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

Percentage Accurate: 69.2% → 99.8%

Time: 18.7s

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: 69.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.1%

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

   1. times-frac 84.4%

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

   2. +-commutative 84.4%

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

   3. +-commutative 84.4%

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

   4. +-commutative 84.4%

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

   5. times-frac 66.1%

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

   6. associate-*l/ 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 56.7%

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

   10. fma-def 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/ 66.1%

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

   2. fma-udef 48.7%

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

   3. cube-mult 48.7%

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

   4. distribute-rgt1-in 66.1%

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

   5. associate-+r+ 66.1%

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

   6. *-commutative 66.1%

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

   7. frac-times 84.4%

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

   8. associate-/r* 99.8%

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

   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. *-commutative 99.8%

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

   2. clear-num 99.8%

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

   3. un-div-inv 99.9%

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

   4. +-commutative 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 67.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + 1}\\ \mathbf{if}\;x \leq -650000:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-157}:\\ \;\;\;\;t_0 \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y 1.0))))
   (if (<= x -650000.0)
     (/ (/ y (+ x y)) (+ x (+ y (+ y 1.0))))
     (if (<= x -2.2e-157) (* t_0 (/ y (* (+ x y) (+ x y)))) (/ t_0 (+ x y))))))

C

double code(double x, double y) {
	double t_0 = x / (y + 1.0);
	double tmp;
	if (x <= -650000.0) {
		tmp = (y / (x + y)) / (x + (y + (y + 1.0)));
	} else if (x <= -2.2e-157) {
		tmp = t_0 * (y / ((x + y) * (x + y)));
	} else {
		tmp = t_0 / (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 + 1.0d0)
    if (x <= (-650000.0d0)) then
        tmp = (y / (x + y)) / (x + (y + (y + 1.0d0)))
    else if (x <= (-2.2d-157)) then
        tmp = t_0 * (y / ((x + y) * (x + y)))
    else
        tmp = t_0 / (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y + 1.0);
	double tmp;
	if (x <= -650000.0) {
		tmp = (y / (x + y)) / (x + (y + (y + 1.0)));
	} else if (x <= -2.2e-157) {
		tmp = t_0 * (y / ((x + y) * (x + y)));
	} else {
		tmp = t_0 / (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + 1.0)
	tmp = 0
	if x <= -650000.0:
		tmp = (y / (x + y)) / (x + (y + (y + 1.0)))
	elif x <= -2.2e-157:
		tmp = t_0 * (y / ((x + y) * (x + y)))
	else:
		tmp = t_0 / (x + y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + 1.0))
	tmp = 0.0
	if (x <= -650000.0)
		tmp = Float64(Float64(y / Float64(x + y)) / Float64(x + Float64(y + Float64(y + 1.0))));
	elseif (x <= -2.2e-157)
		tmp = Float64(t_0 * Float64(y / Float64(Float64(x + y) * Float64(x + y))));
	else
		tmp = Float64(t_0 / Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y + 1.0);
	tmp = 0.0;
	if (x <= -650000.0)
		tmp = (y / (x + y)) / (x + (y + (y + 1.0)));
	elseif (x <= -2.2e-157)
		tmp = t_0 * (y / ((x + y) * (x + y)));
	else
		tmp = t_0 / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -650000.0], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.2e-157], N[(t$95$0 * N[(y / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.5e5

   1. Initial program 54.5%

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

      1. times-frac 79.5%

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

      2. +-commutative 79.5%

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

      3. +-commutative 79.5%

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

      4. +-commutative 79.5%

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

      5. times-frac 54.5%

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

      6. associate-*l/ 69.8%

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

      7. *-commutative 69.8%

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

      8. *-commutative 69.8%

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

      9. distribute-rgt1-in 23.8%

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

      10. fma-def 69.8%

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

      11. +-commutative 69.8%

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

      12. +-commutative 69.8%

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

      13. cube-unmult 69.8%

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

      14. +-commutative 69.8%

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

   3. Simplified 69.8%

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

      1. associate-*r/ 54.5%

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

      2. fma-udef 20.4%

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

      3. cube-mult 20.4%

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

      4. distribute-rgt1-in 54.5%

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

      5. associate-+r+ 54.5%

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

      6. *-commutative 54.5%

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

      7. frac-times 79.6%

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

      8. *-commutative 79.6%

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

      9. clear-num 79.6%

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

      10. associate-/r* 99.8%

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

      11. frac-times 98.7%

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

      12. *-un-lft-identity 98.7%

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

      13. associate-+r+ 98.7%

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

      14. +-commutative 98.7%

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

      15. associate-+l+ 98.7%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in x around -inf 77.2%

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

      1. mul-1-neg 77.2%

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

      2. unsub-neg 77.2%

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

      3. neg-mul-1 77.2%

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

      4. +-commutative 77.2%

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

      5. +-commutative 77.2%

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

      6. neg-mul-1 77.2%

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

      7. unsub-neg 77.2%

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

      8. neg-mul-1 77.2%

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

      9. +-commutative 77.2%

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

      10. distribute-lft-in 77.2%

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

      11. metadata-eval 77.2%

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

      12. neg-mul-1 77.2%

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

      13. unsub-neg 77.2%

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

   9. Simplified 77.2%

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

   if -6.5e5 < x < -2.2000000000000001e-157

   1. Initial program 80.8%

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

      1. associate-/r* 80.7%

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

      2. *-commutative 80.7%

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

      3. +-commutative 80.7%

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

      4. +-commutative 80.7%

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

      5. associate-*l/ 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 98.5%

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

      1. +-commutative 98.5%

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

   7. Simplified 98.5%

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

   if -2.2000000000000001e-157 < x

   1. Initial program 67.5%

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

      1. times-frac 82.7%

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

      2. +-commutative 82.7%

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

      3. +-commutative 82.7%

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

      4. +-commutative 82.7%

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

      5. times-frac 67.5%

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

      6. associate-*l/ 79.4%

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

      7. *-commutative 79.4%

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

      8. *-commutative 79.4%

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

      9. distribute-rgt1-in 63.5%

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

      10. fma-def 79.4%

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

      11. +-commutative 79.4%

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

      12. +-commutative 79.4%

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

      13. cube-unmult 79.4%

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

      14. +-commutative 79.4%

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

   3. Simplified 79.4%

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

      1. associate-*r/ 67.5%

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

      2. fma-udef 55.7%

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

      3. cube-mult 55.6%

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

      4. distribute-rgt1-in 67.5%

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

      5. associate-+r+ 67.5%

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

      6. *-commutative 67.5%

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

      7. frac-times 82.6%

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

      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 x around 0 55.8%

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

      1. +-commutative 55.8%

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

   9. Simplified 55.8%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -650000:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-157}:\\ \;\;\;\;\frac{x}{y + 1} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.52 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x - y \cdot -2}\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-155}:\\ \;\;\;\;\frac{-y}{\left(x + y\right) \cdot \left(y \cdot -2 - \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.52e+154)
   (/ (/ y (+ x y)) (- x (* y -2.0)))
   (if (<= x -2.4e-155)
     (/ (- y) (* (+ x y) (- (* y -2.0) (+ x 1.0))))
     (/ (/ x (+ y 1.0)) (+ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.52e+154) {
		tmp = (y / (x + y)) / (x - (y * -2.0));
	} else if (x <= -2.4e-155) {
		tmp = -y / ((x + y) * ((y * -2.0) - (x + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) / (x + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.52d+154)) then
        tmp = (y / (x + y)) / (x - (y * (-2.0d0)))
    else if (x <= (-2.4d-155)) then
        tmp = -y / ((x + y) * ((y * (-2.0d0)) - (x + 1.0d0)))
    else
        tmp = (x / (y + 1.0d0)) / (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.52e+154) {
		tmp = (y / (x + y)) / (x - (y * -2.0));
	} else if (x <= -2.4e-155) {
		tmp = -y / ((x + y) * ((y * -2.0) - (x + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) / (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.52e+154:
		tmp = (y / (x + y)) / (x - (y * -2.0))
	elif x <= -2.4e-155:
		tmp = -y / ((x + y) * ((y * -2.0) - (x + 1.0)))
	else:
		tmp = (x / (y + 1.0)) / (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.52e+154)
		tmp = Float64(Float64(y / Float64(x + y)) / Float64(x - Float64(y * -2.0)));
	elseif (x <= -2.4e-155)
		tmp = Float64(Float64(-y) / Float64(Float64(x + y) * Float64(Float64(y * -2.0) - Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.52e+154)
		tmp = (y / (x + y)) / (x - (y * -2.0));
	elseif (x <= -2.4e-155)
		tmp = -y / ((x + y) * ((y * -2.0) - (x + 1.0)));
	else
		tmp = (x / (y + 1.0)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.52e+154], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x - N[(y * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.4e-155], N[((-y) / N[(N[(x + y), $MachinePrecision] * N[(N[(y * -2.0), $MachinePrecision] - N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5200000000000001e154

