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

Percentage Accurate: 68.7% → 99.3%

Time: 18.8s

Alternatives: 24

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (/ (/ y (+ y x)) (* (+ y x) (/ (+ y (+ x 1.0)) x))))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (y / (y + x)) / ((y + x) * ((y + (x + 1.0d0)) / x))
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return (y / (y + x)) / ((y + x) * ((y + (x + 1.0)) / x))

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(Float64(y / Float64(y + x)) / Float64(Float64(y + x) * Float64(Float64(y + Float64(x + 1.0)) / x)))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = (y / (y + x)) / ((y + x) * ((y + (x + 1.0)) / x));
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.3%

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

   1. associate-*l* 72.3%

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

   2. +-commutative 72.3%

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

   3. +-commutative 72.3%

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

   4. +-commutative 72.3%

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

   5. associate-*l* 72.3%

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

   6. *-commutative 72.3%

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

   7. associate-*r/ 84.7%

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

   8. *-commutative 84.7%

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

   9. distribute-rgt1-in 67.1%

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

   10. fma-def 84.7%

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

   11. +-commutative 84.7%

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

   12. +-commutative 84.7%

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

   13. cube-unmult 84.7%

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

   14. +-commutative 84.7%

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

3. Simplified 84.7%

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

   1. associate-*r/ 72.3%

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

   2. fma-udef 57.5%

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

   3. cube-mult 57.5%

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

   4. distribute-rgt1-in 72.3%

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

   5. *-commutative 72.3%

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

   6. +-commutative 72.3%

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

   7. associate-+r+ 72.3%

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

   8. frac-times 88.8%

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

   9. *-commutative 88.8%

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

   10. clear-num 88.7%

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

   11. associate-/r* 99.7%

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

   12. frac-times 99.5%

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

   13. *-un-lft-identity 99.5%

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

   14. +-commutative 99.5%

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

   15. +-commutative 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 2: 97.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;y \leq 2 \cdot 10^{-19}:\\ \;\;\;\;t_0 \cdot \frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+153}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{y + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= y 2e-19)
     (* t_0 (/ (/ y (+ x 1.0)) (+ y x)))
     (if (<= y 3.8e+153)
       (* (/ x (* (+ y x) (+ y x))) (/ y (+ y (+ x 1.0))))
       (/ t_0 (+ y (+ x (+ x 1.0))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (y <= 2e-19) {
		tmp = t_0 * ((y / (x + 1.0)) / (y + x));
	} else if (y <= 3.8e+153) {
		tmp = (x / ((y + x) * (y + x))) * (y / (y + (x + 1.0)));
	} else {
		tmp = t_0 / (y + (x + (x + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 + x)
    if (y <= 2d-19) then
        tmp = t_0 * ((y / (x + 1.0d0)) / (y + x))
    else if (y <= 3.8d+153) then
        tmp = (x / ((y + x) * (y + x))) * (y / (y + (x + 1.0d0)))
    else
        tmp = t_0 / (y + (x + (x + 1.0d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (y <= 2e-19) {
		tmp = t_0 * ((y / (x + 1.0)) / (y + x));
	} else if (y <= 3.8e+153) {
		tmp = (x / ((y + x) * (y + x))) * (y / (y + (x + 1.0)));
	} else {
		tmp = t_0 / (y + (x + (x + 1.0)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y + x)
	tmp = 0
	if y <= 2e-19:
		tmp = t_0 * ((y / (x + 1.0)) / (y + x))
	elif y <= 3.8e+153:
		tmp = (x / ((y + x) * (y + x))) * (y / (y + (x + 1.0)))
	else:
		tmp = t_0 / (y + (x + (x + 1.0)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (y <= 2e-19)
		tmp = Float64(t_0 * Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x)));
	elseif (y <= 3.8e+153)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + x))) * Float64(y / Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(t_0 / Float64(y + Float64(x + Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	tmp = 0.0;
	if (y <= 2e-19)
		tmp = t_0 * ((y / (x + 1.0)) / (y + x));
	elseif (y <= 3.8e+153)
		tmp = (x / ((y + x) * (y + x))) * (y / (y + (x + 1.0)));
	else
		tmp = t_0 / (y + (x + (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 2e-19], N[(t$95$0 * N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e+153], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(y + N[(x + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2e-19

   1. Initial program 75.9%

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

      1. associate-*l* 75.9%

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

      2. +-commutative 75.9%

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

      3. +-commutative 75.9%

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

      4. +-commutative 75.9%

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

      5. associate-*l* 75.9%

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

      6. *-commutative 75.9%

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

      7. associate-*r/ 87.7%

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

      8. *-commutative 87.7%

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

      9. distribute-rgt1-in 66.4%

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

      10. fma-def 87.7%

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

      11. +-commutative 87.7%

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

      12. +-commutative 87.7%

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

      13. cube-unmult 87.7%

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

      14. +-commutative 87.7%

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

   3. Simplified 87.7%

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

      1. associate-*r/ 75.9%

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

      2. *-commutative 75.9%

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

      3. fma-udef 56.6%

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

      4. cube-mult 56.6%

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

      5. distribute-rgt1-in 75.9%

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

      6. *-commutative 75.9%

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

      7. +-commutative 75.9%

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

      8. associate-+r+ 75.9%

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

      9. frac-times 91.3%

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

      10. *-commutative 91.3%

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

      11. clear-num 91.2%

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

      12. associate-/r* 99.7%

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

      13. frac-times 99.3%

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

      14. *-un-lft-identity 99.3%

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

      15. +-commutative 99.3%

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

      16. +-commutative 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in y around 0 85.2%

