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

Percentage Accurate: 68.7% → 99.8%

Time: 18.0s

Alternatives: 19

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% 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 68.6%

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

   1. times-frac 87.9%

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

   2. +-commutative 87.9%

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

   3. +-commutative 87.9%

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

   4. +-commutative 87.9%

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

   5. times-frac 68.6%

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

   6. associate-*l/ 79.6%

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

   7. *-commutative 79.6%

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

   8. *-commutative 79.6%

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

   9. distribute-rgt1-in 61.4%

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

   10. fma-def 79.6%

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

   11. +-commutative 79.6%

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

   12. +-commutative 79.6%

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

   13. cube-unmult 79.6%

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

   14. +-commutative 79.6%

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

3. Simplified 79.6%

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

   1. associate-*r/ 68.7%

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

   2. fma-udef 56.2%

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

   3. cube-mult 56.2%

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

   4. distribute-rgt1-in 68.6%

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

   5. associate-+r+ 68.6%

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

   6. *-commutative 68.6%

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

   7. frac-times 87.8%

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

   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 2: 66.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{\frac{y}{y + x}}{y + x}\\ \mathbf{if}\;y \leq -2 \cdot 10^{-294}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{-131}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-17}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{1}{x + \left(y + \left(x + 1\right)\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (/ (/ y (+ y x)) (+ y x)))))
   (if (<= y -2e-294)
     (/ (/ y (+ x 1.0)) (+ y x))
     (if (<= y 4.1e-131)
       t_0
       (if (<= y 5.8e-62)
         (/ (/ y x) (+ x 1.0))
         (if (<= y 7e-17)
           t_0
           (* x (/ (/ 1.0 (+ x (+ y (+ x 1.0)))) (+ y x)))))))))

C

double code(double x, double y) {
	double t_0 = x * ((y / (y + x)) / (y + x));
	double tmp;
	if (y <= -2e-294) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 4.1e-131) {
		tmp = t_0;
	} else if (y <= 5.8e-62) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 7e-17) {
		tmp = t_0;
	} else {
		tmp = x * ((1.0 / (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 = x * ((y / (y + x)) / (y + x))
    if (y <= (-2d-294)) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else if (y <= 4.1d-131) then
        tmp = t_0
    else if (y <= 5.8d-62) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 7d-17) then
        tmp = t_0
    else
        tmp = x * ((1.0d0 / (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 = x * ((y / (y + x)) / (y + x));
	double tmp;
	if (y <= -2e-294) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 4.1e-131) {
		tmp = t_0;
	} else if (y <= 5.8e-62) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 7e-17) {
		tmp = t_0;
	} else {
		tmp = x * ((1.0 / (x + (y + (x + 1.0)))) / (y + x));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * ((y / (y + x)) / (y + x))
	tmp = 0
	if y <= -2e-294:
		tmp = (y / (x + 1.0)) / (y + x)
	elif y <= 4.1e-131:
		tmp = t_0
	elif y <= 5.8e-62:
		tmp = (y / x) / (x + 1.0)
	elif y <= 7e-17:
		tmp = t_0
	else:
		tmp = x * ((1.0 / (x + (y + (x + 1.0)))) / (y + x))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(Float64(y / Float64(y + x)) / Float64(y + x)))
	tmp = 0.0
	if (y <= -2e-294)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	elseif (y <= 4.1e-131)
		tmp = t_0;
	elseif (y <= 5.8e-62)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 7e-17)
		tmp = t_0;
	else
		tmp = Float64(x * Float64(Float64(1.0 / Float64(x + Float64(y + Float64(x + 1.0)))) / Float64(y + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * ((y / (y + x)) / (y + x));
	tmp = 0.0;
	if (y <= -2e-294)
		tmp = (y / (x + 1.0)) / (y + x);
	elseif (y <= 4.1e-131)
		tmp = t_0;
	elseif (y <= 5.8e-62)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 7e-17)
		tmp = t_0;
	else
		tmp = x * ((1.0 / (x + (y + (x + 1.0)))) / (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2e-294], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.1e-131], t$95$0, If[LessEqual[y, 5.8e-62], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7e-17], t$95$0, N[(x * N[(N[(1.0 / N[(x + N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.00000000000000003e-294

   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 85.0%

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

      2. +-commutative 85.0%

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

      3. +-commutative 85.0%

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

      4. +-commutative 85.0%

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

      5. times-frac 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/ 75.8%

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

      7. *-commutative 75.8%

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

      8. *-commutative 75.8%

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

      9. distribute-rgt1-in 50.7%

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

      10. fma-def 75.8%

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

      11. +-commutative 75.8%

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

      12. +-commutative 75.8%

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

      13. cube-unmult 75.8%

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

      14. +-commutative 75.8%

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

   3. Simplified 75.8%

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

      1. associate-*r/ 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 47.0%

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

      3. cube-mult 47.0%

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

      4. distribute-rgt1-in 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 84.9%

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

      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 43.2%

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

      1. +-commutative 43.2%

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

   9. Simplified 43.2%

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

   if -2.00000000000000003e-294 < y < 4.1000000000000002e-131 or 5.79999999999999971e-62 < y < 7.0000000000000003e-17

   1. Initial program 75.8%

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

      1. associate-/r* 77.3%

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

      2. *-commutative 77.3%

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

      3. +-commutative 77.3%

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

      4. +-commutative 77.3%

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

      5. associate-*l/ 84.8%

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

      6. +-commutative 84.8%

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

      7. associate-*r/ 84.8%

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

      8. remove-double-neg 84.8%

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

      9. +-commutative 84.8%

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

      10. +-commutative 84.8%

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

      11. remove-double-neg 84.8%

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

      12. +-commutative 84.8%

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

      13. associate-+l+ 84.8%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in x around 0 78.7%

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

      1. +-commutative 78.7%

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

   7. Simplified 78.7%

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

   8. Taylor expanded in y around 0 78.7%

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

      1. associate-/r* 93.7%

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

      2. div-inv 93.8%

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

   10. Applied egg-rr 93.8%

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

       1. associate-*r/ 93.7%

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

       2. *-rgt-identity 93.7%

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

   12. Simplified 93.7%

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

   if 4.1000000000000002e-131 < y < 5.79999999999999971e-62

   1. Initial program 79.8%

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

      1. associate-/r* 90.9%

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

      2. *-commutative 90.9%

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

      3. +-commutative 90.9%

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

      4. +-commutative 90.9%

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

      5. associate-*l/ 99.4%

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

      6. +-commutative 99.4%

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

      7. associate-*r/ 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 y around 0 88.2%

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

      1. associate-/r* 88.4%

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

      2. +-commutative 88.4%

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

   7. Simplified 88.4%

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

   if 7.0000000000000003e-17 < y

   1. Initial program 70.9%

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

      1. times-frac 92.0%

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

      2. +-commutative 92.0%

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

      3. +-commutative 92.0%

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

      4. +-commutative 92.0%

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

      5. times-frac 70.9%

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

      6. associate-*l/ 79.8%

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

      7. *-commutative 79.8%

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

      8. *-commutative 79.8%

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

      9. distribute-rgt1-in 70.2%

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

      10. fma-def 79.9%

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

      11. +-commutative 79.9%

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

      12. +-commutative 79.9%

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

      13. cube-unmult 79.9%

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

      14. +-commutative 79.9%

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

   3. Simplified 79.9%

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

      1. associate-*r/ 71.0%

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

      2. fma-udef 65.6%

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

      3. cube-mult 65.6%

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

      4. distribute-rgt1-in 70.9%

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

      5. associate-+r+ 70.9%

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

      6. *-commutative 70.9%

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

      7. frac-times 92.0%

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

      8. associate-/r* 99.8%

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

      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. clear-num 99.8%

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

      2. frac-times 99.5%

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

      3. *-un-lft-identity 99.5%

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

      4. +-commutative 99.5%

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

   8. Applied egg-rr 99.5%

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

   9. Taylor expanded in y around -inf 76.5%

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

       1. mul-1-neg 76.5%

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

       2. unsub-neg 76.5%

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

       3. neg-mul-1 76.5%

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

       4. distribute-lft-in 76.5%

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

       5. metadata-eval 76.5%

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

       6. neg-mul-1 76.5%

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

       7. associate-+r+ 76.5%

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

       8. unsub-neg 76.5%

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

       9. +-commutative 76.5%

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

       10. unsub-neg 76.5%

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

   11. Simplified 76.5%

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

       1. div-inv 76.4%

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

       2. +-commutative 76.4%

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

       3. *-un-lft-identity 76.4%

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

       4. times-frac 86.1%

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

       5. /-rgt-identity 86.1%

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

       6. associate--r- 86.1%

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

   13. Applied egg-rr 86.1%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-294}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{-131}:\\ \;\;\;\;x \cdot \frac{\frac{y}{y + x}}{y + x}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-17}:\\ \;\;\;\;x \cdot \frac{\frac{y}{y + x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{1}{x + \left(y + \left(x + 1\right)\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{\frac{y}{y + x}}{y + x}\\ \mathbf{if}\;y \leq -2 \cdot 10^{-294}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-129}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 0.46:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (/ (/ y (+ y x)) (+ y x)))))
   (if (<= y -2e-294)
     (/ (/ y (+ x 1.0)) (+ y x))
     (if (<= y 1.2e-129)
       t_0
       (if (<= y 1.7e-62)
         (/ (/ y x) (+ x 1.0))
         (if (<= y 0.46) t_0 (* (/ y (* (+ y x) (+ y x))) (/ x y))))))))