   1. Initial program 52.5%

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

      1. times-frac 75.3%

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

      2. +-commutative 75.3%

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

      3. +-commutative 75.3%

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

      4. +-commutative 75.3%

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

      5. times-frac 52.5%

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

      6. associate-*l/ 75.3%

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

      7. *-commutative 75.3%

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

      8. *-commutative 75.3%

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

      9. distribute-rgt1-in 5.6%

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

      10. fma-def 75.3%

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

      11. +-commutative 75.3%

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

      12. +-commutative 75.3%

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

      13. cube-unmult 75.3%

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

      14. +-commutative 75.3%

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

   3. Simplified 75.3%

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

      1. associate-*r/ 52.5%

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

      2. fma-udef 0.0%

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

      3. cube-mult 0.0%

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

      4. distribute-rgt1-in 52.5%

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

      5. associate-+r+ 52.5%

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

      6. *-commutative 52.5%

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

      7. frac-times 75.3%

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

      8. *-commutative 75.3%

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

      9. clear-num 75.3%

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

      10. associate-/r* 99.9%

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

      11. frac-times 99.8%

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

      12. *-un-lft-identity 99.8%

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

      13. associate-+r+ 99.8%

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

      14. +-commutative 99.8%

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

      15. associate-+l+ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around -inf 83.7%

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

      1. mul-1-neg 83.7%

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

      2. unsub-neg 83.7%

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

      3. neg-mul-1 83.7%

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

      4. +-commutative 83.7%

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

      5. +-commutative 83.7%

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

      6. neg-mul-1 83.7%

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

      7. unsub-neg 83.7%

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

      8. neg-mul-1 83.7%

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

      9. +-commutative 83.7%

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

      10. distribute-lft-in 83.7%

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

      11. metadata-eval 83.7%

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

      12. neg-mul-1 83.7%

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

      13. unsub-neg 83.7%

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

   9. Simplified 83.7%

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

   10. Taylor expanded in y around inf 83.7%

       \[\leadsto \frac{\frac{y}{x + y}}{x - \color{blue}{-2 \cdot y}} \]
   11. Step-by-step derivation

       1. *-commutative 83.7%

          \[\leadsto \frac{\frac{y}{x + y}}{x - \color{blue}{y \cdot -2}} \]

   12. Simplified 83.7%

       \[\leadsto \frac{\frac{y}{x + y}}{x - \color{blue}{y \cdot -2}} \]

   if -2.5200000000000001e154 < x < -2.4e-155

   1. Initial program 71.3%

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

      1. times-frac 94.1%

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

      2. +-commutative 94.1%

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

      3. +-commutative 94.1%

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

      4. +-commutative 94.1%

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

      5. times-frac 71.3%

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

      6. associate-*l/ 82.6%

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

      7. *-commutative 82.6%

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

      8. *-commutative 82.6%

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

      9. distribute-rgt1-in 72.4%

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

      10. fma-def 82.5%

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

      11. +-commutative 82.5%

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

      12. +-commutative 82.5%

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

      13. cube-unmult 82.5%

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

      14. +-commutative 82.5%

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

   3. Simplified 82.5%

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

      1. associate-*r/ 71.3%

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

      2. fma-udef 62.8%

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

      3. cube-mult 62.7%

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

      4. distribute-rgt1-in 71.3%

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

      5. associate-+r+ 71.3%

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

      6. *-commutative 71.3%

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

      7. frac-times 94.2%

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

      8. *-commutative 94.2%

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

      9. clear-num 94.1%

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

      10. associate-/r* 99.6%

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

      11. frac-times 98.5%

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

      12. *-un-lft-identity 98.5%

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

      13. associate-+r+ 98.5%

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

      14. +-commutative 98.5%

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

      15. associate-+l+ 98.5%

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

   6. Applied egg-rr 98.5%

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

   7. Taylor expanded in x around -inf 54.6%

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

      1. mul-1-neg 54.6%

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

      2. unsub-neg 54.6%

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

      3. neg-mul-1 54.6%

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

      4. +-commutative 54.6%

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

      5. +-commutative 54.6%

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

      6. neg-mul-1 54.6%

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

      7. unsub-neg 54.6%

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

      8. neg-mul-1 54.6%

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

      9. +-commutative 54.6%

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

      10. distribute-lft-in 54.6%

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

      11. metadata-eval 54.6%

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

      12. neg-mul-1 54.6%

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

      13. unsub-neg 54.6%

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

   9. Simplified 54.6%

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

       1. frac-2neg 54.6%

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

       2. div-inv 54.6%

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

       3. distribute-neg-frac 54.6%

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

       4. associate--l- 54.6%

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

       5. associate--r- 54.6%

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

       6. *-un-lft-identity 54.6%

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

       7. *-un-lft-identity 54.6%

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

       8. distribute-rgt-out 54.6%

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

       9. metadata-eval 54.6%

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

   11. Applied egg-rr 54.6%

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

       1. associate-*r/ 54.6%

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

       2. *-rgt-identity 54.6%

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

       3. distribute-frac-neg 54.6%

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

       4. distribute-neg-frac 54.6%

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

       5. associate-/l/ 72.7%

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

       6. distribute-neg-frac 72.7%

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

       7. sub-neg 72.7%

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

       8. metadata-eval 72.7%

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

       9. distribute-neg-in 72.7%

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

       10. +-commutative 72.7%

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

       11. distribute-neg-in 72.7%

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

       12. metadata-eval 72.7%

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

       13. unsub-neg 72.7%

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

       14. distribute-rgt-neg-in 72.7%

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

       15. metadata-eval 72.7%

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

   13. Simplified 72.7%

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

   if -2.4e-155 < x

   1. Initial program 67.5%

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

      1. times-frac 82.7%

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

      2. +-commutative 82.7%

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

      3. +-commutative 82.7%

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

      4. +-commutative 82.7%

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

      5. times-frac 67.5%

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

      6. associate-*l/ 79.4%

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

      7. *-commutative 79.4%

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

      8. *-commutative 79.4%

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

      9. distribute-rgt1-in 63.5%

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

      10. fma-def 79.4%

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

      11. +-commutative 79.4%

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

      12. +-commutative 79.4%

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

      13. cube-unmult 79.4%

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

      14. +-commutative 79.4%

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

   3. Simplified 79.4%

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

      1. associate-*r/ 67.5%

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

      2. fma-udef 55.7%

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

      3. cube-mult 55.6%

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

      4. distribute-rgt1-in 67.5%

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

      5. associate-+r+ 67.5%

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

      6. *-commutative 67.5%

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

      7. frac-times 82.6%

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

      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 x around 0 55.8%

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

      1. +-commutative 55.8%

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

   9. Simplified 55.8%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.52 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x - y \cdot -2}\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-155}:\\ \;\;\;\;\frac{-y}{\left(x + y\right) \cdot \left(y \cdot -2 - \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.07:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + 1}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-155}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.07)
   (/ (/ y (+ x y)) (+ x 1.0))
   (if (<= x -8e-155)
     (* x (/ y (* (+ x y) (+ x y))))
     (/ (/ x (+ y 1.0)) (+ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.07) {
		tmp = (y / (x + y)) / (x + 1.0);
	} else if (x <= -8e-155) {
		tmp = x * (y / ((x + y) * (x + y)));
	} else {
		tmp = (x / (y + 1.0)) / (x + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.07d0)) then
        tmp = (y / (x + y)) / (x + 1.0d0)
    else if (x <= (-8d-155)) then
        tmp = x * (y / ((x + y) * (x + y)))
    else
        tmp = (x / (y + 1.0d0)) / (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.07) {
		tmp = (y / (x + y)) / (x + 1.0);
	} else if (x <= -8e-155) {
		tmp = x * (y / ((x + y) * (x + y)));
	} else {
		tmp = (x / (y + 1.0)) / (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.07:
		tmp = (y / (x + y)) / (x + 1.0)
	elif x <= -8e-155:
		tmp = x * (y / ((x + y) * (x + y)))
	else:
		tmp = (x / (y + 1.0)) / (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.07)
		tmp = Float64(Float64(y / Float64(x + y)) / Float64(x + 1.0));
	elseif (x <= -8e-155)
		tmp = Float64(x * Float64(y / Float64(Float64(x + y) * Float64(x + y))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.07)
		tmp = (y / (x + y)) / (x + 1.0);
	elseif (x <= -8e-155)
		tmp = x * (y / ((x + y) * (x + y)));
	else
		tmp = (x / (y + 1.0)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.07], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8e-155], N[(x * N[(y / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.070000000000000007