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

      1. +-commutative 85.2%

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

   8. Simplified 85.2%

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

      1. clear-num 85.7%

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

      2. associate-/r/ 85.3%

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

      3. associate-/r* 85.8%

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

      4. clear-num 85.8%

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

      5. +-commutative 85.8%

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

   10. Applied egg-rr 85.8%

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

   if 2e-19 < y < 3.79999999999999966e153

   1. Initial program 64.6%

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

      1. times-frac 85.8%

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

      2. +-commutative 85.8%

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

      3. associate-+l+ 85.8%

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

   3. Simplified 85.8%

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

   if 3.79999999999999966e153 < y

   1. Initial program 58.5%

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

      1. associate-*l* 58.5%

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

      2. +-commutative 58.5%

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

      3. +-commutative 58.5%

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

      4. +-commutative 58.5%

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

      5. associate-*l* 58.5%

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

      6. *-commutative 58.5%

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

      7. associate-*r/ 75.9%

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

      8. *-commutative 75.9%

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

      9. distribute-rgt1-in 75.8%

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

      10. fma-def 75.9%

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

      11. +-commutative 75.9%

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

      12. +-commutative 75.9%

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

      13. cube-unmult 75.9%

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

      14. +-commutative 75.9%

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

   3. Simplified 75.9%

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

      1. associate-*r/ 58.5%

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

      2. *-commutative 58.5%

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

      3. fma-udef 58.5%

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

      4. cube-mult 58.5%

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

      5. distribute-rgt1-in 58.5%

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

      6. *-commutative 58.5%

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

      7. +-commutative 58.5%

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

      8. associate-+r+ 58.5%

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

      9. frac-times 75.9%

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

      10. *-commutative 75.9%

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

      11. clear-num 75.9%

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

      12. associate-/r* 99.9%

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

      13. frac-times 99.8%

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

      14. *-un-lft-identity 99.8%

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

      15. +-commutative 99.8%

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

      16. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around -inf 96.6%

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

      1. mul-1-neg 96.6%

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

      2. unsub-neg 96.6%

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

      3. neg-mul-1 96.6%

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

      4. +-commutative 96.6%

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

      5. unsub-neg 96.6%

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

      6. distribute-lft-in 96.6%

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

      7. metadata-eval 96.6%

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

      8. neg-mul-1 96.6%

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

      9. unsub-neg 96.6%

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

   8. Simplified 96.6%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+153}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{y + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \]

Alternative 3: 97.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;y \leq 4.4 \cdot 10^{-20}:\\ \;\;\;\;t_0 \cdot \frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{y + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= y 4.4e-20)
     (* t_0 (/ (/ y (+ x 1.0)) (+ y x)))
     (if (<= y 1.35e+154)
       (* (/ y (* (+ y x) (+ y x))) (/ x (+ y (+ x 1.0))))
       (/ t_0 (+ y (+ x (+ x 1.0))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (y <= 4.4e-20) {
		tmp = t_0 * ((y / (x + 1.0)) / (y + x));
	} else if (y <= 1.35e+154) {
		tmp = (y / ((y + x) * (y + x))) * (x / (y + (x + 1.0)));
	} else {
		tmp = t_0 / (y + (x + (x + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 + x)
    if (y <= 4.4d-20) then
        tmp = t_0 * ((y / (x + 1.0d0)) / (y + x))
    else if (y <= 1.35d+154) then
        tmp = (y / ((y + x) * (y + x))) * (x / (y + (x + 1.0d0)))
    else
        tmp = t_0 / (y + (x + (x + 1.0d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (y <= 4.4e-20) {
		tmp = t_0 * ((y / (x + 1.0)) / (y + x));
	} else if (y <= 1.35e+154) {
		tmp = (y / ((y + x) * (y + x))) * (x / (y + (x + 1.0)));
	} else {
		tmp = t_0 / (y + (x + (x + 1.0)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y + x)
	tmp = 0
	if y <= 4.4e-20:
		tmp = t_0 * ((y / (x + 1.0)) / (y + x))
	elif y <= 1.35e+154:
		tmp = (y / ((y + x) * (y + x))) * (x / (y + (x + 1.0)))
	else:
		tmp = t_0 / (y + (x + (x + 1.0)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (y <= 4.4e-20)
		tmp = Float64(t_0 * Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x)));
	elseif (y <= 1.35e+154)
		tmp = Float64(Float64(y / Float64(Float64(y + x) * Float64(y + x))) * Float64(x / Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(t_0 / Float64(y + Float64(x + Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	tmp = 0.0;
	if (y <= 4.4e-20)
		tmp = t_0 * ((y / (x + 1.0)) / (y + x));
	elseif (y <= 1.35e+154)
		tmp = (y / ((y + x) * (y + x))) * (x / (y + (x + 1.0)));
	else
		tmp = t_0 / (y + (x + (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 4.4e-20], N[(t$95$0 * N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+154], N[(N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(y + N[(x + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.39999999999999982e-20

   1. Initial program 75.9%

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

      1. associate-*l* 75.9%

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

      2. +-commutative 75.9%

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

      3. +-commutative 75.9%

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

      4. +-commutative 75.9%

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

      5. associate-*l* 75.9%

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

      6. *-commutative 75.9%

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

      7. associate-*r/ 87.7%

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

      8. *-commutative 87.7%

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

      9. distribute-rgt1-in 66.4%

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

      10. fma-def 87.7%

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

      11. +-commutative 87.7%

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

      12. +-commutative 87.7%

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

      13. cube-unmult 87.7%

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

      14. +-commutative 87.7%

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

   3. Simplified 87.7%

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

      1. associate-*r/ 75.9%

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

      2. *-commutative 75.9%

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

      3. fma-udef 56.6%

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

      4. cube-mult 56.6%

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

      5. distribute-rgt1-in 75.9%

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

      6. *-commutative 75.9%

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

      7. +-commutative 75.9%

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

      8. associate-+r+ 75.9%

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

      9. frac-times 91.3%

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

      10. *-commutative 91.3%

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

      11. clear-num 91.2%

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

      12. associate-/r* 99.7%

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

      13. frac-times 99.3%

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

      14. *-un-lft-identity 99.3%

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

      15. +-commutative 99.3%

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

      16. +-commutative 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in y around 0 85.2%