C

double code(double x, double y) {
	double t_0 = x * ((y / (y + x)) / (y + x));
	double tmp;
	if (y <= -2e-294) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 1.2e-129) {
		tmp = t_0;
	} else if (y <= 1.7e-62) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 0.46) {
		tmp = t_0;
	} else {
		tmp = (y / ((y + x) * (y + x))) * (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((y / (y + x)) / (y + x))
    if (y <= (-2d-294)) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else if (y <= 1.2d-129) then
        tmp = t_0
    else if (y <= 1.7d-62) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 0.46d0) then
        tmp = t_0
    else
        tmp = (y / ((y + x) * (y + x))) * (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * ((y / (y + x)) / (y + x));
	double tmp;
	if (y <= -2e-294) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 1.2e-129) {
		tmp = t_0;
	} else if (y <= 1.7e-62) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 0.46) {
		tmp = t_0;
	} else {
		tmp = (y / ((y + x) * (y + x))) * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * ((y / (y + x)) / (y + x))
	tmp = 0
	if y <= -2e-294:
		tmp = (y / (x + 1.0)) / (y + x)
	elif y <= 1.2e-129:
		tmp = t_0
	elif y <= 1.7e-62:
		tmp = (y / x) / (x + 1.0)
	elif y <= 0.46:
		tmp = t_0
	else:
		tmp = (y / ((y + x) * (y + x))) * (x / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(Float64(y / Float64(y + x)) / Float64(y + x)))
	tmp = 0.0
	if (y <= -2e-294)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	elseif (y <= 1.2e-129)
		tmp = t_0;
	elseif (y <= 1.7e-62)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 0.46)
		tmp = t_0;
	else
		tmp = Float64(Float64(y / Float64(Float64(y + x) * Float64(y + x))) * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * ((y / (y + x)) / (y + x));
	tmp = 0.0;
	if (y <= -2e-294)
		tmp = (y / (x + 1.0)) / (y + x);
	elseif (y <= 1.2e-129)
		tmp = t_0;
	elseif (y <= 1.7e-62)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 0.46)
		tmp = t_0;
	else
		tmp = (y / ((y + x) * (y + x))) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2e-294], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e-129], t$95$0, If[LessEqual[y, 1.7e-62], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.46], t$95$0, N[(N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.00000000000000003e-294

   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 85.0%

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

      2. +-commutative 85.0%

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

      3. +-commutative 85.0%

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

      4. +-commutative 85.0%

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

      5. times-frac 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/ 75.8%

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

      7. *-commutative 75.8%

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

      8. *-commutative 75.8%

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

      9. distribute-rgt1-in 50.7%

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

      10. fma-def 75.8%

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

      11. +-commutative 75.8%

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

      12. +-commutative 75.8%

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

      13. cube-unmult 75.8%

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

      14. +-commutative 75.8%

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

   3. Simplified 75.8%

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

      1. associate-*r/ 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 47.0%

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

      3. cube-mult 47.0%

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

      4. distribute-rgt1-in 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 84.9%

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

      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 43.2%

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

      1. +-commutative 43.2%

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

   9. Simplified 43.2%

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

   if -2.00000000000000003e-294 < y < 1.19999999999999994e-129 or 1.69999999999999994e-62 < y < 0.46000000000000002

   1. Initial program 76.5%

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

      1. associate-/r* 78.0%

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

      2. *-commutative 78.0%

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

      3. +-commutative 78.0%

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

      4. +-commutative 78.0%

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

      5. associate-*l/ 85.3%

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

      6. +-commutative 85.3%

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

      7. associate-*r/ 85.3%

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

      8. remove-double-neg 85.3%

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

      9. +-commutative 85.3%

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

      10. +-commutative 85.3%

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

      11. remove-double-neg 85.3%

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

      12. +-commutative 85.3%

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

      13. associate-+l+ 85.3%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in x around 0 79.4%

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

      1. +-commutative 79.4%

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

   7. Simplified 79.4%

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

   8. Taylor expanded in y around 0 79.1%

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

      1. associate-/r* 93.6%

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

      2. div-inv 93.7%

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

   10. Applied egg-rr 93.7%

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

       1. associate-*r/ 93.6%

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

       2. *-rgt-identity 93.6%

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

   12. Simplified 93.6%

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

   if 1.19999999999999994e-129 < y < 1.69999999999999994e-62

   1. Initial program 79.8%

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

      1. associate-/r* 90.9%

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

      2. *-commutative 90.9%

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

      3. +-commutative 90.9%

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

      4. +-commutative 90.9%

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

      5. associate-*l/ 99.4%

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

      6. +-commutative 99.4%

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

      7. associate-*r/ 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 y around 0 88.2%

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

      1. associate-/r* 88.4%

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

      2. +-commutative 88.4%

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

   7. Simplified 88.4%

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

   if 0.46000000000000002 < y

   1. Initial program 70.1%

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

      1. associate-/r* 78.1%

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

      2. *-commutative 78.1%

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

      3. +-commutative 78.1%

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

      4. +-commutative 78.1%

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

      5. associate-*l/ 91.8%

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

      6. +-commutative 91.8%

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

      7. associate-*r/ 91.8%

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

      8. remove-double-neg 91.8%

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

      9. +-commutative 91.8%

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

      10. +-commutative 91.8%

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

      11. remove-double-neg 91.8%

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

      12. +-commutative 91.8%

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

      13. associate-+l+ 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in y around inf 85.5%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-294}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-129}:\\ \;\;\;\;x \cdot \frac{\frac{y}{y + x}}{y + x}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 0.46:\\ \;\;\;\;x \cdot \frac{\frac{y}{y + x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{\frac{y}{y + x}}{y + x}\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{-296}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-130}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (/ (/ y (+ y x)) (+ y x)))))
   (if (<= y -5.5e-296)
     (/ (/ y (+ x 1.0)) (+ y x))
     (if (<= y 2.1e-130)
       t_0
       (if (<= y 7.4e-68)
         (/ (/ y x) (+ x 1.0))
         (if (<= y 3.8e-13) t_0 (/ x (* y (+ y (+ x 1.0))))))))))

C

double code(double x, double y) {
	double t_0 = x * ((y / (y + x)) / (y + x));
	double tmp;
	if (y <= -5.5e-296) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 2.1e-130) {
		tmp = t_0;
	} else if (y <= 7.4e-68) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 3.8e-13) {
		tmp = t_0;
	} else {
		tmp = x / (y * (y + (x + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((y / (y + x)) / (y + x))
    if (y <= (-5.5d-296)) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else if (y <= 2.1d-130) then
        tmp = t_0
    else if (y <= 7.4d-68) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 3.8d-13) then
        tmp = t_0
    else
        tmp = x / (y * (y + (x + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * ((y / (y + x)) / (y + x));
	double tmp;
	if (y <= -5.5e-296) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 2.1e-130) {
		tmp = t_0;
	} else if (y <= 7.4e-68) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 3.8e-13) {
		tmp = t_0;
	} else {
		tmp = x / (y * (y + (x + 1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * ((y / (y + x)) / (y + x))
	tmp = 0
	if y <= -5.5e-296:
		tmp = (y / (x + 1.0)) / (y + x)
	elif y <= 2.1e-130:
		tmp = t_0
	elif y <= 7.4e-68:
		tmp = (y / x) / (x + 1.0)
	elif y <= 3.8e-13:
		tmp = t_0
	else:
		tmp = x / (y * (y + (x + 1.0)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(Float64(y / Float64(y + x)) / Float64(y + x)))
	tmp = 0.0
	if (y <= -5.5e-296)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	elseif (y <= 2.1e-130)
		tmp = t_0;
	elseif (y <= 7.4e-68)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 3.8e-13)
		tmp = t_0;
	else
		tmp = Float64(x / Float64(y * Float64(y + Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * ((y / (y + x)) / (y + x));
	tmp = 0.0;
	if (y <= -5.5e-296)
		tmp = (y / (x + 1.0)) / (y + x);
	elseif (y <= 2.1e-130)
		tmp = t_0;
	elseif (y <= 7.4e-68)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 3.8e-13)
		tmp = t_0;
	else
		tmp = x / (y * (y + (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.5e-296], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.1e-130], t$95$0, If[LessEqual[y, 7.4e-68], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e-13], t$95$0, N[(x / N[(y * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.5000000000000004e-296