   1. Initial program 54.5%

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

      1. times-frac 79.5%

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

      2. +-commutative 79.5%

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

      3. +-commutative 79.5%

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

      4. +-commutative 79.5%

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

      5. times-frac 54.5%

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

      6. associate-*l/ 69.8%

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

      7. *-commutative 69.8%

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

      8. *-commutative 69.8%

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

      9. distribute-rgt1-in 23.8%

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

      10. fma-def 69.8%

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

      11. +-commutative 69.8%

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

      12. +-commutative 69.8%

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

      13. cube-unmult 69.8%

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

      14. +-commutative 69.8%

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

   3. Simplified 69.8%

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

      1. associate-*r/ 54.5%

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

      2. fma-udef 20.4%

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

      3. cube-mult 20.4%

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

      4. distribute-rgt1-in 54.5%

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

      5. associate-+r+ 54.5%

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

      6. *-commutative 54.5%

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

      7. frac-times 79.6%

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

      8. *-commutative 79.6%

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

      9. clear-num 79.6%

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

      10. associate-/r* 99.8%

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

      11. frac-times 98.7%

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

      12. *-un-lft-identity 98.7%

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

      13. associate-+r+ 98.7%

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

      14. +-commutative 98.7%

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

      15. associate-+l+ 98.7%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in y around 0 75.7%

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

      1. +-commutative 75.7%

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

   9. Simplified 75.7%

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

   if -0.070000000000000007 < x < -8.00000000000000011e-155

   1. Initial program 80.8%

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

      1. associate-/r* 80.7%

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

      2. *-commutative 80.7%

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

      3. +-commutative 80.7%

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

      4. +-commutative 80.7%

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

      5. associate-*l/ 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 98.5%

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

      1. +-commutative 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in y around 0 80.4%

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

   if -8.00000000000000011e-155 < x

   1. Initial program 67.5%

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

      1. times-frac 82.7%

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

      2. +-commutative 82.7%

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

      3. +-commutative 82.7%

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

      4. +-commutative 82.7%

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

      5. times-frac 67.5%

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

      6. associate-*l/ 79.4%

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

      7. *-commutative 79.4%

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

      8. *-commutative 79.4%

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

      9. distribute-rgt1-in 63.5%

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

      10. fma-def 79.4%

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

      11. +-commutative 79.4%

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

      12. +-commutative 79.4%

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

      13. cube-unmult 79.4%

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

      14. +-commutative 79.4%

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

   3. Simplified 79.4%

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

      1. associate-*r/ 67.5%

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

      2. fma-udef 55.7%

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

      3. cube-mult 55.6%

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

      4. distribute-rgt1-in 67.5%

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

      5. associate-+r+ 67.5%

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

      6. *-commutative 67.5%

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

      7. frac-times 82.6%

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

      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 x around 0 55.8%

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

      1. +-commutative 55.8%

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

   9. Simplified 55.8%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.07:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + 1}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-155}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2600:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x - y \cdot -2}\\ \mathbf{elif}\;x \leq -2.35 \cdot 10^{-153}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2600.0)
   (/ (/ y (+ x y)) (- x (* y -2.0)))
   (if (<= x -2.35e-153)
     (* x (/ y (* (+ x y) (+ x y))))
     (/ (/ x (+ y 1.0)) (+ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2600.0) {
		tmp = (y / (x + y)) / (x - (y * -2.0));
	} else if (x <= -2.35e-153) {
		tmp = x * (y / ((x + y) * (x + y)));
	} else {
		tmp = (x / (y + 1.0)) / (x + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2600.0d0)) then
        tmp = (y / (x + y)) / (x - (y * (-2.0d0)))
    else if (x <= (-2.35d-153)) then
        tmp = x * (y / ((x + y) * (x + y)))
    else
        tmp = (x / (y + 1.0d0)) / (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2600.0) {
		tmp = (y / (x + y)) / (x - (y * -2.0));
	} else if (x <= -2.35e-153) {
		tmp = x * (y / ((x + y) * (x + y)));
	} else {
		tmp = (x / (y + 1.0)) / (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2600.0:
		tmp = (y / (x + y)) / (x - (y * -2.0))
	elif x <= -2.35e-153:
		tmp = x * (y / ((x + y) * (x + y)))
	else:
		tmp = (x / (y + 1.0)) / (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2600.0)
		tmp = Float64(Float64(y / Float64(x + y)) / Float64(x - Float64(y * -2.0)));
	elseif (x <= -2.35e-153)
		tmp = Float64(x * Float64(y / Float64(Float64(x + y) * Float64(x + y))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2600.0)
		tmp = (y / (x + y)) / (x - (y * -2.0));
	elseif (x <= -2.35e-153)
		tmp = x * (y / ((x + y) * (x + y)));
	else
		tmp = (x / (y + 1.0)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2600.0], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x - N[(y * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.35e-153], N[(x * N[(y / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2600

   1. Initial program 54.5%

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

      1. times-frac 79.5%

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

      2. +-commutative 79.5%

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

      3. +-commutative 79.5%

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

      4. +-commutative 79.5%

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

      5. times-frac 54.5%

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

      6. associate-*l/ 69.8%

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

      7. *-commutative 69.8%

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

      8. *-commutative 69.8%

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

      9. distribute-rgt1-in 23.8%

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

      10. fma-def 69.8%

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

      11. +-commutative 69.8%

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

      12. +-commutative 69.8%

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

      13. cube-unmult 69.8%

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

      14. +-commutative 69.8%

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

   3. Simplified 69.8%

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

      1. associate-*r/ 54.5%

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

      2. fma-udef 20.4%

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

      3. cube-mult 20.4%

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

      4. distribute-rgt1-in 54.5%

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

      5. associate-+r+ 54.5%

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

      6. *-commutative 54.5%

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

      7. frac-times 79.6%

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

      8. *-commutative 79.6%

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

      9. clear-num 79.6%

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

      10. associate-/r* 99.8%

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

      11. frac-times 98.7%

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

      12. *-un-lft-identity 98.7%

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

      13. associate-+r+ 98.7%

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

      14. +-commutative 98.7%

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

      15. associate-+l+ 98.7%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in x around -inf 77.2%

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

      1. mul-1-neg 77.2%

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

      2. unsub-neg 77.2%

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

      3. neg-mul-1 77.2%

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

      4. +-commutative 77.2%

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

      5. +-commutative 77.2%

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

      6. neg-mul-1 77.2%

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

      7. unsub-neg 77.2%

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

      8. neg-mul-1 77.2%

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

      9. +-commutative 77.2%

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

      10. distribute-lft-in 77.2%

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

      11. metadata-eval 77.2%

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

      12. neg-mul-1 77.2%

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

      13. unsub-neg 77.2%

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

   9. Simplified 77.2%

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

   10. Taylor expanded in y around inf 76.8%

       \[\leadsto \frac{\frac{y}{x + y}}{x - \color{blue}{-2 \cdot y}} \]
   11. Step-by-step derivation

       1. *-commutative 76.8%

          \[\leadsto \frac{\frac{y}{x + y}}{x - \color{blue}{y \cdot -2}} \]

   12. Simplified 76.8%

       \[\leadsto \frac{\frac{y}{x + y}}{x - \color{blue}{y \cdot -2}} \]

   if -2600 < x < -2.35e-153

   1. Initial program 80.8%

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

      1. associate-/r* 80.7%

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

      2. *-commutative 80.7%

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

      3. +-commutative 80.7%

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

      4. +-commutative 80.7%

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

      5. associate-*l/ 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 98.5%

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

      1. +-commutative 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in y around 0 80.4%