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

      1. +-commutative 85.2%

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

   8. Simplified 85.2%

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

      1. clear-num 85.7%

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

      2. associate-/r/ 85.3%

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

      3. associate-/r* 85.8%

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

      4. clear-num 85.8%

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

      5. +-commutative 85.8%

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

   10. Applied egg-rr 85.8%

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

   if 4.39999999999999982e-20 < y < 1.35000000000000003e154

   1. Initial program 64.6%

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

      1. associate-*l* 64.6%

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

      2. +-commutative 64.6%

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

      3. +-commutative 64.6%

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

      4. +-commutative 64.6%

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

      5. *-commutative 64.6%

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

      6. associate-*l* 64.6%

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

      7. times-frac 85.9%

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

      8. +-commutative 85.9%

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

      9. +-commutative 85.9%

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

      10. associate-+l+ 85.9%

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

   3. Simplified 85.9%

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

   if 1.35000000000000003e154 < y

   1. Initial program 58.5%

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

      1. associate-*l* 58.5%

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

      2. +-commutative 58.5%

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

      3. +-commutative 58.5%

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

      4. +-commutative 58.5%

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

      5. associate-*l* 58.5%

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

      6. *-commutative 58.5%

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

      7. associate-*r/ 75.9%

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

      8. *-commutative 75.9%

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

      9. distribute-rgt1-in 75.8%

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

      10. fma-def 75.9%

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

      11. +-commutative 75.9%

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

      12. +-commutative 75.9%

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

      13. cube-unmult 75.9%

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

      14. +-commutative 75.9%

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

   3. Simplified 75.9%

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

      1. associate-*r/ 58.5%

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

      2. *-commutative 58.5%

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

      3. fma-udef 58.5%

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

      4. cube-mult 58.5%

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

      5. distribute-rgt1-in 58.5%

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

      6. *-commutative 58.5%

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

      7. +-commutative 58.5%

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

      8. associate-+r+ 58.5%

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

      9. frac-times 75.9%

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

      10. *-commutative 75.9%

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

      11. clear-num 75.9%

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

      12. associate-/r* 99.9%

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

      13. frac-times 99.8%

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

      14. *-un-lft-identity 99.8%

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

      15. +-commutative 99.8%

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

      16. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around -inf 96.6%

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

      1. mul-1-neg 96.6%

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

      2. unsub-neg 96.6%

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

      3. neg-mul-1 96.6%

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

      4. +-commutative 96.6%

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

      5. unsub-neg 96.6%

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

      6. distribute-lft-in 96.6%

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

      7. metadata-eval 96.6%

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

      8. neg-mul-1 96.6%

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

      9. unsub-neg 96.6%

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

   8. Simplified 96.6%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{y + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \]

Alternative 4: 90.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;x \leq -2.05 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y - \left(-1 - y\right)\right)}\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-179}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{y + 1}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-231}:\\ \;\;\;\;\frac{t_0}{\left(y + x\right) \cdot \frac{1}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= x -2.05e-10)
     (/ (/ y (+ y x)) (+ x (- y (- -1.0 y))))
     (if (<= x -6.5e-179)
       (* (/ x (* (+ y x) (+ y x))) (/ y (+ y 1.0)))
       (if (<= x -2.2e-231)
         (/ t_0 (* (+ y x) (/ 1.0 y)))
         (/ t_0 (+ y (+ x (+ x 1.0)))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -2.05e-10) {
		tmp = (y / (y + x)) / (x + (y - (-1.0 - y)));
	} else if (x <= -6.5e-179) {
		tmp = (x / ((y + x) * (y + x))) * (y / (y + 1.0));
	} else if (x <= -2.2e-231) {
		tmp = t_0 / ((y + x) * (1.0 / y));
	} else {
		tmp = t_0 / (y + (x + (x + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 + x)
    if (x <= (-2.05d-10)) then
        tmp = (y / (y + x)) / (x + (y - ((-1.0d0) - y)))
    else if (x <= (-6.5d-179)) then
        tmp = (x / ((y + x) * (y + x))) * (y / (y + 1.0d0))
    else if (x <= (-2.2d-231)) then
        tmp = t_0 / ((y + x) * (1.0d0 / y))
    else
        tmp = t_0 / (y + (x + (x + 1.0d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -2.05e-10) {
		tmp = (y / (y + x)) / (x + (y - (-1.0 - y)));
	} else if (x <= -6.5e-179) {
		tmp = (x / ((y + x) * (y + x))) * (y / (y + 1.0));
	} else if (x <= -2.2e-231) {
		tmp = t_0 / ((y + x) * (1.0 / y));
	} else {
		tmp = t_0 / (y + (x + (x + 1.0)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y + x)
	tmp = 0
	if x <= -2.05e-10:
		tmp = (y / (y + x)) / (x + (y - (-1.0 - y)))
	elif x <= -6.5e-179:
		tmp = (x / ((y + x) * (y + x))) * (y / (y + 1.0))
	elif x <= -2.2e-231:
		tmp = t_0 / ((y + x) * (1.0 / y))
	else:
		tmp = t_0 / (y + (x + (x + 1.0)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (x <= -2.05e-10)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + Float64(y - Float64(-1.0 - y))));
	elseif (x <= -6.5e-179)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + x))) * Float64(y / Float64(y + 1.0)));
	elseif (x <= -2.2e-231)
		tmp = Float64(t_0 / Float64(Float64(y + x) * Float64(1.0 / y)));
	else
		tmp = Float64(t_0 / Float64(y + Float64(x + Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	tmp = 0.0;
	if (x <= -2.05e-10)
		tmp = (y / (y + x)) / (x + (y - (-1.0 - y)));
	elseif (x <= -6.5e-179)
		tmp = (x / ((y + x) * (y + x))) * (y / (y + 1.0));
	elseif (x <= -2.2e-231)
		tmp = t_0 / ((y + x) * (1.0 / y));
	else
		tmp = t_0 / (y + (x + (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.05e-10], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y - N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6.5e-179], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.2e-231], N[(t$95$0 / N[(N[(y + x), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(y + N[(x + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.0499999999999999e-10