   1. Initial program 60.1%

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

      1. times-frac 83.5%

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

      2. +-commutative 83.5%

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

      3. +-commutative 83.5%

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

      4. +-commutative 83.5%

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

      5. times-frac 60.1%

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

      6. associate-*l/ 74.4%

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

      7. *-commutative 74.4%

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

      8. *-commutative 74.4%

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

      9. distribute-rgt1-in 49.9%

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

      10. fma-def 74.5%

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

      11. +-commutative 74.5%

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

      12. +-commutative 74.5%

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

      13. cube-unmult 74.5%

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

      14. +-commutative 74.5%

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

   3. Simplified 74.5%

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

      1. associate-*r/ 60.1%

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

      2. fma-udef 46.1%

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

      3. cube-mult 46.1%

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

      4. distribute-rgt1-in 60.1%

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

      5. associate-+r+ 60.1%

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

      6. *-commutative 60.1%

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

      7. frac-times 83.4%

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

      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 42.6%

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

      1. +-commutative 42.6%

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

   9. Simplified 42.6%

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

   if -5.5000000000000004e-296 < y < 2.10000000000000002e-130 or 7.40000000000000004e-68 < y < 3.8e-13

   1. Initial program 78.7%

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

      1. associate-/r* 80.2%

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

      2. *-commutative 80.2%

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

      3. +-commutative 80.2%

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

      4. +-commutative 80.2%

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

      5. associate-*l/ 87.8%

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

      6. +-commutative 87.8%

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

      7. associate-*r/ 87.8%

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

      8. remove-double-neg 87.8%

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

      9. +-commutative 87.8%

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

      10. +-commutative 87.8%

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

      11. remove-double-neg 87.8%

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

      12. +-commutative 87.8%

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

      13. associate-+l+ 87.8%

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

   3. Simplified 87.8%

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

   5. Taylor expanded in x around 0 81.6%

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

      1. +-commutative 81.6%

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

   7. Simplified 81.6%

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

   8. Taylor expanded in y around 0 81.6%

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

      1. associate-/r* 93.6%

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

      2. div-inv 93.7%

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

   10. Applied egg-rr 93.7%

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

       1. associate-*r/ 93.6%

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

       2. *-rgt-identity 93.6%

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

   12. Simplified 93.6%

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

   if 2.10000000000000002e-130 < y < 7.40000000000000004e-68

   1. Initial program 79.8%

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

      1. associate-/r* 90.9%

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

      2. *-commutative 90.9%

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

      3. +-commutative 90.9%

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

      4. +-commutative 90.9%

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

      5. associate-*l/ 99.4%

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

      6. +-commutative 99.4%

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

      7. associate-*r/ 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 y around 0 88.2%

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

      1. associate-/r* 88.4%

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

      2. +-commutative 88.4%

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

   7. Simplified 88.4%

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

   if 3.8e-13 < y

   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* 78.4%

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

      2. *-commutative 78.4%

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

      3. +-commutative 78.4%

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

      4. +-commutative 78.4%

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

      5. associate-*l/ 91.8%

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

      6. +-commutative 91.8%

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

      7. associate-*r/ 91.9%

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

      8. remove-double-neg 91.9%

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

      9. +-commutative 91.9%

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

      10. +-commutative 91.9%

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

      11. remove-double-neg 91.9%

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

      12. +-commutative 91.9%

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

      13. associate-+l+ 91.9%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in y around inf 76.5%

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

      1. expm1-log1p-u 76.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{y} \cdot \frac{x}{x + \left(y + 1\right)}\right)\right)} \]

      2. expm1-udef 60.2%

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

      3. frac-times 60.2%

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

      4. *-un-lft-identity 60.2%

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

      5. associate-+r+ 60.2%

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

      6. +-commutative 60.2%

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

      7. associate-+r+ 60.2%

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

   7. Applied egg-rr 60.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{y \cdot \left(y + \left(x + 1\right)\right)}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 77.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{y \cdot \left(y + \left(x + 1\right)\right)}\right)\right)} \]

      2. expm1-log1p 77.7%

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

   9. Simplified 77.7%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-296}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-130}:\\ \;\;\;\;x \cdot \frac{\frac{y}{y + x}}{y + x}\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \frac{\frac{y}{y + x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-294}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-130}:\\ \;\;\;\;x \cdot \frac{\frac{y}{y + x}}{y + x}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.1e-294)
   (/ (/ y (+ x 1.0)) (+ y x))
   (if (<= y 1.6e-130)
     (* x (/ (/ y (+ y x)) (+ y x)))
     (if (<= y 9e-67)
       (/ (/ y x) (+ x 1.0))
       (* (/ y (* (+ y x) (+ y x))) (/ x (+ y 1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.1e-294) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 1.6e-130) {
		tmp = x * ((y / (y + x)) / (y + x));
	} else if (y <= 9e-67) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (y / ((y + x) * (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 <= (-2.1d-294)) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else if (y <= 1.6d-130) then
        tmp = x * ((y / (y + x)) / (y + x))
    else if (y <= 9d-67) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (y / ((y + x) * (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 <= -2.1e-294) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 1.6e-130) {
		tmp = x * ((y / (y + x)) / (y + x));
	} else if (y <= 9e-67) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (y / ((y + x) * (y + x))) * (x / (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.1e-294:
		tmp = (y / (x + 1.0)) / (y + x)
	elif y <= 1.6e-130:
		tmp = x * ((y / (y + x)) / (y + x))
	elif y <= 9e-67:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (y / ((y + x) * (y + x))) * (x / (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.1e-294)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	elseif (y <= 1.6e-130)
		tmp = Float64(x * Float64(Float64(y / Float64(y + x)) / Float64(y + x)));
	elseif (y <= 9e-67)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(y / Float64(Float64(y + x) * Float64(y + x))) * Float64(x / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.1e-294)
		tmp = (y / (x + 1.0)) / (y + x);
	elseif (y <= 1.6e-130)
		tmp = x * ((y / (y + x)) / (y + x));
	elseif (y <= 9e-67)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (y / ((y + x) * (y + x))) * (x / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.1e-294], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.6e-130], N[(x * N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9e-67], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.09999999999999984e-294

   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 85.0%

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

      2. +-commutative 85.0%

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

      3. +-commutative 85.0%

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

      4. +-commutative 85.0%

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

      5. times-frac 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/ 75.8%

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

      7. *-commutative 75.8%

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

      8. *-commutative 75.8%

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

      9. distribute-rgt1-in 50.7%

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

      10. fma-def 75.8%

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

      11. +-commutative 75.8%

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

      12. +-commutative 75.8%

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

      13. cube-unmult 75.8%

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

      14. +-commutative 75.8%

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

   3. Simplified 75.8%

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

      1. associate-*r/ 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 47.0%

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

      3. cube-mult 47.0%

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

      4. distribute-rgt1-in 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 84.9%

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

      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 43.2%

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

      1. +-commutative 43.2%

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

   9. Simplified 43.2%

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

   if -2.09999999999999984e-294 < y < 1.6e-130

   1. Initial program 72.4%

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

      1. associate-/r* 72.4%

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

      2. *-commutative 72.4%

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

      3. +-commutative 72.4%

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

      4. +-commutative 72.4%

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

      5. associate-*l/ 81.5%

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

      6. +-commutative 81.5%

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

      7. associate-*r/ 81.5%

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

      8. remove-double-neg 81.5%

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

      9. +-commutative 81.5%

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

      10. +-commutative 81.5%

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

      11. remove-double-neg 81.5%

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

      12. +-commutative 81.5%

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

      13. associate-+l+ 81.5%

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

   3. Simplified 81.5%

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

   5. Taylor expanded in x around 0 76.0%

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

      1. +-commutative 76.0%

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

   7. Simplified 76.0%

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

   8. Taylor expanded in y around 0 76.0%

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

      1. associate-/r* 94.3%

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

      2. div-inv 94.3%

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

   10. Applied egg-rr 94.3%

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

       1. associate-*r/ 94.3%

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

       2. *-rgt-identity 94.3%

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

   12. Simplified 94.3%

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

   if 1.6e-130 < y < 9.00000000000000031e-67

   1. Initial program 79.8%

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

      1. associate-/r* 90.9%

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

      2. *-commutative 90.9%

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

      3. +-commutative 90.9%

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

      4. +-commutative 90.9%

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

      5. associate-*l/ 99.4%

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

      6. +-commutative 99.4%

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

      7. associate-*r/ 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 y around 0 88.2%

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

      1. associate-/r* 88.4%

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

      2. +-commutative 88.4%

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

   7. Simplified 88.4%

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

   if 9.00000000000000031e-67 < y

   1. Initial program 73.6%

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

      1. associate-/r* 81.5%

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

      2. *-commutative 81.5%

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

      3. +-commutative 81.5%

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

      4. +-commutative 81.5%

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

      5. associate-*l/ 93.0%

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

      6. +-commutative 93.0%

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

      7. associate-*r/ 93.0%

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

      8. remove-double-neg 93.0%

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

      9. +-commutative 93.0%

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

      10. +-commutative 93.0%

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

      11. remove-double-neg 93.0%

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

      12. +-commutative 93.0%

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

      13. associate-+l+ 93.0%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in x around 0 86.6%