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

   if -2.35e-153 < x

   1. Initial program 67.5%

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

      1. times-frac 82.7%

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

      2. +-commutative 82.7%

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

      3. +-commutative 82.7%

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

      4. +-commutative 82.7%

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

      5. times-frac 67.5%

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

      6. associate-*l/ 79.4%

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

      7. *-commutative 79.4%

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

      8. *-commutative 79.4%

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

      9. distribute-rgt1-in 63.5%

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

      10. fma-def 79.4%

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

      11. +-commutative 79.4%

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

      12. +-commutative 79.4%

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

      13. cube-unmult 79.4%

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

      14. +-commutative 79.4%

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

   3. Simplified 79.4%

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

      1. associate-*r/ 67.5%

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

      2. fma-udef 55.7%

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

      3. cube-mult 55.6%

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

      4. distribute-rgt1-in 67.5%

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

      5. associate-+r+ 67.5%

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

      6. *-commutative 67.5%

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

      7. frac-times 82.6%

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

      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 x around 0 55.8%

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

      1. +-commutative 55.8%

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

   9. Simplified 55.8%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2600:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x - y \cdot -2}\\ \mathbf{elif}\;x \leq -2.35 \cdot 10^{-153}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -70:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-155}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -70.0)
   (/ (/ y (+ x y)) (+ x (+ y (+ y 1.0))))
   (if (<= x -3.5e-155)
     (* x (/ y (* (+ x y) (+ x y))))
     (/ (/ x (+ y 1.0)) (+ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -70.0) {
		tmp = (y / (x + y)) / (x + (y + (y + 1.0)));
	} else if (x <= -3.5e-155) {
		tmp = x * (y / ((x + y) * (x + y)));
	} else {
		tmp = (x / (y + 1.0)) / (x + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-70.0d0)) then
        tmp = (y / (x + y)) / (x + (y + (y + 1.0d0)))
    else if (x <= (-3.5d-155)) then
        tmp = x * (y / ((x + y) * (x + y)))
    else
        tmp = (x / (y + 1.0d0)) / (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -70.0) {
		tmp = (y / (x + y)) / (x + (y + (y + 1.0)));
	} else if (x <= -3.5e-155) {
		tmp = x * (y / ((x + y) * (x + y)));
	} else {
		tmp = (x / (y + 1.0)) / (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -70.0:
		tmp = (y / (x + y)) / (x + (y + (y + 1.0)))
	elif x <= -3.5e-155:
		tmp = x * (y / ((x + y) * (x + y)))
	else:
		tmp = (x / (y + 1.0)) / (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -70.0)
		tmp = Float64(Float64(y / Float64(x + y)) / Float64(x + Float64(y + Float64(y + 1.0))));
	elseif (x <= -3.5e-155)
		tmp = Float64(x * Float64(y / Float64(Float64(x + y) * Float64(x + y))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -70.0)
		tmp = (y / (x + y)) / (x + (y + (y + 1.0)));
	elseif (x <= -3.5e-155)
		tmp = x * (y / ((x + y) * (x + y)));
	else
		tmp = (x / (y + 1.0)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -70.0], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.5e-155], N[(x * N[(y / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -70

   1. Initial program 54.5%

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

      1. times-frac 79.5%

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

      2. +-commutative 79.5%

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

      3. +-commutative 79.5%

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

      4. +-commutative 79.5%

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

      5. times-frac 54.5%

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

      6. associate-*l/ 69.8%

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

      7. *-commutative 69.8%

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

      8. *-commutative 69.8%

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

      9. distribute-rgt1-in 23.8%

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

      10. fma-def 69.8%

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

      11. +-commutative 69.8%

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

      12. +-commutative 69.8%

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

      13. cube-unmult 69.8%

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

      14. +-commutative 69.8%

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

   3. Simplified 69.8%

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

      1. associate-*r/ 54.5%

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

      2. fma-udef 20.4%

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

      3. cube-mult 20.4%

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

      4. distribute-rgt1-in 54.5%

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

      5. associate-+r+ 54.5%

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

      6. *-commutative 54.5%

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

      7. frac-times 79.6%

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

      8. *-commutative 79.6%

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

      9. clear-num 79.6%

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

      10. associate-/r* 99.8%

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

      11. frac-times 98.7%

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

      12. *-un-lft-identity 98.7%

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

      13. associate-+r+ 98.7%

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

      14. +-commutative 98.7%

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

      15. associate-+l+ 98.7%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in x around -inf 77.2%

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

      1. mul-1-neg 77.2%

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

      2. unsub-neg 77.2%

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

      3. neg-mul-1 77.2%

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

      4. +-commutative 77.2%

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

      5. +-commutative 77.2%

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

      6. neg-mul-1 77.2%

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

      7. unsub-neg 77.2%

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

      8. neg-mul-1 77.2%

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

      9. +-commutative 77.2%

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

      10. distribute-lft-in 77.2%

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

      11. metadata-eval 77.2%

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

      12. neg-mul-1 77.2%

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

      13. unsub-neg 77.2%

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

   9. Simplified 77.2%

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

   if -70 < x < -3.50000000000000015e-155

   1. Initial program 80.8%

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

      1. associate-/r* 80.7%

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

      2. *-commutative 80.7%

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

      3. +-commutative 80.7%

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

      4. +-commutative 80.7%

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

      5. associate-*l/ 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 98.5%

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

      1. +-commutative 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in y around 0 80.4%

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

   if -3.50000000000000015e-155 < x

   1. Initial program 67.5%

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

      1. times-frac 82.7%

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

      2. +-commutative 82.7%

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

      3. +-commutative 82.7%

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

      4. +-commutative 82.7%

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

      5. times-frac 67.5%

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

      6. associate-*l/ 79.4%

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

      7. *-commutative 79.4%

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

      8. *-commutative 79.4%

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

      9. distribute-rgt1-in 63.5%

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

      10. fma-def 79.4%

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

      11. +-commutative 79.4%

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

      12. +-commutative 79.4%

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

      13. cube-unmult 79.4%

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

      14. +-commutative 79.4%

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

   3. Simplified 79.4%

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

      1. associate-*r/ 67.5%

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

      2. fma-udef 55.7%

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

      3. cube-mult 55.6%

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

      4. distribute-rgt1-in 67.5%

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

      5. associate-+r+ 67.5%

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

      6. *-commutative 67.5%

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

      7. frac-times 82.6%

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

      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 x around 0 55.8%

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

      1. +-commutative 55.8%

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

   9. Simplified 55.8%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -70:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-155}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x + y}\\ \mathbf{if}\;x \leq -0.245:\\ \;\;\;\;\frac{t_0}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x + y} \cdot \frac{x}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ x y))))
   (if (<= x -0.245)
     (/ t_0 (+ x (+ y (+ y 1.0))))
     (* (/ t_0 (+ x y)) (/ x (+ y 1.0))))))

C

double code(double x, double y) {
	double t_0 = y / (x + y);
	double tmp;
	if (x <= -0.245) {
		tmp = t_0 / (x + (y + (y + 1.0)));
	} else {
		tmp = (t_0 / (x + y)) * (x / (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (x + y)
    if (x <= (-0.245d0)) then
        tmp = t_0 / (x + (y + (y + 1.0d0)))
    else
        tmp = (t_0 / (x + y)) * (x / (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y / (x + y);
	double tmp;
	if (x <= -0.245) {
		tmp = t_0 / (x + (y + (y + 1.0)));
	} else {
		tmp = (t_0 / (x + y)) * (x / (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (x + y)
	tmp = 0
	if x <= -0.245:
		tmp = t_0 / (x + (y + (y + 1.0)))
	else:
		tmp = (t_0 / (x + y)) * (x / (y + 1.0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(x + y))
	tmp = 0.0
	if (x <= -0.245)
		tmp = Float64(t_0 / Float64(x + Float64(y + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(t_0 / Float64(x + y)) * Float64(x / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (x + y);
	tmp = 0.0;
	if (x <= -0.245)
		tmp = t_0 / (x + (y + (y + 1.0)));
	else
		tmp = (t_0 / (x + y)) * (x / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.245], N[(t$95$0 / N[(x + N[(y + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.245