   1. Initial program 64.1%

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

      1. associate-*l* 64.1%

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

      2. +-commutative 64.1%

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

      3. +-commutative 64.1%

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

      4. +-commutative 64.1%

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

      5. associate-*l* 64.1%

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

      6. *-commutative 64.1%

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

      7. associate-*r/ 79.7%

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

      8. *-commutative 79.7%

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

      9. distribute-rgt1-in 31.7%

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

      10. fma-def 79.7%

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

      11. +-commutative 79.7%

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

      12. +-commutative 79.7%

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

      13. cube-unmult 79.7%

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

      14. +-commutative 79.7%

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

   3. Simplified 79.7%

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

      1. associate-*r/ 64.1%

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

      2. fma-udef 26.3%

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

      3. cube-mult 26.3%

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

      4. distribute-rgt1-in 64.1%

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

      5. *-commutative 64.1%

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

      6. +-commutative 64.1%

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

      7. associate-+r+ 64.1%

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

      8. frac-times 81.2%

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

      9. *-commutative 81.2%

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

      10. clear-num 81.2%

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

      11. associate-/r* 99.7%

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

      12. frac-times 99.8%

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

      13. *-un-lft-identity 99.8%

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

      14. +-commutative 99.8%

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

      15. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around -inf 81.6%

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

      1. mul-1-neg 81.6%

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

      2. unsub-neg 81.6%

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

      3. neg-mul-1 81.6%

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

      4. +-commutative 81.6%

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

      5. unsub-neg 81.6%

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

      6. distribute-lft-in 81.6%

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

      7. metadata-eval 81.6%

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

      8. neg-mul-1 81.6%

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

      9. unsub-neg 81.6%

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

   8. Simplified 81.6%

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

   if -2.0499999999999999e-10 < x < -6.49999999999999996e-179

   1. Initial program 81.1%

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

      1. times-frac 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. +-commutative 99.7%

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

   6. Simplified 99.7%

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

   if -6.49999999999999996e-179 < x < -2.20000000000000009e-231

   1. Initial program 72.7%

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

      1. associate-*l* 72.7%

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

      2. +-commutative 72.7%

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

      3. +-commutative 72.7%

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

      4. +-commutative 72.7%

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

      5. associate-*l* 72.7%

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

      6. *-commutative 72.7%

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

      7. associate-*r/ 80.2%

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

      8. *-commutative 80.2%

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

      9. distribute-rgt1-in 71.1%

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

      10. fma-def 80.2%

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

      11. +-commutative 80.2%

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

      12. +-commutative 80.2%

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

      13. cube-unmult 80.2%

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

      14. +-commutative 80.2%

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

   3. Simplified 80.2%

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

      1. associate-*r/ 72.7%

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

      2. *-commutative 72.7%

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

      3. fma-udef 63.7%

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

      4. cube-mult 63.7%

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

      5. distribute-rgt1-in 72.7%

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

      6. *-commutative 72.7%

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

      7. +-commutative 72.7%

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

      8. associate-+r+ 72.7%

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

      9. frac-times 80.2%

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

      10. *-commutative 80.2%

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

      11. clear-num 80.2%

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

      12. associate-/r* 99.9%

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

      13. frac-times 99.7%

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

      14. *-un-lft-identity 99.7%

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

      15. +-commutative 99.7%

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

      16. +-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in y around 0 90.4%

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

      1. +-commutative 90.4%

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

   8. Simplified 90.4%

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

   9. Taylor expanded in x around 0 90.4%

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

   if -2.20000000000000009e-231 < x

   1. Initial program 73.7%

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

      1. associate-*l* 73.7%

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

      2. +-commutative 73.7%

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

      3. +-commutative 73.7%

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

      4. +-commutative 73.7%

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

      5. associate-*l* 73.7%

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

      6. *-commutative 73.7%

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

      7. associate-*r/ 85.0%

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

      8. *-commutative 85.0%

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

      9. distribute-rgt1-in 77.2%

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

      10. fma-def 85.0%

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

      11. +-commutative 85.0%

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

      12. +-commutative 85.0%

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

      13. cube-unmult 85.0%

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

      14. +-commutative 85.0%

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

   3. Simplified 85.0%

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

      1. associate-*r/ 73.7%

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

      2. *-commutative 73.7%

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

      3. fma-udef 66.4%

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

      4. cube-mult 66.4%

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

      5. distribute-rgt1-in 73.7%

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

      6. *-commutative 73.7%

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

      7. +-commutative 73.7%

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

      8. associate-+r+ 73.7%

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

      9. frac-times 90.0%

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

      10. *-commutative 90.0%

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

      11. clear-num 89.9%

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

      12. associate-/r* 99.7%

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

      13. frac-times 99.4%

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

      14. *-un-lft-identity 99.4%

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

      15. +-commutative 99.4%

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

      16. +-commutative 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in y around -inf 59.4%

