \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{y + \left(x + 1\right)}}}{x + y} \]

   6. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot \frac{x}{y + \left(x + 1\right)}}{x + y}} \]

   7. Taylor expanded in x around inf 76.3%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \]

   8. Taylor expanded in y around inf 6.7%

      \[\leadsto \color{blue}{\frac{1}{x}} \]

   if -2.54999999999999992e-11 < x

   1. Initial program 70.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 71.0%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. *-commutative 71.0%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 71.0%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      4. +-commutative 71.0%

         \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      5. associate-*l/ 82.4%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 82.4%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 82.4%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 82.4%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 82.4%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 82.4%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 82.4%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 82.4%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 82.4%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 82.4%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 56.6%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 56.6%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   7. Simplified 56.6%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   8. Taylor expanded in y around 0 35.9%

      \[\leadsto \frac{x}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 47.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y \cdot \left(y + 1\right)} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ x (* y (+ y 1.0))))

C

double code(double x, double y) {
	return x / (y * (y + 1.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / (y * (y + 1.0d0))
end function

Java

public static double code(double x, double y) {
	return x / (y * (y + 1.0));
}

Python

def code(x, y):
	return x / (y * (y + 1.0))

Julia

function code(x, y)
	return Float64(x / Float64(y * Float64(y + 1.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = x / (y * (y + 1.0));
end

Wolfram

code[x_, y_] := N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.0%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 70.9%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

   2. *-commutative 70.9%

      \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

   3. +-commutative 70.9%

      \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

   4. +-commutative 70.9%

      \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

   5. associate-*l/ 83.8%

      \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

   6. +-commutative 83.8%

      \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

   7. associate-*r/ 83.8%

      \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

   8. remove-double-neg 83.8%

      \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

   9. +-commutative 83.8%

      \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

   10. +-commutative 83.8%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

   11. remove-double-neg 83.8%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

   12. +-commutative 83.8%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

   13. associate-+l+ 83.8%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

3. Simplified 83.8%

   \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 47.4%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
6. Step-by-step derivation

   1. +-commutative 47.4%

      \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

7. Simplified 47.4%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

8. Final simplification 47.4%

   \[\leadsto \frac{x}{y \cdot \left(y + 1\right)} \]
9. Add Preprocessing

Alternative 20: 3.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{-0.25}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ -0.25 y))

C

double code(double x, double y) {
	return -0.25 / y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (-0.25d0) / y
end function

Java

public static double code(double x, double y) {
	return -0.25 / y;
}

Python

def code(x, y):
	return -0.25 / y

Julia

function code(x, y)
	return Float64(-0.25 / y)
end

MATLAB

function tmp = code(x, y)
	tmp = -0.25 / y;
end

Wolfram

code[x_, y_] := N[(-0.25 / y), $MachinePrecision]

Derivation

1. Initial program 66.0%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 70.9%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

   2. *-commutative 70.9%

      \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

   3. +-commutative 70.9%

      \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

   4. +-commutative 70.9%

      \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

   5. associate-*l/ 83.8%

      \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

   6. +-commutative 83.8%

      \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

   7. associate-*r/ 83.8%

      \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

   8. remove-double-neg 83.8%

      \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

   9. +-commutative 83.8%

      \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

   10. +-commutative 83.8%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

   11. remove-double-neg 83.8%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

   12. +-commutative 83.8%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

   13. associate-+l+ 83.8%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

3. Simplified 83.8%

   \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 52.9%

   \[\leadsto \frac{y}{\color{blue}{2 \cdot \left(x \cdot y\right) + {x}^{2}}} \cdot \frac{x}{x + \left(y + 1\right)} \]

6. Taylor expanded in y around inf 14.6%

   \[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{x} - 0.25 \cdot \frac{1}{y}\right)} \cdot \frac{x}{x + \left(y + 1\right)} \]
7. Step-by-step derivation

   1. associate-*r/ 14.6%

      \[\leadsto \left(\color{blue}{\frac{0.5 \cdot 1}{x}} - 0.25 \cdot \frac{1}{y}\right) \cdot \frac{x}{x + \left(y + 1\right)} \]

   2. metadata-eval 14.6%

      \[\leadsto \left(\frac{\color{blue}{0.5}}{x} - 0.25 \cdot \frac{1}{y}\right) \cdot \frac{x}{x + \left(y + 1\right)} \]

   3. associate-*r/ 14.6%

      \[\leadsto \left(\frac{0.5}{x} - \color{blue}{\frac{0.25 \cdot 1}{y}}\right) \cdot \frac{x}{x + \left(y + 1\right)} \]

   4. metadata-eval 14.6%

      \[\leadsto \left(\frac{0.5}{x} - \frac{\color{blue}{0.25}}{y}\right) \cdot \frac{x}{x + \left(y + 1\right)} \]