   1. Initial program 54.5%

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

      1. times-frac 79.5%

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

      2. +-commutative 79.5%

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

      3. +-commutative 79.5%

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

      4. +-commutative 79.5%

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

      5. times-frac 54.5%

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

      6. associate-*l/ 69.8%

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

      7. *-commutative 69.8%

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

      8. *-commutative 69.8%

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

      9. distribute-rgt1-in 23.8%

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

      10. fma-def 69.8%

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

      11. +-commutative 69.8%

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

      12. +-commutative 69.8%

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

      13. cube-unmult 69.8%

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

      14. +-commutative 69.8%

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

   3. Simplified 69.8%

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

      1. associate-*r/ 54.5%

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

      2. fma-udef 20.4%

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

      3. cube-mult 20.4%

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

      4. distribute-rgt1-in 54.5%

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

      5. associate-+r+ 54.5%

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

      6. *-commutative 54.5%

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

      7. frac-times 79.6%

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

      8. *-commutative 79.6%

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

      9. clear-num 79.6%

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

      10. associate-/r* 99.8%

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

      11. frac-times 98.7%

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

      12. *-un-lft-identity 98.7%

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

      13. associate-+r+ 98.7%

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

      14. +-commutative 98.7%

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

      15. associate-+l+ 98.7%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in x around -inf 77.2%

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

      1. mul-1-neg 77.2%

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

      2. unsub-neg 77.2%

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

      3. neg-mul-1 77.2%

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

      4. +-commutative 77.2%

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

      5. +-commutative 77.2%

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

      6. neg-mul-1 77.2%

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

      7. unsub-neg 77.2%

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

      8. neg-mul-1 77.2%

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

      9. +-commutative 77.2%

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

      10. distribute-lft-in 77.2%

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

      11. metadata-eval 77.2%

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

      12. neg-mul-1 77.2%

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

      13. unsub-neg 77.2%

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

   9. Simplified 77.2%

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

   if -0.245 < x

   1. Initial program 70.2%

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

      1. associate-/r* 71.2%

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

      2. *-commutative 71.2%

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

      3. +-commutative 71.2%

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

      4. +-commutative 71.2%

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

      5. associate-*l/ 86.1%

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

      6. +-commutative 86.1%

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

      7. associate-*r/ 86.0%

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

      8. remove-double-neg 86.0%

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

      9. +-commutative 86.0%

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

      10. +-commutative 86.0%

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

      11. remove-double-neg 86.0%

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

      12. +-commutative 86.0%

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

      13. associate-+l+ 86.0%

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

   3. Simplified 86.0%

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

   5. Taylor expanded in x around 0 80.0%

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

      1. +-commutative 80.0%

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

   7. Simplified 80.0%

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

      1. associate-/r* 91.3%

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

      2. div-inv 91.2%

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

   9. Applied egg-rr 91.2%

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

       1. associate-*r/ 91.3%

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

       2. *-rgt-identity 91.3%

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

   11. Simplified 91.3%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.245:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + y} \cdot \frac{x}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-203}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;y \leq 19500000:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6.8e-203)
   (- (/ y x) y)
   (if (<= y 19500000.0) (- (/ x y) x) (* (/ x y) (/ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.8e-203) {
		tmp = (y / x) - y;
	} else if (y <= 19500000.0) {
		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 <= 6.8d-203) then
        tmp = (y / x) - y
    else if (y <= 19500000.0d0) 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 <= 6.8e-203) {
		tmp = (y / x) - y;
	} else if (y <= 19500000.0) {
		tmp = (x / y) - x;
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6.8e-203:
		tmp = (y / x) - y
	elif y <= 19500000.0:
		tmp = (x / y) - x
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.8e-203)
		tmp = Float64(Float64(y / x) - y);
	elseif (y <= 19500000.0)
		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 <= 6.8e-203)
		tmp = (y / x) - y;
	elseif (y <= 19500000.0)
		tmp = (x / y) - x;
	else
		tmp = (x / y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.8e-203], N[(N[(y / x), $MachinePrecision] - y), $MachinePrecision], If[LessEqual[y, 19500000.0], 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 < 6.7999999999999998e-203

   1. Initial program 65.3%

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

      1. associate-/r* 67.1%

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

      2. *-commutative 67.1%

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

      3. +-commutative 67.1%

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

      4. +-commutative 67.1%

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

      5. associate-*l/ 84.2%

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

      6. +-commutative 84.2%

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

      7. associate-*r/ 84.2%

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

      8. remove-double-neg 84.2%

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

      9. +-commutative 84.2%

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

      10. +-commutative 84.2%

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

      11. remove-double-neg 84.2%

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

      12. +-commutative 84.2%

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

      13. associate-+l+ 84.2%

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

   3. Simplified 84.2%

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

   5. Taylor expanded in y around 0 55.9%

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

      1. associate-/r* 55.4%

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

      2. +-commutative 55.4%

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

   7. Simplified 55.4%

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

   8. Taylor expanded in x around 0 22.1%

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

      1. neg-mul-1 22.1%

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

      2. +-commutative 22.1%

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

      3. unsub-neg 22.1%

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

   10. Simplified 22.1%

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

   if 6.7999999999999998e-203 < y < 1.95e7

   1. Initial program 74.7%

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

      1. associate-/r* 81.3%

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

      2. *-commutative 81.3%

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

      3. +-commutative 81.3%

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

      4. +-commutative 81.3%

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

      5. associate-*l/ 93.2%

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

      6. +-commutative 93.2%

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

      7. associate-*r/ 93.2%

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

      8. remove-double-neg 93.2%

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

      9. +-commutative 93.2%

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

      10. +-commutative 93.2%

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

      11. remove-double-neg 93.2%

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

      12. +-commutative 93.2%

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

      13. associate-+l+ 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in x around 0 23.4%

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

      1. +-commutative 23.4%

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

   7. Simplified 23.4%

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

   8. Taylor expanded in y around 0 23.5%

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

      1. neg-mul-1 23.5%

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

      2. +-commutative 23.5%

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

      3. unsub-neg 23.5%

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

   10. Simplified 23.5%

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

   if 1.95e7 < y

   1. Initial program 62.5%

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

      1. associate-/r* 66.9%

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

      2. *-commutative 66.9%

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

      3. +-commutative 66.9%

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

      4. +-commutative 66.9%

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

      5. associate-*l/ 79.1%

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

      6. +-commutative 79.1%

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

      7. associate-*r/ 79.1%

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

      8. remove-double-neg 79.1%

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

      9. +-commutative 79.1%

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

      10. +-commutative 79.1%

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

      11. remove-double-neg 79.1%

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

      12. +-commutative 79.1%

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

      13. associate-+l+ 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in y around inf 71.0%

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

   6. Taylor expanded in y around inf 70.0%

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

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-203}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;y \leq 19500000:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-202}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{+166}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.15e-202)
   (- (/ y x) y)
   (if (<= y 2.75e+166) (/ x (* y (+ y 1.0))) (* (/ x y) (/ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.15e-202) {
		tmp = (y / x) - y;
	} else if (y <= 2.75e+166) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.15d-202) then
        tmp = (y / x) - y
    else if (y <= 2.75d+166) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.15e-202) {
		tmp = (y / x) - y;
	} else if (y <= 2.75e+166) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.15e-202:
		tmp = (y / x) - y
	elif y <= 2.75e+166:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.15e-202)
		tmp = Float64(Float64(y / x) - y);
	elseif (y <= 2.75e+166)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.15e-202)
		tmp = (y / x) - y;
	elseif (y <= 2.75e+166)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.15e-202], N[(N[(y / x), $MachinePrecision] - y), $MachinePrecision], If[LessEqual[y, 2.75e+166], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.1499999999999999e-202