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

      1. mul-1-neg 59.4%

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

      2. unsub-neg 59.4%

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

      3. neg-mul-1 59.4%

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

      4. +-commutative 59.4%

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

      5. unsub-neg 59.4%

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

      6. distribute-lft-in 59.4%

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

      7. metadata-eval 59.4%

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

      8. neg-mul-1 59.4%

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

      9. unsub-neg 59.4%

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

   8. Simplified 59.4%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y - \left(-1 - y\right)\right)}\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-179}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{y + 1}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-231}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{\left(y + x\right) \cdot \frac{1}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \]

Alternative 5: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-231}:\\ \;\;\;\;\frac{t_0}{\left(y + x\right) \cdot \frac{1}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= x -1.6e-14)
     (/ (/ y (+ y x)) (+ x 1.0))
     (if (<= x -2.2e-231) (/ t_0 (* (+ y x) (/ 1.0 y))) (/ t_0 (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -1.6e-14) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else if (x <= -2.2e-231) {
		tmp = t_0 / ((y + x) * (1.0 / y));
	} else {
		tmp = t_0 / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 + x)
    if (x <= (-1.6d-14)) then
        tmp = (y / (y + x)) / (x + 1.0d0)
    else if (x <= (-2.2d-231)) then
        tmp = t_0 / ((y + x) * (1.0d0 / y))
    else
        tmp = t_0 / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -1.6e-14) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else if (x <= -2.2e-231) {
		tmp = t_0 / ((y + x) * (1.0 / y));
	} else {
		tmp = t_0 / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y + x)
	tmp = 0
	if x <= -1.6e-14:
		tmp = (y / (y + x)) / (x + 1.0)
	elif x <= -2.2e-231:
		tmp = t_0 / ((y + x) * (1.0 / y))
	else:
		tmp = t_0 / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (x <= -1.6e-14)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + 1.0));
	elseif (x <= -2.2e-231)
		tmp = Float64(t_0 / Float64(Float64(y + x) * Float64(1.0 / y)));
	else
		tmp = Float64(t_0 / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	tmp = 0.0;
	if (x <= -1.6e-14)
		tmp = (y / (y + x)) / (x + 1.0);
	elseif (x <= -2.2e-231)
		tmp = t_0 / ((y + x) * (1.0 / y));
	else
		tmp = t_0 / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.6e-14], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.2e-231], N[(t$95$0 / N[(N[(y + x), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6000000000000001e-14

   1. Initial program 64.6%

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

      1. associate-*l* 64.6%

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

      2. +-commutative 64.6%

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

      3. +-commutative 64.6%

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

      4. +-commutative 64.6%

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

      5. associate-*l* 64.6%

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

      6. *-commutative 64.6%

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

      7. associate-*r/ 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 31.2%

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

      10. fma-def 80.0%

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

      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. Step-by-step derivation

      1. associate-*r/ 64.6%

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

      2. fma-udef 25.9%

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

      3. cube-mult 25.9%

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

      4. distribute-rgt1-in 64.6%

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

      5. *-commutative 64.6%

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

      6. +-commutative 64.6%

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

      7. associate-+r+ 64.6%

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

      8. frac-times 81.5%

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

      9. *-commutative 81.5%

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

      10. clear-num 81.5%

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

      11. associate-/r* 99.7%

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

      12. frac-times 99.8%

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

      13. *-un-lft-identity 99.8%

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

      14. +-commutative 99.8%

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

      15. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around 0 80.1%

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

      1. +-commutative 80.1%

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

   8. Simplified 80.1%

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

   if -1.6000000000000001e-14 < x < -2.20000000000000009e-231

   1. Initial program 78.5%

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

      1. associate-*l* 78.5%

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

      2. +-commutative 78.5%

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

      3. +-commutative 78.5%

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

      4. +-commutative 78.5%

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

      5. associate-*l* 78.5%

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

      6. *-commutative 78.5%

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

      7. associate-*r/ 90.3%

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

      8. *-commutative 90.3%

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

      9. distribute-rgt1-in 83.3%

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

      10. fma-def 90.3%

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

      11. +-commutative 90.3%

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

      12. +-commutative 90.3%

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

      13. cube-unmult 90.3%

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

      14. +-commutative 90.3%

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

   3. Simplified 90.3%

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

      1. associate-*r/ 78.5%

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

      2. *-commutative 78.5%

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

      3. fma-udef 71.6%

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

      4. cube-mult 71.6%

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

      5. distribute-rgt1-in 78.5%

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

      6. *-commutative 78.5%

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

      7. +-commutative 78.5%

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

      8. associate-+r+ 78.5%

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

      9. frac-times 94.7%

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

      10. *-commutative 94.7%

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

      11. clear-num 94.7%

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

      12. associate-/r* 99.7%

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

      13. frac-times 99.6%

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

      14. *-un-lft-identity 99.6%

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

      15. +-commutative 99.6%

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

      16. +-commutative 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in y around 0 73.3%