8. Simplified 14.6%

   \[\leadsto \color{blue}{\left(\frac{0.5}{x} - \frac{0.25}{y}\right)} \cdot \frac{x}{x + \left(y + 1\right)} \]

9. Taylor expanded in x around inf 3.8%

   \[\leadsto \color{blue}{\frac{-0.25}{y}} \]

10. Final simplification 3.8%

    \[\leadsto \frac{-0.25}{y} \]
11. Add Preprocessing

Alternative 21: 4.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 0.5 y))

C

double code(double x, double y) {
	return 0.5 / y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.5d0 / y
end function

Java

public static double code(double x, double y) {
	return 0.5 / y;
}

Python

def code(x, y):
	return 0.5 / y

Julia

function code(x, y)
	return Float64(0.5 / y)
end

MATLAB

function tmp = code(x, y)
	tmp = 0.5 / y;
end

Wolfram

code[x_, y_] := N[(0.5 / y), $MachinePrecision]

Derivation

1. Initial program 66.0%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 70.9%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

   2. *-commutative 70.9%

      \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

   3. +-commutative 70.9%

      \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

   4. +-commutative 70.9%

      \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

   5. associate-*l/ 83.8%

      \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

   6. +-commutative 83.8%

      \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

   7. associate-*r/ 83.8%

      \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

   8. remove-double-neg 83.8%

      \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

   9. +-commutative 83.8%

      \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

   10. +-commutative 83.8%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

   11. remove-double-neg 83.8%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

   12. +-commutative 83.8%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

   13. associate-+l+ 83.8%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

3. Simplified 83.8%

   \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 52.9%

   \[\leadsto \frac{y}{\color{blue}{2 \cdot \left(x \cdot y\right) + {x}^{2}}} \cdot \frac{x}{x + \left(y + 1\right)} \]

6. Taylor expanded in y around inf 4.1%

   \[\leadsto \color{blue}{\frac{0.5}{y}} \]

7. Final simplification 4.1%

   \[\leadsto \frac{0.5}{y} \]
8. Add Preprocessing

Alternative 22: 4.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 x))

C

double code(double x, double y) {
	return 1.0 / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / x
end function

Java

public static double code(double x, double y) {
	return 1.0 / x;
}

Python

def code(x, y):
	return 1.0 / x

Julia

function code(x, y)
	return Float64(1.0 / x)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / x;
end

Wolfram

code[x_, y_] := N[(1.0 / x), $MachinePrecision]

Derivation

1. Initial program 66.0%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. times-frac 83.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. +-commutative 83.9%

      \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1} \]

   3. +-commutative 83.9%

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

   4. +-commutative 83.9%

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   5. times-frac 66.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

   6. associate-*l/ 77.7%

      \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

   7. *-commutative 77.7%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

   8. *-commutative 77.7%

      \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   9. distribute-rgt1-in 59.5%

      \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   10. fma-def 77.8%

       \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

   11. +-commutative 77.8%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   12. +-commutative 77.8%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   13. cube-unmult 77.8%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

   14. +-commutative 77.8%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

3. Simplified 77.8%

   \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r/ 66.0%

      \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   2. fma-udef 52.3%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   3. cube-mult 52.2%

      \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   4. distribute-rgt1-in 66.0%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   5. associate-+r+ 66.0%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

   6. *-commutative 66.0%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

   7. frac-times 83.8%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   8. associate-/r* 99.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \cdot \frac{x}{x + \left(y + 1\right)} \]

   9. associate-*l/ 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot \frac{x}{x + \left(y + 1\right)}}{x + y}} \]

   10. associate-+r+ 99.8%

       \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

   11. +-commutative 99.8%

       \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{\left(y + x\right)} + 1}}{x + y} \]

   12. associate-+l+ 99.8%

       \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{y + \left(x + 1\right)}}}{x + y} \]

6. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot \frac{x}{y + \left(x + 1\right)}}{x + y}} \]

7. Taylor expanded in x around inf 37.9%

   \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \]

8. Taylor expanded in y around inf 4.6%

   \[\leadsto \color{blue}{\frac{1}{x}} \]

9. Final simplification 4.6%

   \[\leadsto \frac{1}{x} \]
10. Add Preprocessing

Developer target: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x)))))

C

double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x / ((y + 1.0d0) + x)) / (y + x)) / (1.0d0 / (y / (y + x)))
end function

Java

public static double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Python

def code(x, y):
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)))

Julia

function code(x, y)
	return Float64(Float64(Float64(x / Float64(Float64(y + 1.0) + x)) / Float64(y + x)) / Float64(1.0 / Float64(y / Float64(y + x))))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
end

Wolfram

code[x_, y_] := N[(N[(N[(x / N[(N[(y + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024036 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))