   1. Initial program 65.3%

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

      1. associate-/r* 67.1%

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

      2. *-commutative 67.1%

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

      3. +-commutative 67.1%

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

      4. +-commutative 67.1%

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

      5. associate-*l/ 84.2%

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

      6. +-commutative 84.2%

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

      7. associate-*r/ 84.2%

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

      8. remove-double-neg 84.2%

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

      9. +-commutative 84.2%

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

      10. +-commutative 84.2%

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

      11. remove-double-neg 84.2%

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

      12. +-commutative 84.2%

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

      13. associate-+l+ 84.2%

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

   3. Simplified 84.2%

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

   5. Taylor expanded in y around 0 55.9%

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

      1. associate-/r* 55.4%

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

      2. +-commutative 55.4%

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

   7. Simplified 55.4%

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

   8. Taylor expanded in x around 0 22.1%

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

      1. neg-mul-1 22.1%

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

      2. +-commutative 22.1%

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

      3. unsub-neg 22.1%

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

   10. Simplified 22.1%

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

   if 1.1499999999999999e-202 < y < 2.75000000000000004e166

   1. Initial program 68.0%

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

      1. associate-/r* 76.3%

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

      2. *-commutative 76.3%

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

      3. +-commutative 76.3%

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

      4. +-commutative 76.3%

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

      5. associate-*l/ 84.9%

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

      6. +-commutative 84.9%

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

      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 32.4%

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

      1. +-commutative 32.4%

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

   7. Simplified 32.4%

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

   if 2.75000000000000004e166 < y

   1. Initial program 66.2%

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

      1. associate-/r* 66.2%

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

      2. *-commutative 66.2%

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

      3. +-commutative 66.2%

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

      4. +-commutative 66.2%

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

      5. associate-*l/ 84.4%

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

      6. +-commutative 84.4%

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

      7. associate-*r/ 84.4%

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

      8. remove-double-neg 84.4%

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

      9. +-commutative 84.4%

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

      10. +-commutative 84.4%

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

      11. remove-double-neg 84.4%

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

      12. +-commutative 84.4%

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

      13. associate-+l+ 84.4%

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

   3. Simplified 84.4%

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

   5. Taylor expanded in y around inf 85.4%

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

   6. Taylor expanded in y around inf 85.3%

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

4. Final simplification 34.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-202}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{+166}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-110}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-45}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.5e-110)
   (/ y (* x (+ x 1.0)))
   (if (<= x 7.2e-45) (/ x (* y (+ y 1.0))) (* (/ x y) (/ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.5e-110) {
		tmp = y / (x * (x + 1.0));
	} else if (x <= 7.2e-45) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.5d-110)) then
        tmp = y / (x * (x + 1.0d0))
    else if (x <= 7.2d-45) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.5e-110) {
		tmp = y / (x * (x + 1.0));
	} else if (x <= 7.2e-45) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.5e-110:
		tmp = y / (x * (x + 1.0))
	elif x <= 7.2e-45:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.5e-110)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (x <= 7.2e-45)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.5e-110)
		tmp = y / (x * (x + 1.0));
	elseif (x <= 7.2e-45)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.5e-110], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.2e-45], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.4999999999999998e-110

   1. Initial program 62.8%

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

      1. associate-/r* 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/ 85.7%

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

      6. +-commutative 85.7%

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

      7. associate-*r/ 85.7%

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

      8. remove-double-neg 85.7%

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

      9. +-commutative 85.7%

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

      10. +-commutative 85.7%

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

      11. remove-double-neg 85.7%

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

      12. +-commutative 85.7%

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

      13. associate-+l+ 85.7%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in y around 0 63.9%

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

   if -5.4999999999999998e-110 < x < 7.20000000000000001e-45

   1. Initial program 64.0%

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

      1. associate-/r* 64.1%

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

      2. *-commutative 64.1%

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

      3. +-commutative 64.1%

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

      4. +-commutative 64.1%

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

      5. associate-*l/ 79.9%

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

      6. +-commutative 79.9%

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

      7. associate-*r/ 79.9%

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

      8. remove-double-neg 79.9%

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

      9. +-commutative 79.9%

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

      10. +-commutative 79.9%

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

      11. remove-double-neg 79.9%

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

      12. +-commutative 79.9%

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

      13. associate-+l+ 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in x around 0 74.9%

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

      1. +-commutative 74.9%

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

   7. Simplified 74.9%

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

   if 7.20000000000000001e-45 < x

   1. Initial program 72.9%

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

      1. associate-/r* 75.4%

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

      2. *-commutative 75.4%

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

      3. +-commutative 75.4%

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

      4. +-commutative 75.4%

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

      5. associate-*l/ 87.9%

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

      6. +-commutative 87.9%

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

      7. associate-*r/ 87.9%

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

      8. remove-double-neg 87.9%

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

      9. +-commutative 87.9%

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

      10. +-commutative 87.9%

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

      11. remove-double-neg 87.9%

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

      12. +-commutative 87.9%

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

      13. associate-+l+ 87.9%

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

   3. Simplified 87.9%

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

   5. Taylor expanded in y around inf 35.1%

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

   6. Taylor expanded in y around inf 34.5%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-110}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-45}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.1%

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

   1. associate-/r* 69.4%

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

   2. *-commutative 69.4%

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

   3. +-commutative 69.4%

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

   4. +-commutative 69.4%

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

   5. associate-*l/ 84.4%

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

   6. +-commutative 84.4%

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

   7. associate-*r/ 84.4%

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

   8. remove-double-neg 84.4%

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

   9. +-commutative 84.4%

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

   10. +-commutative 84.4%

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

   11. remove-double-neg 84.4%

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

   12. +-commutative 84.4%

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

   13. associate-+l+ 84.4%

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

3. Simplified 84.4%

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

   1. associate-/r* 84.3%

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

   2. div-inv 84.3%

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

6. Applied egg-rr 99.7%

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

   1. associate-*r/ 84.3%

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

   2. *-rgt-identity 84.3%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 12: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.1%

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

   1. times-frac 84.4%

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

   2. +-commutative 84.4%

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

   3. +-commutative 84.4%

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

   4. +-commutative 84.4%

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

   5. times-frac 66.1%

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

   6. associate-*l/ 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 56.7%

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

   10. fma-def 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/ 66.1%

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

   2. fma-udef 48.7%

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

   3. cube-mult 48.7%

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

   4. distribute-rgt1-in 66.1%

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

   5. associate-+r+ 66.1%

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

   6. *-commutative 66.1%

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

   7. frac-times 84.4%

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

   8. associate-/r* 99.8%

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

   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. Final simplification 99.8%

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

Alternative 13: 61.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-109}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.7e-109) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ y (+ x 1.0))) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.7e-109) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + (x + 1.0))) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.7d-109)) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (y + (x + 1.0d0))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.7e-109) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + (x + 1.0))) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.7e-109:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (y + (x + 1.0))) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.7e-109)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.7e-109)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (y + (x + 1.0))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.7e-109], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.70000000000000006e-109

   1. Initial program 62.8%

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

      1. associate-/r* 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/ 85.7%

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

      6. +-commutative 85.7%

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

      7. associate-*r/ 85.7%

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

      8. remove-double-neg 85.7%

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

      9. +-commutative 85.7%

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

      10. +-commutative 85.7%

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

      11. remove-double-neg 85.7%

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

      12. +-commutative 85.7%

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

      13. associate-+l+ 85.7%

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

   3. Simplified 85.7%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 63.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 66.8%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 66.8%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   7. Simplified 66.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if -1.70000000000000006e-109 < x

   1. Initial program 68.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 69.3%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. *-commutative 69.3%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 69.3%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      4. +-commutative 69.3%

         \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      5. associate-*l/ 83.6%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 83.6%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 83.6%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 83.6%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 83.6%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 83.6%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 83.6%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 83.6%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 83.6%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 83.6%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 56.6%

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{x + \left(y + 1\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ 56.6%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + \left(y + 1\right)}}{y}} \]

      2. *-un-lft-identity 56.6%

         \[\leadsto \frac{\color{blue}{\frac{x}{x + \left(y + 1\right)}}}{y} \]

      3. associate-+r+ 56.6%

         \[\leadsto \frac{\frac{x}{\color{blue}{\left(x + y\right) + 1}}}{y} \]

      4. +-commutative 56.6%

         \[\leadsto \frac{\frac{x}{\color{blue}{\left(y + x\right)} + 1}}{y} \]

      5. associate-+r+ 56.6%

         \[\leadsto \frac{\frac{x}{\color{blue}{y + \left(x + 1\right)}}}{y} \]