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

      1. +-commutative 73.3%

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

   8. Simplified 73.3%

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

   9. Taylor expanded in x around 0 73.3%

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

   if -2.20000000000000009e-231 < x

   1. Initial program 73.7%

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

      1. associate-*l* 73.7%

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

      2. +-commutative 73.7%

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

      3. +-commutative 73.7%

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

      4. +-commutative 73.7%

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

      5. associate-*l* 73.7%

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

      6. *-commutative 73.7%

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

      7. associate-*r/ 85.0%

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

      8. *-commutative 85.0%

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

      9. distribute-rgt1-in 77.2%

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

      10. fma-def 85.0%

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

      11. +-commutative 85.0%

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

      12. +-commutative 85.0%

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

      13. cube-unmult 85.0%

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

      14. +-commutative 85.0%

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

   3. Simplified 85.0%

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

      1. associate-*r/ 73.7%

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

      2. *-commutative 73.7%

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

      3. fma-udef 66.4%

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

      4. cube-mult 66.4%

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

      5. distribute-rgt1-in 73.7%

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

      6. *-commutative 73.7%

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

      7. +-commutative 73.7%

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

      8. associate-+r+ 73.7%

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

      9. frac-times 90.0%

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

      10. *-commutative 90.0%

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

      11. clear-num 89.9%

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

      12. associate-/r* 99.7%

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

      13. frac-times 99.4%

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

      14. *-un-lft-identity 99.4%

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

      15. +-commutative 99.4%

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

      16. +-commutative 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in x around 0 58.3%

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

      1. +-commutative 58.3%

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

   8. Simplified 58.3%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-231}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{\left(y + x\right) \cdot \frac{1}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \]

Alternative 6: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-231}:\\ \;\;\;\;\frac{t_0}{\left(y + x\right) \cdot \frac{1}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= x -5.5e-14)
     (/ (/ y (+ y x)) (+ x 1.0))
     (if (<= x -1.7e-231)
       (/ t_0 (* (+ y x) (/ 1.0 y)))
       (/ t_0 (+ y (+ x (+ x 1.0))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -5.5e-14) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else if (x <= -1.7e-231) {
		tmp = t_0 / ((y + x) * (1.0 / y));
	} else {
		tmp = t_0 / (y + (x + (x + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 + x)
    if (x <= (-5.5d-14)) then
        tmp = (y / (y + x)) / (x + 1.0d0)
    else if (x <= (-1.7d-231)) then
        tmp = t_0 / ((y + x) * (1.0d0 / y))
    else
        tmp = t_0 / (y + (x + (x + 1.0d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -5.5e-14) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else if (x <= -1.7e-231) {
		tmp = t_0 / ((y + x) * (1.0 / y));
	} else {
		tmp = t_0 / (y + (x + (x + 1.0)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y + x)
	tmp = 0
	if x <= -5.5e-14:
		tmp = (y / (y + x)) / (x + 1.0)
	elif x <= -1.7e-231:
		tmp = t_0 / ((y + x) * (1.0 / y))
	else:
		tmp = t_0 / (y + (x + (x + 1.0)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (x <= -5.5e-14)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + 1.0));
	elseif (x <= -1.7e-231)
		tmp = Float64(t_0 / Float64(Float64(y + x) * Float64(1.0 / y)));
	else
		tmp = Float64(t_0 / Float64(y + Float64(x + Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	tmp = 0.0;
	if (x <= -5.5e-14)
		tmp = (y / (y + x)) / (x + 1.0);
	elseif (x <= -1.7e-231)
		tmp = t_0 / ((y + x) * (1.0 / y));
	else
		tmp = t_0 / (y + (x + (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.5e-14], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.7e-231], N[(t$95$0 / N[(N[(y + x), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(y + N[(x + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.49999999999999991e-14

   1. Initial program 64.6%

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

      1. associate-*l* 64.6%

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

      2. +-commutative 64.6%

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

      3. +-commutative 64.6%

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

      4. +-commutative 64.6%

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

      5. associate-*l* 64.6%

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

      6. *-commutative 64.6%

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

      7. associate-*r/ 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 31.2%

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

      10. fma-def 80.0%

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

      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. Step-by-step derivation

      1. associate-*r/ 64.6%

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

      2. fma-udef 25.9%

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

      3. cube-mult 25.9%

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

      4. distribute-rgt1-in 64.6%

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

      5. *-commutative 64.6%

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

      6. +-commutative 64.6%

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

      7. associate-+r+ 64.6%

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

      8. frac-times 81.5%

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

      9. *-commutative 81.5%

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

      10. clear-num 81.5%

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

      11. associate-/r* 99.7%

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

      12. frac-times 99.8%

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

      13. *-un-lft-identity 99.8%

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

      14. +-commutative 99.8%

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

      15. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around 0 80.1%

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

      1. +-commutative 80.1%

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

   8. Simplified 80.1%

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

   if -5.49999999999999991e-14 < x < -1.7e-231

   1. Initial program 78.5%

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

      1. associate-*l* 78.5%

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

      2. +-commutative 78.5%

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

      3. +-commutative 78.5%

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

      4. +-commutative 78.5%

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

      5. associate-*l* 78.5%

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

      6. *-commutative 78.5%

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

      7. associate-*r/ 90.3%

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

      8. *-commutative 90.3%

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

      9. distribute-rgt1-in 83.3%

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

      10. fma-def 90.3%

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

      11. +-commutative 90.3%

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

      12. +-commutative 90.3%

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

      13. cube-unmult 90.3%

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

      14. +-commutative 90.3%

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

   3. Simplified 90.3%

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

      1. associate-*r/ 78.5%

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

      2. *-commutative 78.5%

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

      3. fma-udef 71.6%

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

      4. cube-mult 71.6%

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

      5. distribute-rgt1-in 78.5%

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

      6. *-commutative 78.5%

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

      7. +-commutative 78.5%

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

      8. associate-+r+ 78.5%

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

      9. frac-times 94.7%

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

      10. *-commutative 94.7%

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

      11. clear-num 94.7%

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

      12. associate-/r* 99.7%

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

      13. frac-times 99.6%

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

      14. *-un-lft-identity 99.6%

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

      15. +-commutative 99.6%

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

      16. +-commutative 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in y around 0 73.3%