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

      1. +-commutative 86.6%

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

   7. Simplified 86.6%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-294}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-130}:\\ \;\;\;\;x \cdot \frac{\frac{y}{y + x}}{y + x}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot \left(y + x\right)}\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-59}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 3800000000:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y (+ y x)))))
   (if (<= y -8.5e-8)
     t_0
     (if (<= y 4.5e-59)
       (/ y x)
       (if (<= y 3800000000.0) (/ x (* y (+ y 1.0))) t_0)))))

C

double code(double x, double y) {
	double t_0 = x / (y * (y + x));
	double tmp;
	if (y <= -8.5e-8) {
		tmp = t_0;
	} else if (y <= 4.5e-59) {
		tmp = y / x;
	} else if (y <= 3800000000.0) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y * (y + x))
    if (y <= (-8.5d-8)) then
        tmp = t_0
    else if (y <= 4.5d-59) then
        tmp = y / x
    else if (y <= 3800000000.0d0) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y * (y + x));
	double tmp;
	if (y <= -8.5e-8) {
		tmp = t_0;
	} else if (y <= 4.5e-59) {
		tmp = y / x;
	} else if (y <= 3800000000.0) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y * (y + x))
	tmp = 0
	if y <= -8.5e-8:
		tmp = t_0
	elif y <= 4.5e-59:
		tmp = y / x
	elif y <= 3800000000.0:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * Float64(y + x)))
	tmp = 0.0
	if (y <= -8.5e-8)
		tmp = t_0;
	elseif (y <= 4.5e-59)
		tmp = Float64(y / x);
	elseif (y <= 3800000000.0)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y * (y + x));
	tmp = 0.0;
	if (y <= -8.5e-8)
		tmp = t_0;
	elseif (y <= 4.5e-59)
		tmp = y / x;
	elseif (y <= 3800000000.0)
		tmp = x / (y * (y + 1.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8.5e-8], t$95$0, If[LessEqual[y, 4.5e-59], N[(y / x), $MachinePrecision], If[LessEqual[y, 3800000000.0], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.49999999999999935e-8 or 3.8e9 < y

   1. Initial program 61.7%

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

      1. *-un-lft-identity 61.7%

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

      2. associate-+r+ 61.7%

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

      3. associate-*l* 61.7%

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

      4. times-frac 68.6%

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

      5. associate-+r+ 68.6%

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

      6. +-commutative 68.6%

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

      7. associate-+l+ 68.6%

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

   4. Applied egg-rr 68.6%

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

      1. associate-/l* 88.9%

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

      2. frac-times 86.7%

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

      3. *-un-lft-identity 86.7%

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

      4. +-commutative 86.7%

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

      5. +-commutative 86.7%

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

   6. Applied egg-rr 86.7%

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

   7. Taylor expanded in y around inf 79.4%

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

   if -8.49999999999999935e-8 < y < 4.50000000000000012e-59

   1. Initial program 72.7%

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

      1. times-frac 85.4%

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

      2. +-commutative 85.4%

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

      3. +-commutative 85.4%

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

      4. +-commutative 85.4%

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

      5. times-frac 72.7%

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

      6. associate-*l/ 83.0%

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

      7. *-commutative 83.0%

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

      8. *-commutative 83.0%

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

      9. distribute-rgt1-in 67.9%

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

      10. fma-def 83.1%

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

      11. +-commutative 83.1%

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

      12. +-commutative 83.1%

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

      13. cube-unmult 83.1%

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

      14. +-commutative 83.1%

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

   3. Simplified 83.1%

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

      1. associate-*r/ 72.7%

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

      2. fma-udef 57.6%

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

      3. cube-mult 57.6%

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

      4. distribute-rgt1-in 72.7%

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

      5. associate-+r+ 72.7%

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

      6. *-commutative 72.7%

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

      7. frac-times 85.3%

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

      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/ 99.9%

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

      10. associate-+r+ 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. clear-num 99.7%

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

      2. frac-times 99.7%

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

      3. *-un-lft-identity 99.7%

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

      4. +-commutative 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in y around 0 79.2%

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

       1. +-commutative 79.2%

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

   11. Simplified 79.2%

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

   12. Taylor expanded in x around 0 54.2%

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

   if 4.50000000000000012e-59 < y < 3.8e9

   1. Initial program 93.4%

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

      1. associate-/r* 99.4%

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

      2. *-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. +-commutative 99.4%

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

      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/ 99.5%

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

      8. remove-double-neg 99.5%

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

      9. +-commutative 99.5%

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

      10. +-commutative 99.5%

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

      11. remove-double-neg 99.5%

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

      12. +-commutative 99.5%

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

      13. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 28.3%

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

      1. +-commutative 28.3%

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

   7. Simplified 28.3%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + x\right)}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-59}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 3800000000:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{x + \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.4e-32)
   (/ (* (/ y (+ y x)) (/ x (+ x 1.0))) (+ y x))
   (* (/ y (* (+ y x) (+ y x))) (/ x (+ x (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.4e-32) {
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x);
	} else {
		tmp = (y / ((y + x) * (y + x))) * (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 <= 1.4d-32) then
        tmp = ((y / (y + x)) * (x / (x + 1.0d0))) / (y + x)
    else
        tmp = (y / ((y + x) * (y + x))) * (x / (x + (y + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.4e-32) {
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x);
	} else {
		tmp = (y / ((y + x) * (y + x))) * (x / (x + (y + 1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.4e-32:
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x)
	else:
		tmp = (y / ((y + x) * (y + x))) * (x / (x + (y + 1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.4e-32)
		tmp = Float64(Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(x + 1.0))) / Float64(y + x));
	else
		tmp = Float64(Float64(y / Float64(Float64(y + x) * Float64(y + x))) * Float64(x / Float64(x + Float64(y + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.4e-32)
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x);
	else
		tmp = (y / ((y + x) * (y + x))) * (x / (x + (y + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.4e-32], N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.3999999999999999e-32

   1. Initial program 67.9%

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

      1. times-frac 86.2%

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

      2. +-commutative 86.2%

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

      3. +-commutative 86.2%

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

      4. +-commutative 86.2%

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

      5. times-frac 67.9%

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

      6. associate-*l/ 79.8%

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

      7. *-commutative 79.8%

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

      8. *-commutative 79.8%

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

      9. distribute-rgt1-in 58.6%

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

      10. fma-def 79.8%

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

      11. +-commutative 79.8%

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

      12. +-commutative 79.8%

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

      13. cube-unmult 79.8%

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

      14. +-commutative 79.8%

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

   3. Simplified 79.8%

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

      1. associate-*r/ 67.9%

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

      2. fma-udef 53.1%

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

      3. cube-mult 53.0%

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

      4. distribute-rgt1-in 67.9%

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

      5. associate-+r+ 67.9%

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

      6. *-commutative 67.9%

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

      7. frac-times 86.1%

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

      8. associate-/r* 99.7%

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

      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 0 81.3%

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

      1. +-commutative 81.3%

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

   9. Simplified 81.3%

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

   if 1.3999999999999999e-32 < y

   1. Initial program 70.4%

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

      1. associate-/r* 79.3%

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

      2. *-commutative 79.3%

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

      3. +-commutative 79.3%

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

      4. +-commutative 79.3%

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

      5. associate-*l/ 92.2%

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

      6. +-commutative 92.2%

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

      7. associate-*r/ 92.2%

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

      8. remove-double-neg 92.2%

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

      9. +-commutative 92.2%

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

      10. +-commutative 92.2%

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

      11. remove-double-neg 92.2%

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

      12. +-commutative 92.2%

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

      13. associate-+l+ 92.2%

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

   3. Simplified 92.2%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{x + \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.05e-66)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 6.6e-13)
     (* x (/ y (* (+ y x) (+ y x))))
     (/ x (* y (+ y (+ x 1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.05e-66) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 6.6e-13) {
		tmp = x * (y / ((y + x) * (y + x)));
	} else {
		tmp = x / (y * (y + (x + 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 <= 1.05d-66) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 6.6d-13) then
        tmp = x * (y / ((y + x) * (y + x)))
    else
        tmp = x / (y * (y + (x + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.05e-66) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 6.6e-13) {
		tmp = x * (y / ((y + x) * (y + x)));
	} else {
		tmp = x / (y * (y + (x + 1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.05e-66:
		tmp = (y / x) / (x + 1.0)
	elif y <= 6.6e-13:
		tmp = x * (y / ((y + x) * (y + x)))
	else:
		tmp = x / (y * (y + (x + 1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.05e-66)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 6.6e-13)
		tmp = Float64(x * Float64(y / Float64(Float64(y + x) * Float64(y + x))));
	else
		tmp = Float64(x / Float64(y * Float64(y + Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.05e-66)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 6.6e-13)
		tmp = x * (y / ((y + x) * (y + x)));
	else
		tmp = x / (y * (y + (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.05e-66], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.6e-13], N[(x * N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.05e-66