   7. Applied egg-rr 56.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + \left(x + 1\right)}}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-109}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-109}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.7e-109) (/ (/ y (+ x y)) (+ x 1.0)) (/ (/ x (+ y (+ x 1.0))) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.7e-109) {
		tmp = (y / (x + y)) / (x + 1.0);
	} else {
		tmp = (x / (y + (x + 1.0))) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.7d-109)) then
        tmp = (y / (x + y)) / (x + 1.0d0)
    else
        tmp = (x / (y + (x + 1.0d0))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.7e-109) {
		tmp = (y / (x + y)) / (x + 1.0);
	} else {
		tmp = (x / (y + (x + 1.0))) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.7e-109:
		tmp = (y / (x + y)) / (x + 1.0)
	else:
		tmp = (x / (y + (x + 1.0))) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.7e-109)
		tmp = Float64(Float64(y / Float64(x + y)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.7e-109)
		tmp = (y / (x + y)) / (x + 1.0);
	else
		tmp = (x / (y + (x + 1.0))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.7e-109], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.70000000000000006e-109

   1. Initial program 62.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 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 62.8%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. associate-*l/ 78.9%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

      7. *-commutative 78.9%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 78.9%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 43.8%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 78.9%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 78.9%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 78.9%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 78.9%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 78.9%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 78.9%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 62.7%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 35.9%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 35.9%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 62.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. associate-+r+ 62.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 62.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. frac-times 85.7%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

      8. *-commutative 85.7%

         \[\leadsto \color{blue}{\frac{x}{x + \left(y + 1\right)} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      9. clear-num 85.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + \left(y + 1\right)}{x}}} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \]

      10. associate-/r* 99.8%

          \[\leadsto \frac{1}{\frac{x + \left(y + 1\right)}{x}} \cdot \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \]

      11. frac-times 99.0%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{y}{x + y}}{\frac{x + \left(y + 1\right)}{x} \cdot \left(x + y\right)}} \]

      12. *-un-lft-identity 99.0%

          \[\leadsto \frac{\color{blue}{\frac{y}{x + y}}}{\frac{x + \left(y + 1\right)}{x} \cdot \left(x + y\right)} \]

      13. associate-+r+ 99.0%

          \[\leadsto \frac{\frac{y}{x + y}}{\frac{\color{blue}{\left(x + y\right) + 1}}{x} \cdot \left(x + y\right)} \]

      14. +-commutative 99.0%

          \[\leadsto \frac{\frac{y}{x + y}}{\frac{\color{blue}{\left(y + x\right)} + 1}{x} \cdot \left(x + y\right)} \]

      15. associate-+l+ 99.0%

          \[\leadsto \frac{\frac{y}{x + y}}{\frac{\color{blue}{y + \left(x + 1\right)}}{x} \cdot \left(x + y\right)} \]

   6. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)}} \]

   7. Taylor expanded in y around 0 67.1%

      \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{1 + x}} \]
   8. Step-by-step derivation

      1. +-commutative 67.1%

         \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{x + 1}} \]

   9. Simplified 67.1%

      \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{x + 1}} \]

   if -1.70000000000000006e-109 < x

   1. Initial program 68.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 69.3%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. *-commutative 69.3%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 69.3%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      4. +-commutative 69.3%

         \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      5. associate-*l/ 83.6%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 83.6%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 83.6%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 83.6%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 83.6%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 83.6%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 83.6%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 83.6%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 83.6%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 83.6%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 56.6%

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{x + \left(y + 1\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ 56.6%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + \left(y + 1\right)}}{y}} \]

      2. *-un-lft-identity 56.6%

         \[\leadsto \frac{\color{blue}{\frac{x}{x + \left(y + 1\right)}}}{y} \]

      3. associate-+r+ 56.6%

         \[\leadsto \frac{\frac{x}{\color{blue}{\left(x + y\right) + 1}}}{y} \]

      4. +-commutative 56.6%

         \[\leadsto \frac{\frac{x}{\color{blue}{\left(y + x\right)} + 1}}{y} \]

      5. associate-+r+ 56.6%

         \[\leadsto \frac{\frac{x}{\color{blue}{y + \left(x + 1\right)}}}{y} \]

   7. Applied egg-rr 56.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + \left(x + 1\right)}}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-109}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 60.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-109}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.7e-109) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.7e-109) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.7d-109)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.7e-109) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.7e-109:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.7e-109)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.7e-109)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.7e-109], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.70000000000000006e-109

   1. Initial program 62.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 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/ 85.7%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 85.7%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 85.7%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 85.7%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 85.7%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 85.7%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 85.7%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 85.7%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 85.7%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 85.7%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 63.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   if -1.70000000000000006e-109 < x

   1. Initial program 68.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 83.6%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 83.6%

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1} \]

      3. +-commutative 83.6%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      4. +-commutative 83.6%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      5. times-frac 68.1%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. associate-*l/ 80.0%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

      7. *-commutative 80.0%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 80.0%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 64.3%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 80.0%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 80.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 80.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 80.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 80.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 80.0%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 68.1%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 56.3%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 56.3%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 68.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. associate-+r+ 68.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 68.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. frac-times 83.6%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

      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. Step-by-step derivation

      1. *-commutative 99.9%

         \[\leadsto \frac{\color{blue}{\frac{x}{y + \left(x + 1\right)} \cdot \frac{y}{x + y}}}{x + y} \]

      2. clear-num 99.8%

         \[\leadsto \frac{\frac{x}{y + \left(x + 1\right)} \cdot \color{blue}{\frac{1}{\frac{x + y}{y}}}}{x + y} \]

      3. un-div-inv 99.9%

         \[\leadsto \frac{\color{blue}{\frac{\frac{x}{y + \left(x + 1\right)}}{\frac{x + y}{y}}}}{x + y} \]

      4. +-commutative 99.9%

         \[\leadsto \frac{\frac{\frac{x}{y + \left(x + 1\right)}}{\frac{\color{blue}{y + x}}{y}}}{x + y} \]

   8. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{\frac{\frac{x}{y + \left(x + 1\right)}}{\frac{y + x}{y}}}}{x + y} \]

   9. Taylor expanded in x around 0 54.4%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   10. Step-by-step derivation

       1. associate-/r* 56.5%

          \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

   11. Simplified 56.5%

       \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-109}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 60.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-109}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.7e-109) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.7e-109) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.7d-109)) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.7e-109) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.7e-109:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.7e-109)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.7e-109)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.7e-109], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.70000000000000006e-109

   1. Initial program 62.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 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/ 85.7%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 85.7%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 85.7%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 85.7%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 85.7%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 85.7%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 85.7%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 85.7%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 85.7%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 85.7%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 63.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 66.8%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 66.8%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   7. Simplified 66.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if -1.70000000000000006e-109 < x

   1. Initial program 68.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 83.6%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 83.6%

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1} \]

      3. +-commutative 83.6%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      4. +-commutative 83.6%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      5. times-frac 68.1%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. associate-*l/ 80.0%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

      7. *-commutative 80.0%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 80.0%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 64.3%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 80.0%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 80.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 80.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 80.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 80.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 80.0%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 68.1%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 56.3%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 56.3%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 68.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. associate-+r+ 68.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 68.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. frac-times 83.6%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

      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. Step-by-step derivation

      1. *-commutative 99.9%

         \[\leadsto \frac{\color{blue}{\frac{x}{y + \left(x + 1\right)} \cdot \frac{y}{x + y}}}{x + y} \]

      2. clear-num 99.8%

         \[\leadsto \frac{\frac{x}{y + \left(x + 1\right)} \cdot \color{blue}{\frac{1}{\frac{x + y}{y}}}}{x + y} \]

      3. un-div-inv 99.9%

         \[\leadsto \frac{\color{blue}{\frac{\frac{x}{y + \left(x + 1\right)}}{\frac{x + y}{y}}}}{x + y} \]

      4. +-commutative 99.9%

         \[\leadsto \frac{\frac{\frac{x}{y + \left(x + 1\right)}}{\frac{\color{blue}{y + x}}{y}}}{x + y} \]

   8. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{\frac{\frac{x}{y + \left(x + 1\right)}}{\frac{y + x}{y}}}}{x + y} \]

   9. Taylor expanded in x around 0 54.4%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   10. Step-by-step derivation

       1. associate-/r* 56.5%

          \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

   11. Simplified 56.5%

       \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-109}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 23.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-203}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 6.8e-203) (- (/ y x) y) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.8e-203) {
		tmp = (y / x) - y;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 6.8d-203) then
        tmp = (y / x) - y
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6.8e-203) {
		tmp = (y / x) - y;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6.8e-203:
		tmp = (y / x) - y
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.8e-203)
		tmp = Float64(Float64(y / x) - y);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.8e-203)
		tmp = (y / x) - y;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.8e-203], N[(N[(y / x), $MachinePrecision] - y), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.7999999999999998e-203