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

      1. +-commutative 73.3%

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

   8. Simplified 73.3%

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

   9. Taylor expanded in x around 0 73.3%

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

   if -1.7e-231 < x

   1. Initial program 73.7%

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

      1. associate-*l* 73.7%

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

      2. +-commutative 73.7%

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

      3. +-commutative 73.7%

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

      4. +-commutative 73.7%

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

      5. associate-*l* 73.7%

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

      6. *-commutative 73.7%

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

      7. associate-*r/ 85.0%

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

      8. *-commutative 85.0%

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

      9. distribute-rgt1-in 77.2%

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

      10. fma-def 85.0%

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

      11. +-commutative 85.0%

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

      12. +-commutative 85.0%

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

      13. cube-unmult 85.0%

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

      14. +-commutative 85.0%

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

   3. Simplified 85.0%

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

      1. associate-*r/ 73.7%

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

      2. *-commutative 73.7%

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

      3. fma-udef 66.4%

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

      4. cube-mult 66.4%

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

      5. distribute-rgt1-in 73.7%

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

      6. *-commutative 73.7%

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

      7. +-commutative 73.7%

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

      8. associate-+r+ 73.7%

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

      9. frac-times 90.0%

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

      10. *-commutative 90.0%

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

      11. clear-num 89.9%

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

      12. associate-/r* 99.7%

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

      13. frac-times 99.4%

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

      14. *-un-lft-identity 99.4%

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

      15. +-commutative 99.4%

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

      16. +-commutative 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in y around -inf 59.4%

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

      1. mul-1-neg 59.4%

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

      2. unsub-neg 59.4%

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

      3. neg-mul-1 59.4%

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

      4. +-commutative 59.4%

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

      5. unsub-neg 59.4%

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

      6. distribute-lft-in 59.4%

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

      7. metadata-eval 59.4%

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

      8. neg-mul-1 59.4%

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

      9. unsub-neg 59.4%

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

   8. Simplified 59.4%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-231}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{\left(y + x\right) \cdot \frac{1}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \]

Alternative 7: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;x \leq -4.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y - \left(-1 - y\right)\right)}\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-231}:\\ \;\;\;\;\frac{t_0}{\left(y + x\right) \cdot \frac{1}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= x -4.6e-16)
     (/ (/ y (+ y x)) (+ x (- y (- -1.0 y))))
     (if (<= x -1.8e-231)
       (/ t_0 (* (+ y x) (/ 1.0 y)))
       (/ t_0 (+ y (+ x (+ x 1.0))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -4.6e-16) {
		tmp = (y / (y + x)) / (x + (y - (-1.0 - y)));
	} else if (x <= -1.8e-231) {
		tmp = t_0 / ((y + x) * (1.0 / y));
	} else {
		tmp = t_0 / (y + (x + (x + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 + x)
    if (x <= (-4.6d-16)) then
        tmp = (y / (y + x)) / (x + (y - ((-1.0d0) - y)))
    else if (x <= (-1.8d-231)) then
        tmp = t_0 / ((y + x) * (1.0d0 / y))
    else
        tmp = t_0 / (y + (x + (x + 1.0d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -4.6e-16) {
		tmp = (y / (y + x)) / (x + (y - (-1.0 - y)));
	} else if (x <= -1.8e-231) {
		tmp = t_0 / ((y + x) * (1.0 / y));
	} else {
		tmp = t_0 / (y + (x + (x + 1.0)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y + x)
	tmp = 0
	if x <= -4.6e-16:
		tmp = (y / (y + x)) / (x + (y - (-1.0 - y)))
	elif x <= -1.8e-231:
		tmp = t_0 / ((y + x) * (1.0 / y))
	else:
		tmp = t_0 / (y + (x + (x + 1.0)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (x <= -4.6e-16)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + Float64(y - Float64(-1.0 - y))));
	elseif (x <= -1.8e-231)
		tmp = Float64(t_0 / Float64(Float64(y + x) * Float64(1.0 / y)));
	else
		tmp = Float64(t_0 / Float64(y + Float64(x + Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	tmp = 0.0;
	if (x <= -4.6e-16)
		tmp = (y / (y + x)) / (x + (y - (-1.0 - y)));
	elseif (x <= -1.8e-231)
		tmp = t_0 / ((y + x) * (1.0 / y));
	else
		tmp = t_0 / (y + (x + (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.6e-16], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y - N[(-1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.8e-231], N[(t$95$0 / N[(N[(y + x), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(y + N[(x + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.5999999999999998e-16