   1. Initial program 66.3%

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

      1. associate-/r* 69.5%

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

      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/ 85.4%

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

      6. +-commutative 85.4%

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

      7. associate-*r/ 85.4%

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

      8. remove-double-neg 85.4%

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

      9. +-commutative 85.4%

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

      10. +-commutative 85.4%

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

      11. remove-double-neg 85.4%

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

      12. +-commutative 85.4%

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

      13. associate-+l+ 85.4%

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

   3. Simplified 85.4%

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

   5. Taylor expanded in y around 0 61.6%

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

      1. associate-/r* 60.6%

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

      2. +-commutative 60.6%

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

   7. Simplified 60.6%

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

   if 1.05e-66 < y < 6.6000000000000001e-13

   1. Initial program 92.0%

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

      1. associate-/r* 99.6%

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

      2. *-commutative 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. associate-*l/ 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-*r/ 99.7%

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

      8. remove-double-neg 99.7%

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

      9. +-commutative 99.7%

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

      10. +-commutative 99.7%

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

      11. remove-double-neg 99.7%

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

      12. +-commutative 99.7%

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

      13. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 91.7%

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

      1. +-commutative 91.7%

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

   7. Simplified 91.7%

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

   8. Taylor expanded in y around 0 91.7%

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

   if 6.6000000000000001e-13 < y

   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* 78.4%

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

      2. *-commutative 78.4%

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

      3. +-commutative 78.4%

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

      4. +-commutative 78.4%

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

      5. associate-*l/ 91.8%

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

      6. +-commutative 91.8%

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

      7. associate-*r/ 91.9%

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

      8. remove-double-neg 91.9%

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

      9. +-commutative 91.9%

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

      10. +-commutative 91.9%

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

      11. remove-double-neg 91.9%

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

      12. +-commutative 91.9%

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

      13. associate-+l+ 91.9%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in y around inf 76.5%

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

      1. expm1-log1p-u 76.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{y} \cdot \frac{x}{x + \left(y + 1\right)}\right)\right)} \]

      2. expm1-udef 60.2%

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

      3. frac-times 60.2%

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

      4. *-un-lft-identity 60.2%

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

      5. associate-+r+ 60.2%

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

      6. +-commutative 60.2%

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

      7. associate-+r+ 60.2%

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

   7. Applied egg-rr 60.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{y \cdot \left(y + \left(x + 1\right)\right)}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 77.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{y \cdot \left(y + \left(x + 1\right)\right)}\right)\right)} \]

      2. expm1-log1p 77.7%

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

   9. Simplified 77.7%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{1}{x + \left(y + \left(x + 1\right)\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7e-17)
   (/ (* (/ y (+ y x)) (/ x (+ x 1.0))) (+ y x))
   (* x (/ (/ 1.0 (+ x (+ y (+ x 1.0)))) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7e-17) {
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x);
	} else {
		tmp = x * ((1.0 / (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) :: tmp
    if (y <= 7d-17) then
        tmp = ((y / (y + x)) * (x / (x + 1.0d0))) / (y + x)
    else
        tmp = x * ((1.0d0 / (x + (y + (x + 1.0d0)))) / (y + x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7e-17) {
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x);
	} else {
		tmp = x * ((1.0 / (x + (y + (x + 1.0)))) / (y + x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7e-17:
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x)
	else:
		tmp = x * ((1.0 / (x + (y + (x + 1.0)))) / (y + x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7e-17)
		tmp = Float64(Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(x + 1.0))) / Float64(y + x));
	else
		tmp = Float64(x * Float64(Float64(1.0 / Float64(x + Float64(y + Float64(x + 1.0)))) / Float64(y + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7e-17)
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x);
	else
		tmp = x * ((1.0 / (x + (y + (x + 1.0)))) / (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7e-17], N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(1.0 / N[(x + N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.0000000000000003e-17

   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 86.3%

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

      2. +-commutative 86.3%

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

      3. +-commutative 86.3%

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

      4. +-commutative 86.3%

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

      5. times-frac 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/ 79.5%

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

      7. *-commutative 79.5%

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

      8. *-commutative 79.5%

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

      9. distribute-rgt1-in 58.0%

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

      10. fma-def 79.5%

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

      11. +-commutative 79.5%

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

      12. +-commutative 79.5%

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

      13. cube-unmult 79.5%

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

      14. +-commutative 79.5%

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

   3. Simplified 79.5%

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

      1. associate-*r/ 67.8%

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

      2. fma-udef 52.5%

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

      3. cube-mult 52.5%

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

      4. distribute-rgt1-in 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 86.2%

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

      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 0 81.5%

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

      1. +-commutative 81.5%

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

   9. Simplified 81.5%

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

   if 7.0000000000000003e-17 < y

   1. Initial program 70.9%

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

      1. times-frac 92.0%

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

      2. +-commutative 92.0%

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

      3. +-commutative 92.0%

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

      4. +-commutative 92.0%

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

      5. times-frac 70.9%

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

      6. associate-*l/ 79.8%

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

      7. *-commutative 79.8%

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

      8. *-commutative 79.8%

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

      9. distribute-rgt1-in 70.2%

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

      10. fma-def 79.9%

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

      11. +-commutative 79.9%

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

      12. +-commutative 79.9%

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

      13. cube-unmult 79.9%

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

      14. +-commutative 79.9%

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

   3. Simplified 79.9%

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

      1. associate-*r/ 71.0%

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

      2. fma-udef 65.6%

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

      3. cube-mult 65.6%

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

      4. distribute-rgt1-in 70.9%

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

      5. associate-+r+ 70.9%

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

      6. *-commutative 70.9%

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

      7. frac-times 92.0%

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

      8. associate-/r* 99.8%

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

      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. clear-num 99.8%

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

      2. frac-times 99.5%

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

      3. *-un-lft-identity 99.5%

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

      4. +-commutative 99.5%

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

   8. Applied egg-rr 99.5%

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

   9. Taylor expanded in y around -inf 76.5%

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

       1. mul-1-neg 76.5%

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

       2. unsub-neg 76.5%

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

       3. neg-mul-1 76.5%

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

       4. distribute-lft-in 76.5%

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

       5. metadata-eval 76.5%

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

       6. neg-mul-1 76.5%

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

       7. associate-+r+ 76.5%

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

       8. unsub-neg 76.5%

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

       9. +-commutative 76.5%

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

       10. unsub-neg 76.5%

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

   11. Simplified 76.5%

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

       1. div-inv 76.4%

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

       2. +-commutative 76.4%

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

       3. *-un-lft-identity 76.4%

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

       4. times-frac 86.1%

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

       5. /-rgt-identity 86.1%

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

       6. associate--r- 86.1%

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

   13. Applied egg-rr 86.1%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{1}{x + \left(y + \left(x + 1\right)\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-20} \lor \neg \left(y \leq 2.7 \cdot 10^{-59}\right):\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4e-20) (not (<= y 2.7e-59))) (/ x (* y (+ y 1.0))) (/ y x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4e-20) || !(y <= 2.7e-59)) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = y / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-4d-20)) .or. (.not. (y <= 2.7d-59))) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = y / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4e-20) || !(y <= 2.7e-59)) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = y / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4e-20) or not (y <= 2.7e-59):
		tmp = x / (y * (y + 1.0))
	else:
		tmp = y / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4e-20) || !(y <= 2.7e-59))
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(y / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4e-20) || ~((y <= 2.7e-59)))
		tmp = x / (y * (y + 1.0));
	else
		tmp = y / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4e-20], N[Not[LessEqual[y, 2.7e-59]], $MachinePrecision]], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.99999999999999978e-20 or 2.6999999999999999e-59 < y

   1. Initial program 65.6%

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

      1. associate-/r* 72.3%

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

      2. *-commutative 72.3%

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

      3. +-commutative 72.3%

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

      4. +-commutative 72.3%

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

      5. associate-*l/ 90.2%

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

      6. +-commutative 90.2%

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

      7. associate-*r/ 90.2%

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

      8. remove-double-neg 90.2%

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

      9. +-commutative 90.2%

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

      10. +-commutative 90.2%

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

      11. remove-double-neg 90.2%

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

      12. +-commutative 90.2%

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

      13. associate-+l+ 90.2%

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

   3. Simplified 90.2%

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

   5. Taylor expanded in x around 0 71.9%

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

      1. +-commutative 71.9%

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

   7. Simplified 71.9%

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

   if -3.99999999999999978e-20 < y < 2.6999999999999999e-59

   1. Initial program 72.2%

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

      1. times-frac 85.1%

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

      2. +-commutative 85.1%

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

      3. +-commutative 85.1%

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

      4. +-commutative 85.1%

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

      5. times-frac 72.2%

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

      6. associate-*l/ 82.8%

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

      7. *-commutative 82.8%

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

      8. *-commutative 82.8%

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

      9. distribute-rgt1-in 67.4%

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

      10. fma-def 82.8%

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

      11. +-commutative 82.8%

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

      12. +-commutative 82.8%

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

      13. cube-unmult 82.8%

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

      14. +-commutative 82.8%

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

   3. Simplified 82.8%

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

      1. associate-*r/ 72.2%

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

      2. fma-udef 56.9%

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

      3. cube-mult 56.8%

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

      4. distribute-rgt1-in 72.2%

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

      5. associate-+r+ 72.2%

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

      6. *-commutative 72.2%

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

      7. frac-times 85.0%

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

      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/ 99.9%

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

      10. associate-+r+ 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. clear-num 99.7%