   1. Initial program 65.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 67.1%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. *-commutative 67.1%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 67.1%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      4. +-commutative 67.1%

         \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      5. associate-*l/ 84.2%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 84.2%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 84.2%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 84.2%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 84.2%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 84.2%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 84.2%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 84.2%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 84.2%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 84.2%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 55.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 55.4%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 55.4%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   7. Simplified 55.4%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   8. Taylor expanded in x around 0 22.1%

      \[\leadsto \color{blue}{-1 \cdot y + \frac{y}{x}} \]
   9. Step-by-step derivation

      1. neg-mul-1 22.1%

         \[\leadsto \color{blue}{\left(-y\right)} + \frac{y}{x} \]

      2. +-commutative 22.1%

         \[\leadsto \color{blue}{\frac{y}{x} + \left(-y\right)} \]

      3. unsub-neg 22.1%

         \[\leadsto \color{blue}{\frac{y}{x} - y} \]

   10. Simplified 22.1%

       \[\leadsto \color{blue}{\frac{y}{x} - y} \]

   if 6.7999999999999998e-203 < y

   1. Initial program 67.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 72.6%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. *-commutative 72.6%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 72.6%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      4. +-commutative 72.6%

         \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      5. associate-*l/ 84.7%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 84.7%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 84.6%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 84.6%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 84.6%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 84.6%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 84.6%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 84.6%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 84.6%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 84.6%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 51.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 51.5%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   7. Simplified 51.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   8. Taylor expanded in y around 0 27.3%

      \[\leadsto \frac{x}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-203}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 4.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 0.5 y))

C

double code(double x, double y) {
	return 0.5 / y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.5d0 / y
end function

Java

public static double code(double x, double y) {
	return 0.5 / y;
}

Python

def code(x, y):
	return 0.5 / y

Julia

function code(x, y)
	return Float64(0.5 / y)
end

MATLAB

function tmp = code(x, y)
	tmp = 0.5 / y;
end

Wolfram

code[x_, y_] := N[(0.5 / y), $MachinePrecision]

Derivation

1. Initial program 66.1%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. times-frac 84.4%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. +-commutative 84.4%

      \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1} \]

   3. +-commutative 84.4%

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

   4. +-commutative 84.4%

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   5. times-frac 66.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

   6. associate-*l/ 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 56.7%

      \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   10. fma-def 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/ 66.1%

      \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   2. fma-udef 48.7%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   3. cube-mult 48.7%

      \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   4. distribute-rgt1-in 66.1%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   5. associate-+r+ 66.1%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

   6. *-commutative 66.1%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

   7. frac-times 84.4%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   8. *-commutative 84.4%

      \[\leadsto \color{blue}{\frac{x}{x + \left(y + 1\right)} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   9. clear-num 84.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{x + \left(y + 1\right)}{x}}} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \]

   10. associate-/r* 99.8%

       \[\leadsto \frac{1}{\frac{x + \left(y + 1\right)}{x}} \cdot \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \]

   11. frac-times 99.3%

       \[\leadsto \color{blue}{\frac{1 \cdot \frac{y}{x + y}}{\frac{x + \left(y + 1\right)}{x} \cdot \left(x + y\right)}} \]

   12. *-un-lft-identity 99.3%

       \[\leadsto \frac{\color{blue}{\frac{y}{x + y}}}{\frac{x + \left(y + 1\right)}{x} \cdot \left(x + y\right)} \]

   13. associate-+r+ 99.3%

       \[\leadsto \frac{\frac{y}{x + y}}{\frac{\color{blue}{\left(x + y\right) + 1}}{x} \cdot \left(x + y\right)} \]

   14. +-commutative 99.3%

       \[\leadsto \frac{\frac{y}{x + y}}{\frac{\color{blue}{\left(y + x\right)} + 1}{x} \cdot \left(x + y\right)} \]

   15. associate-+l+ 99.3%

       \[\leadsto \frac{\frac{y}{x + y}}{\frac{\color{blue}{y + \left(x + 1\right)}}{x} \cdot \left(x + y\right)} \]

6. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)}} \]

7. Taylor expanded in x around -inf 54.8%

   \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{x + -1 \cdot \left(-1 \cdot y + -1 \cdot \left(1 + y\right)\right)}} \]
8. Step-by-step derivation

   1. mul-1-neg 54.8%

      \[\leadsto \frac{\frac{y}{x + y}}{x + \color{blue}{\left(-\left(-1 \cdot y + -1 \cdot \left(1 + y\right)\right)\right)}} \]

   2. unsub-neg 54.8%

      \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{x - \left(-1 \cdot y + -1 \cdot \left(1 + y\right)\right)}} \]

   3. neg-mul-1 54.8%

      \[\leadsto \frac{\frac{y}{x + y}}{x - \left(\color{blue}{\left(-y\right)} + -1 \cdot \left(1 + y\right)\right)} \]

   4. +-commutative 54.8%

      \[\leadsto \frac{\frac{y}{x + y}}{x - \color{blue}{\left(-1 \cdot \left(1 + y\right) + \left(-y\right)\right)}} \]

   5. +-commutative 54.8%

      \[\leadsto \frac{\frac{y}{x + y}}{x - \left(-1 \cdot \color{blue}{\left(y + 1\right)} + \left(-y\right)\right)} \]

   6. neg-mul-1 54.8%

      \[\leadsto \frac{\frac{y}{x + y}}{x - \left(\color{blue}{\left(-\left(y + 1\right)\right)} + \left(-y\right)\right)} \]

   7. unsub-neg 54.8%

      \[\leadsto \frac{\frac{y}{x + y}}{x - \color{blue}{\left(\left(-\left(y + 1\right)\right) - y\right)}} \]

   8. neg-mul-1 54.8%

      \[\leadsto \frac{\frac{y}{x + y}}{x - \left(\color{blue}{-1 \cdot \left(y + 1\right)} - y\right)} \]

   9. +-commutative 54.8%

      \[\leadsto \frac{\frac{y}{x + y}}{x - \left(-1 \cdot \color{blue}{\left(1 + y\right)} - y\right)} \]

   10. distribute-lft-in 54.8%

       \[\leadsto \frac{\frac{y}{x + y}}{x - \left(\color{blue}{\left(-1 \cdot 1 + -1 \cdot y\right)} - y\right)} \]

   11. metadata-eval 54.8%

       \[\leadsto \frac{\frac{y}{x + y}}{x - \left(\left(\color{blue}{-1} + -1 \cdot y\right) - y\right)} \]

   12. neg-mul-1 54.8%

       \[\leadsto \frac{\frac{y}{x + y}}{x - \left(\left(-1 + \color{blue}{\left(-y\right)}\right) - y\right)} \]

   13. unsub-neg 54.8%

       \[\leadsto \frac{\frac{y}{x + y}}{x - \left(\color{blue}{\left(-1 - y\right)} - y\right)} \]

9. Simplified 54.8%

   \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{x - \left(\left(-1 - y\right) - y\right)}} \]

10. Taylor expanded in y around inf 4.3%

    \[\leadsto \color{blue}{\frac{0.5}{y}} \]

11. Final simplification 4.3%

    \[\leadsto \frac{0.5}{y} \]
12. Add Preprocessing

Alternative 19: 26.1% 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 66.1%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 69.4%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

   2. *-commutative 69.4%

      \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

   3. +-commutative 69.4%

      \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

   4. +-commutative 69.4%

      \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

   5. associate-*l/ 84.4%

      \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

   6. +-commutative 84.4%

      \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

   7. associate-*r/ 84.4%

      \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

   8. remove-double-neg 84.4%

      \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

   9. +-commutative 84.4%

      \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

   10. +-commutative 84.4%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

   11. remove-double-neg 84.4%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

   12. +-commutative 84.4%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

   13. associate-+l+ 84.4%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

3. Simplified 84.4%

   \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 46.2%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
6. Step-by-step derivation

   1. +-commutative 46.2%

      \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

7. Simplified 46.2%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

8. Taylor expanded in y around 0 24.2%

   \[\leadsto \frac{x}{\color{blue}{y}} \]

9. Final simplification 24.2%

   \[\leadsto \frac{x}{y} \]
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 2024024 
(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))))