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

      2. frac-times 99.7%

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

      3. *-un-lft-identity 99.7%

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

      4. +-commutative 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in y around 0 79.7%

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

       1. +-commutative 79.7%

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

   11. Simplified 79.7%

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

   12. Taylor expanded in x around 0 55.1%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-20} \lor \neg \left(y \leq 2.7 \cdot 10^{-59}\right):\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 44.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-59}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.76:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.7e-59)
   (/ y x)
   (if (<= y 0.76) (- (/ x y) x) (* (/ x y) (/ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.7e-59) {
		tmp = y / x;
	} else if (y <= 0.76) {
		tmp = (x / y) - x;
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.7d-59) then
        tmp = y / x
    else if (y <= 0.76d0) then
        tmp = (x / y) - x
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.7e-59) {
		tmp = y / x;
	} else if (y <= 0.76) {
		tmp = (x / y) - x;
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.7e-59:
		tmp = y / x
	elif y <= 0.76:
		tmp = (x / y) - x
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.7e-59)
		tmp = Float64(y / x);
	elseif (y <= 0.76)
		tmp = Float64(Float64(x / y) - x);
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.7e-59)
		tmp = y / x;
	elseif (y <= 0.76)
		tmp = (x / y) - x;
	else
		tmp = (x / y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.7e-59], N[(y / x), $MachinePrecision], If[LessEqual[y, 0.76], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.6999999999999999e-59

   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. times-frac 85.6%

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

      2. +-commutative 85.6%

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

      3. +-commutative 85.6%

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

      4. +-commutative 85.6%

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

      5. times-frac 66.6%

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

      6. associate-*l/ 79.0%

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

      7. *-commutative 79.0%

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

      8. *-commutative 79.0%

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

      9. distribute-rgt1-in 57.5%

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

      10. fma-def 79.0%

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

      11. +-commutative 79.0%

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

      12. +-commutative 79.0%

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

      13. cube-unmult 79.0%

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

      14. +-commutative 79.0%

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

   3. Simplified 79.0%

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

      1. associate-*r/ 66.7%

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

      2. fma-udef 51.8%

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

      3. cube-mult 51.7%

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

      4. distribute-rgt1-in 66.6%

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

      5. associate-+r+ 66.6%

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

      6. *-commutative 66.6%

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

      7. frac-times 85.5%

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

      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. clear-num 99.7%

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

      2. frac-times 99.5%

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

      3. *-un-lft-identity 99.5%

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

      4. +-commutative 99.5%

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

   8. Applied egg-rr 99.5%

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

   9. Taylor expanded in y around 0 62.0%

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

       1. +-commutative 62.0%

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

   11. Simplified 62.0%

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

   12. Taylor expanded in x around 0 37.6%

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

   if 2.6999999999999999e-59 < y < 0.76000000000000001

   1. Initial program 91.2%

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

      1. associate-/r* 99.4%

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

      2. *-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. +-commutative 99.4%

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

      5. associate-*l/ 99.6%

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

      6. +-commutative 99.6%

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

      7. associate-*r/ 99.6%

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

      8. remove-double-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. remove-double-neg 99.6%

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

      12. +-commutative 99.6%

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

      13. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 37.5%

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

      1. +-commutative 37.5%

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

   7. Simplified 37.5%

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

   8. Taylor expanded in y around 0 37.5%

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

      1. neg-mul-1 37.5%

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

      2. +-commutative 37.5%

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

      3. unsub-neg 37.5%

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

   10. Simplified 37.5%

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

   if 0.76000000000000001 < y

   1. Initial program 70.1%

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

      1. associate-/r* 78.1%

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

      2. *-commutative 78.1%

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

      3. +-commutative 78.1%

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

      4. +-commutative 78.1%

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

      5. associate-*l/ 91.8%

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

      6. +-commutative 91.8%

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

      7. associate-*r/ 91.8%

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

      8. remove-double-neg 91.8%

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

      9. +-commutative 91.8%

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

      10. +-commutative 91.8%

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

      11. remove-double-neg 91.8%

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

      12. +-commutative 91.8%

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

      13. associate-+l+ 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in y around inf 76.2%

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

   6. Taylor expanded in y around inf 75.6%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-59}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.76:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 99.8% accurate, 1.0× speedup

Math

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

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(y / Float64(y + x)) * Float64(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[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.6%

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

   1. associate-+r+ 68.6%

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

   2. *-commutative 68.6%

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

   3. frac-times 87.8%

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

   4. associate-*l/ 83.1%

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

   5. times-frac 99.8%

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

   6. associate-+r+ 99.8%

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

   7. +-commutative 99.8%

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

   8. associate-+l+ 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 13: 63.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3e-13) (/ (/ y x) (+ x 1.0)) (/ x (* y (+ y (+ x 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3e-13) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = x / (y * (y + (x + 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 <= 3d-13) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = x / (y * (y + (x + 1.0d0)))
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if y <= 3e-13:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = x / (y * (y + (x + 1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3e-13)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(x / Float64(y * Float64(y + Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3e-13)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = x / (y * (y + (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3e-13], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.99999999999999984e-13

   1. Initial program 67.9%

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

      1. associate-/r* 71.5%

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

      2. *-commutative 71.5%

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

      3. +-commutative 71.5%

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

      4. +-commutative 71.5%

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

      5. associate-*l/ 86.3%

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

      6. +-commutative 86.3%

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

      7. associate-*r/ 86.3%

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

      8. remove-double-neg 86.3%

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

      9. +-commutative 86.3%

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

      10. +-commutative 86.3%

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

      11. remove-double-neg 86.3%

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

      12. +-commutative 86.3%

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

      13. associate-+l+ 86.3%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in y around 0 62.5%

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

      1. associate-/r* 61.6%

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

      2. +-commutative 61.6%

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

   7. Simplified 61.6%

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

   if 2.99999999999999984e-13 < y

   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* 78.4%

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

      2. *-commutative 78.4%

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

      3. +-commutative 78.4%

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

      4. +-commutative 78.4%

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

      5. associate-*l/ 91.8%

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

      6. +-commutative 91.8%

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

      7. associate-*r/ 91.9%

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

      8. remove-double-neg 91.9%

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

      9. +-commutative 91.9%

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

      10. +-commutative 91.9%

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

      11. remove-double-neg 91.9%

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

      12. +-commutative 91.9%

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

      13. associate-+l+ 91.9%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in y around inf 76.5%

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

      1. expm1-log1p-u 76.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{y} \cdot \frac{x}{x + \left(y + 1\right)}\right)\right)} \]

      2. expm1-udef 60.2%

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

      3. frac-times 60.2%

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

      4. *-un-lft-identity 60.2%

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

      5. associate-+r+ 60.2%

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

      6. +-commutative 60.2%

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

      7. associate-+r+ 60.2%

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

   7. Applied egg-rr 60.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{y \cdot \left(y + \left(x + 1\right)\right)}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 77.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{y \cdot \left(y + \left(x + 1\right)\right)}\right)\right)} \]

      2. expm1-log1p 77.7%

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

   9. Simplified 77.7%

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

4. Final simplification 66.1%

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

Alternative 14: 62.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.5e+17) (/ y (* x (+ x 1.0))) (/ x (* y (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.5e+17) {
		tmp = y / (x * (x + 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 (y <= 2.5d+17) then
        tmp = y / (x * (x + 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 (y <= 2.5e+17) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + x));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[y, 2.5e+17], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.5e17

   1. Initial program 69.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 72.4%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. *-commutative 72.4%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 72.4%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      4. +-commutative 72.4%

         \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      5. associate-*l/ 86.7%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 86.7%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 86.7%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 86.7%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 86.7%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 86.7%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 86.7%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 86.7%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 86.7%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 86.7%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 63.3%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   if 2.5e17 < y

   1. Initial program 67.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 67.6%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(x \cdot y\right)}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. associate-+r+ 67.6%

         \[\leadsto \frac{1 \cdot \left(x \cdot y\right)}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      3. associate-*l* 67.6%

         \[\leadsto \frac{1 \cdot \left(x \cdot y\right)}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]

      4. times-frac 76.3%

         \[\leadsto \color{blue}{\frac{1}{x + y} \cdot \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      5. associate-+r+ 76.3%

         \[\leadsto \frac{1}{x + y} \cdot \frac{x \cdot y}{\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      6. +-commutative 76.3%

         \[\leadsto \frac{1}{x + y} \cdot \frac{x \cdot y}{\left(x + y\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      7. associate-+l+ 76.3%

         \[\leadsto \frac{1}{x + y} \cdot \frac{x \cdot y}{\left(x + y\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

   4. Applied egg-rr 76.3%

      \[\leadsto \color{blue}{\frac{1}{x + y} \cdot \frac{x \cdot y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-/l* 91.3%

         \[\leadsto \frac{1}{x + y} \cdot \color{blue}{\frac{x}{\frac{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}{y}}} \]

      2. frac-times 88.4%

         \[\leadsto \color{blue}{\frac{1 \cdot x}{\left(x + y\right) \cdot \frac{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}{y}}} \]

      3. *-un-lft-identity 88.4%

         \[\leadsto \frac{\color{blue}{x}}{\left(x + y\right) \cdot \frac{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}{y}} \]

      4. +-commutative 88.4%

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \frac{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}{y}} \]

      5. +-commutative 88.4%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \frac{\color{blue}{\left(y + x\right)} \cdot \left(y + \left(x + 1\right)\right)}{y}} \]

   6. Applied egg-rr 88.4%

      \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) \cdot \frac{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}{y}}} \]

   7. Taylor expanded in y around inf 84.3%

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.6e+16) (/ (/ y x) (+ x 1.0)) (/ x (* y (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.6e+16) {
		tmp = (y / x) / (x + 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 (y <= 2.6d+16) then
        tmp = (y / x) / (x + 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 (y <= 2.6e+16) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = x / (y * (y + x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.6e+16:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = x / (y * (y + x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.6e+16)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(x / Float64(y * Float64(y + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.6e+16)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = x / (y * (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.6e+16], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.6e16

   1. Initial program 69.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 72.4%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. *-commutative 72.4%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 72.4%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      4. +-commutative 72.4%

         \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      5. associate-*l/ 86.7%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 86.7%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 86.7%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 86.7%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 86.7%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 86.7%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 86.7%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 86.7%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 86.7%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 86.7%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 63.3%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 62.5%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 62.5%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   7. Simplified 62.5%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if 2.6e16 < y

   1. Initial program 67.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 67.6%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(x \cdot y\right)}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. associate-+r+ 67.6%

         \[\leadsto \frac{1 \cdot \left(x \cdot y\right)}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      3. associate-*l* 67.6%

         \[\leadsto \frac{1 \cdot \left(x \cdot y\right)}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]

      4. times-frac 76.3%

         \[\leadsto \color{blue}{\frac{1}{x + y} \cdot \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      5. associate-+r+ 76.3%

         \[\leadsto \frac{1}{x + y} \cdot \frac{x \cdot y}{\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      6. +-commutative 76.3%

         \[\leadsto \frac{1}{x + y} \cdot \frac{x \cdot y}{\left(x + y\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      7. associate-+l+ 76.3%

         \[\leadsto \frac{1}{x + y} \cdot \frac{x \cdot y}{\left(x + y\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

   4. Applied egg-rr 76.3%

      \[\leadsto \color{blue}{\frac{1}{x + y} \cdot \frac{x \cdot y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-/l* 91.3%

         \[\leadsto \frac{1}{x + y} \cdot \color{blue}{\frac{x}{\frac{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}{y}}} \]

      2. frac-times 88.4%

         \[\leadsto \color{blue}{\frac{1 \cdot x}{\left(x + y\right) \cdot \frac{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}{y}}} \]

      3. *-un-lft-identity 88.4%

         \[\leadsto \frac{\color{blue}{x}}{\left(x + y\right) \cdot \frac{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}{y}} \]

      4. +-commutative 88.4%

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \frac{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}{y}} \]

      5. +-commutative 88.4%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \frac{\color{blue}{\left(y + x\right)} \cdot \left(y + \left(x + 1\right)\right)}{y}} \]

   6. Applied egg-rr 88.4%

      \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) \cdot \frac{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}{y}}} \]

   7. Taylor expanded in y around inf 84.3%

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 33.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-59}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.7e-59) (/ y x) (/ x (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.7e-59) {
		tmp = y / x;
	} else {
		tmp = x / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.7d-59) then
        tmp = y / x
    else
        tmp = x / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.7e-59) {
		tmp = y / x;
	} else {
		tmp = x / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.7e-59:
		tmp = y / x
	else:
		tmp = x / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.7e-59)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.7e-59)
		tmp = y / x;
	else
		tmp = x / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.7e-59], N[(y / x), $MachinePrecision], N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.6999999999999999e-59

   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. times-frac 85.6%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 85.6%

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1} \]

      3. +-commutative 85.6%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      4. +-commutative 85.6%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      5. times-frac 66.6%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. associate-*l/ 79.0%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

      7. *-commutative 79.0%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 79.0%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 57.5%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 79.0%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 79.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 79.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 79.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 79.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 79.0%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 66.7%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 51.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 51.7%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 66.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. associate-+r+ 66.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 66.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. frac-times 85.5%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

      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. clear-num 99.7%

         \[\leadsto \frac{\color{blue}{\frac{1}{\frac{x + y}{y}}} \cdot \frac{x}{y + \left(x + 1\right)}}{x + y} \]

      2. frac-times 99.5%

         \[\leadsto \frac{\color{blue}{\frac{1 \cdot x}{\frac{x + y}{y} \cdot \left(y + \left(x + 1\right)\right)}}}{x + y} \]

      3. *-un-lft-identity 99.5%

         \[\leadsto \frac{\frac{\color{blue}{x}}{\frac{x + y}{y} \cdot \left(y + \left(x + 1\right)\right)}}{x + y} \]

      4. +-commutative 99.5%

         \[\leadsto \frac{\frac{x}{\frac{\color{blue}{y + x}}{y} \cdot \left(y + \left(x + 1\right)\right)}}{x + y} \]

   8. Applied egg-rr 99.5%

      \[\leadsto \frac{\color{blue}{\frac{x}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)}}}{x + y} \]

   9. Taylor expanded in y around 0 62.0%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   10. Step-by-step derivation

       1. +-commutative 62.0%

          \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   11. Simplified 62.0%

       \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]

   12. Taylor expanded in x around 0 37.6%

       \[\leadsto \color{blue}{\frac{y}{x}} \]

   if 2.6999999999999999e-59 < y

   1. Initial program 73.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 92.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 92.9%

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1} \]

      3. +-commutative 92.9%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      4. +-commutative 92.9%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      5. times-frac 73.0%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. associate-*l/ 80.9%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

      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 69.9%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 80.9%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 80.9%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 80.9%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 81.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 81.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 81.0%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 73.0%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 65.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 65.8%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 73.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. associate-+r+ 73.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 73.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. frac-times 92.9%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

      8. associate-/r* 99.8%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \cdot \frac{x}{x + \left(y + 1\right)} \]

      9. associate-*l/ 99.9%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot \frac{x}{x + \left(y + 1\right)}}{x + y}} \]

      10. associate-+r+ 99.9%

          \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      11. +-commutative 99.9%

          \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{\left(y + x\right)} + 1}}{x + y} \]

      12. associate-+l+ 99.9%

          \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{y + \left(x + 1\right)}}}{x + y} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot \frac{x}{y + \left(x + 1\right)}}{x + y}} \]

   7. Taylor expanded in y around 0 60.5%

      \[\leadsto \frac{\frac{y}{x + y} \cdot \color{blue}{\frac{x}{1 + x}}}{x + y} \]
   8. Step-by-step derivation

      1. +-commutative 60.5%

         \[\leadsto \frac{\frac{y}{x + y} \cdot \frac{x}{\color{blue}{x + 1}}}{x + y} \]

   9. Simplified 60.5%

      \[\leadsto \frac{\frac{y}{x + y} \cdot \color{blue}{\frac{x}{x + 1}}}{x + y} \]

   10. Taylor expanded in x around 0 33.4%

       \[\leadsto \frac{\color{blue}{x}}{x + y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-59}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 33.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.45 \cdot 10^{-58}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.45e-58) (/ y x) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.45e-58) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.45d-58) then
        tmp = y / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.45e-58) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.45e-58:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.45e-58)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.45e-58)
		tmp = y / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.45e-58], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.45000000000000015e-58

   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. times-frac 85.6%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 85.6%

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1} \]

      3. +-commutative 85.6%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      4. +-commutative 85.6%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      5. times-frac 66.6%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. associate-*l/ 79.0%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

      7. *-commutative 79.0%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 79.0%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 57.5%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 79.0%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 79.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 79.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 79.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 79.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 79.0%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 66.7%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 51.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 51.7%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 66.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. associate-+r+ 66.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 66.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. frac-times 85.5%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

      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. clear-num 99.7%

         \[\leadsto \frac{\color{blue}{\frac{1}{\frac{x + y}{y}}} \cdot \frac{x}{y + \left(x + 1\right)}}{x + y} \]

      2. frac-times 99.5%

         \[\leadsto \frac{\color{blue}{\frac{1 \cdot x}{\frac{x + y}{y} \cdot \left(y + \left(x + 1\right)\right)}}}{x + y} \]

      3. *-un-lft-identity 99.5%

         \[\leadsto \frac{\frac{\color{blue}{x}}{\frac{x + y}{y} \cdot \left(y + \left(x + 1\right)\right)}}{x + y} \]

      4. +-commutative 99.5%

         \[\leadsto \frac{\frac{x}{\frac{\color{blue}{y + x}}{y} \cdot \left(y + \left(x + 1\right)\right)}}{x + y} \]

   8. Applied egg-rr 99.5%

      \[\leadsto \frac{\color{blue}{\frac{x}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)}}}{x + y} \]

   9. Taylor expanded in y around 0 62.0%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   10. Step-by-step derivation

       1. +-commutative 62.0%

          \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   11. Simplified 62.0%

       \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]

   12. Taylor expanded in x around 0 37.6%

       \[\leadsto \color{blue}{\frac{y}{x}} \]

   if 2.45000000000000015e-58 < y

   1. Initial program 73.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 81.0%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. *-commutative 81.0%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 81.0%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      4. +-commutative 81.0%

         \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      5. associate-*l/ 92.8%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 92.8%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 92.9%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 92.9%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 92.9%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 92.9%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 92.9%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 92.9%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 92.9%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 92.9%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 69.4%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 69.4%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   7. Simplified 69.4%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   8. Taylor expanded in y around 0 33.1%

      \[\leadsto \frac{x}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.45 \cdot 10^{-58}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 4.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 0.5 x))

C

double code(double x, double y) {
	return 0.5 / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.5d0 / x
end function

Java

public static double code(double x, double y) {
	return 0.5 / x;
}

Python

def code(x, y):
	return 0.5 / x

Julia

function code(x, y)
	return Float64(0.5 / x)
end

MATLAB

function tmp = code(x, y)
	tmp = 0.5 / x;
end

Wolfram

code[x_, y_] := N[(0.5 / x), $MachinePrecision]

Derivation

1. Initial program 68.6%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. times-frac 87.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. +-commutative 87.9%

      \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1} \]

   3. +-commutative 87.9%

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

   4. +-commutative 87.9%

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   5. times-frac 68.6%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

   6. associate-*l/ 79.6%

      \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

   7. *-commutative 79.6%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

   8. *-commutative 79.6%

      \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   9. distribute-rgt1-in 61.4%

      \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   10. fma-def 79.6%

       \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

   11. +-commutative 79.6%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   12. +-commutative 79.6%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   13. cube-unmult 79.6%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

   14. +-commutative 79.6%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

3. Simplified 79.6%

   \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r/ 68.7%

      \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   2. fma-udef 56.2%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   3. cube-mult 56.2%

      \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   4. distribute-rgt1-in 68.6%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   5. associate-+r+ 68.6%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

   6. *-commutative 68.6%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

   7. frac-times 87.8%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   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. clear-num 99.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{x + y}{y}}} \cdot \frac{x}{y + \left(x + 1\right)}}{x + y} \]

   2. frac-times 99.5%

      \[\leadsto \frac{\color{blue}{\frac{1 \cdot x}{\frac{x + y}{y} \cdot \left(y + \left(x + 1\right)\right)}}}{x + y} \]

   3. *-un-lft-identity 99.5%

      \[\leadsto \frac{\frac{\color{blue}{x}}{\frac{x + y}{y} \cdot \left(y + \left(x + 1\right)\right)}}{x + y} \]

   4. +-commutative 99.5%

      \[\leadsto \frac{\frac{x}{\frac{\color{blue}{y + x}}{y} \cdot \left(y + \left(x + 1\right)\right)}}{x + y} \]

8. Applied egg-rr 99.5%

   \[\leadsto \frac{\color{blue}{\frac{x}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)}}}{x + y} \]

9. Taylor expanded in y around -inf 50.7%

   \[\leadsto \frac{\frac{x}{\color{blue}{y + -1 \cdot \left(-1 \cdot x + -1 \cdot \left(1 + x\right)\right)}}}{x + y} \]
10. Step-by-step derivation

    1. mul-1-neg 50.7%

       \[\leadsto \frac{\frac{x}{y + \color{blue}{\left(-\left(-1 \cdot x + -1 \cdot \left(1 + x\right)\right)\right)}}}{x + y} \]

    2. unsub-neg 50.7%

       \[\leadsto \frac{\frac{x}{\color{blue}{y - \left(-1 \cdot x + -1 \cdot \left(1 + x\right)\right)}}}{x + y} \]

    3. neg-mul-1 50.7%

       \[\leadsto \frac{\frac{x}{y - \left(\color{blue}{\left(-x\right)} + -1 \cdot \left(1 + x\right)\right)}}{x + y} \]

    4. distribute-lft-in 50.7%

       \[\leadsto \frac{\frac{x}{y - \left(\left(-x\right) + \color{blue}{\left(-1 \cdot 1 + -1 \cdot x\right)}\right)}}{x + y} \]

    5. metadata-eval 50.7%

       \[\leadsto \frac{\frac{x}{y - \left(\left(-x\right) + \left(\color{blue}{-1} + -1 \cdot x\right)\right)}}{x + y} \]

    6. neg-mul-1 50.7%

       \[\leadsto \frac{\frac{x}{y - \left(\left(-x\right) + \left(-1 + \color{blue}{\left(-x\right)}\right)\right)}}{x + y} \]

    7. associate-+r+ 50.7%

       \[\leadsto \frac{\frac{x}{y - \color{blue}{\left(\left(\left(-x\right) + -1\right) + \left(-x\right)\right)}}}{x + y} \]

    8. unsub-neg 50.7%

       \[\leadsto \frac{\frac{x}{y - \color{blue}{\left(\left(\left(-x\right) + -1\right) - x\right)}}}{x + y} \]

    9. +-commutative 50.7%

       \[\leadsto \frac{\frac{x}{y - \left(\color{blue}{\left(-1 + \left(-x\right)\right)} - x\right)}}{x + y} \]

    10. unsub-neg 50.7%

        \[\leadsto \frac{\frac{x}{y - \left(\color{blue}{\left(-1 - x\right)} - x\right)}}{x + y} \]

11. Simplified 50.7%

    \[\leadsto \frac{\frac{x}{\color{blue}{y - \left(\left(-1 - x\right) - x\right)}}}{x + y} \]

12. Taylor expanded in x around inf 4.2%

    \[\leadsto \color{blue}{\frac{0.5}{x}} \]

13. Final simplification 4.2%

    \[\leadsto \frac{0.5}{x} \]
14. Add Preprocessing

Alternative 19: 26.7% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ x y))

C

double code(double x, double y) {
	return x / y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / y
end function

Java

public static double code(double x, double y) {
	return x / y;
}

Python

def code(x, y):
	return x / y

Julia

function code(x, y)
	return Float64(x / y)
end

MATLAB

function tmp = code(x, y)
	tmp = x / y;
end

Wolfram

code[x_, y_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 68.6%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 73.4%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

   2. *-commutative 73.4%

      \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

   3. +-commutative 73.4%

      \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

   4. +-commutative 73.4%

      \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

   5. associate-*l/ 87.8%

      \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

   6. +-commutative 87.8%

      \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

   7. associate-*r/ 87.8%

      \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

   8. remove-double-neg 87.8%

      \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

   9. +-commutative 87.8%

      \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

   10. +-commutative 87.8%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

   11. remove-double-neg 87.8%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

   12. +-commutative 87.8%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

   13. associate-+l+ 87.8%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

3. Simplified 87.8%

   \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 48.8%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
6. Step-by-step derivation

   1. +-commutative 48.8%

      \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

7. Simplified 48.8%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

8. Taylor expanded in y around 0 24.7%

   \[\leadsto \frac{x}{\color{blue}{y}} \]

9. Final simplification 24.7%

   \[\leadsto \frac{x}{y} \]
10. Add Preprocessing

Developer target: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x)))))

C

double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x / ((y + 1.0d0) + x)) / (y + x)) / (1.0d0 / (y / (y + x)))
end function

Java

public static double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Python

def code(x, y):
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)))

Julia

function code(x, y)
	return Float64(Float64(Float64(x / Float64(Float64(y + 1.0) + x)) / Float64(y + x)) / Float64(1.0 / Float64(y / Float64(y + x))))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
end

Wolfram

code[x_, y_] := N[(N[(N[(x / N[(N[(y + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024020 
(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))))