   1. Initial program 64.6%

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

      1. associate-*l* 64.6%

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

      2. +-commutative 64.6%

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

      3. +-commutative 64.6%

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

      4. +-commutative 64.6%

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

      5. associate-*l* 64.6%

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

      6. *-commutative 64.6%

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

      7. associate-*r/ 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 31.2%

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

      10. fma-def 80.0%

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

      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. Step-by-step derivation

      1. associate-*r/ 64.6%

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

      2. fma-udef 25.9%

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

      3. cube-mult 25.9%

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

      4. distribute-rgt1-in 64.6%

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

      5. *-commutative 64.6%

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

      6. +-commutative 64.6%

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

      7. associate-+r+ 64.6%

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

      8. frac-times 81.5%

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

      9. *-commutative 81.5%

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

      10. clear-num 81.5%

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

      11. associate-/r* 99.7%

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

      12. frac-times 99.8%

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

      13. *-un-lft-identity 99.8%

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

      14. +-commutative 99.8%

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

      15. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around -inf 80.5%

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

      1. mul-1-neg 80.5%

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

      2. unsub-neg 80.5%

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

      3. neg-mul-1 80.5%

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

      4. +-commutative 80.5%

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

      5. unsub-neg 80.5%

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

      6. distribute-lft-in 80.5%

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

      7. metadata-eval 80.5%

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

      8. neg-mul-1 80.5%

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

      9. unsub-neg 80.5%

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

   8. Simplified 80.5%

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

   if -4.5999999999999998e-16 < x < -1.79999999999999987e-231

   1. Initial program 78.5%

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

      1. associate-*l* 78.5%

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

      2. +-commutative 78.5%

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

      3. +-commutative 78.5%

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

      4. +-commutative 78.5%

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

      5. associate-*l* 78.5%

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

      6. *-commutative 78.5%

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

      7. associate-*r/ 90.3%

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

      8. *-commutative 90.3%

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

      9. distribute-rgt1-in 83.3%

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

      10. fma-def 90.3%

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

      11. +-commutative 90.3%

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

      12. +-commutative 90.3%

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

      13. cube-unmult 90.3%

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

      14. +-commutative 90.3%

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

   3. Simplified 90.3%

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

      1. associate-*r/ 78.5%

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

      2. *-commutative 78.5%

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

      3. fma-udef 71.6%

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

      4. cube-mult 71.6%

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

      5. distribute-rgt1-in 78.5%

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

      6. *-commutative 78.5%

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

      7. +-commutative 78.5%

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

      8. associate-+r+ 78.5%

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

      9. frac-times 94.7%

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

      10. *-commutative 94.7%

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

      11. clear-num 94.7%

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

      12. associate-/r* 99.7%

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

      13. frac-times 99.6%

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

      14. *-un-lft-identity 99.6%

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

      15. +-commutative 99.6%

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

      16. +-commutative 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in y around 0 73.3%

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

      1. +-commutative 73.3%

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

   8. Simplified 73.3%

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

   9. Taylor expanded in x around 0 73.3%

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

   if -1.79999999999999987e-231 < x

   1. Initial program 73.7%

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

      1. associate-*l* 73.7%

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

      2. +-commutative 73.7%

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

      3. +-commutative 73.7%

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

      4. +-commutative 73.7%

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

      5. associate-*l* 73.7%

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

      6. *-commutative 73.7%

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

      7. associate-*r/ 85.0%

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

      8. *-commutative 85.0%

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

      9. distribute-rgt1-in 77.2%

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

      10. fma-def 85.0%

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

      11. +-commutative 85.0%

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

      12. +-commutative 85.0%

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

      13. cube-unmult 85.0%

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

      14. +-commutative 85.0%

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

   3. Simplified 85.0%

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

      1. associate-*r/ 73.7%

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

      2. *-commutative 73.7%

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

      3. fma-udef 66.4%

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

      4. cube-mult 66.4%

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

      5. distribute-rgt1-in 73.7%

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

      6. *-commutative 73.7%

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

      7. +-commutative 73.7%

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

      8. associate-+r+ 73.7%

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

      9. frac-times 90.0%

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

      10. *-commutative 90.0%

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

      11. clear-num 89.9%

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

      12. associate-/r* 99.7%

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

      13. frac-times 99.4%

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

      14. *-un-lft-identity 99.4%

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

      15. +-commutative 99.4%

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

      16. +-commutative 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in y around -inf 59.4%

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

      1. mul-1-neg 59.4%

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

      2. unsub-neg 59.4%

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

      3. neg-mul-1 59.4%

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

      4. +-commutative 59.4%

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

      5. unsub-neg 59.4%

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

      6. distribute-lft-in 59.4%

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

      7. metadata-eval 59.4%

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

      8. neg-mul-1 59.4%

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

      9. unsub-neg 59.4%

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

   8. Simplified 59.4%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y - \left(-1 - y\right)\right)}\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-231}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{\left(y + x\right) \cdot \frac{1}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \]

Alternative 8: 91.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;y \leq 7.5 \cdot 10^{+29}:\\ \;\;\;\;t_0 \cdot \frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= y 7.5e+29)
     (* t_0 (/ (/ y (+ x 1.0)) (+ y x)))
     (/ t_0 (+ y (+ x (+ x 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (y <= 7.5e+29) {
		tmp = t_0 * ((y / (x + 1.0)) / (y + x));
	} else {
		tmp = t_0 / (y + (x + (x + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 + x)
    if (y <= 7.5d+29) then
        tmp = t_0 * ((y / (x + 1.0d0)) / (y + x))
    else
        tmp = t_0 / (y + (x + (x + 1.0d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (y <= 7.5e+29) {
		tmp = t_0 * ((y / (x + 1.0)) / (y + x));
	} else {
		tmp = t_0 / (y + (x + (x + 1.0)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y + x)
	tmp = 0
	if y <= 7.5e+29:
		tmp = t_0 * ((y / (x + 1.0)) / (y + x))
	else:
		tmp = t_0 / (y + (x + (x + 1.0)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (y <= 7.5e+29)
		tmp = Float64(t_0 * Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x)));
	else
		tmp = Float64(t_0 / Float64(y + Float64(x + Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	tmp = 0.0;
	if (y <= 7.5e+29)
		tmp = t_0 * ((y / (x + 1.0)) / (y + x));
	else
		tmp = t_0 / (y + (x + (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 7.5e+29], N[(t$95$0 * N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(y + N[(x + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 7.49999999999999945e29

   1. Initial program 76.4%

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

      1. associate-*l* 76.4%

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

      2. +-commutative 76.4%

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

      3. +-commutative 76.4%

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

      4. +-commutative 76.4%

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

      5. associate-*l* 76.4%

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

      6. *-commutative 76.4%

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

      7. associate-*r/ 87.7%

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

      8. *-commutative 87.7%

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

      9. distribute-rgt1-in 66.8%

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

      10. fma-def 87.7%

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

      11. +-commutative 87.7%

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

      12. +-commutative 87.7%

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

      13. cube-unmult 87.8%

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

      14. +-commutative 87.8%

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

   3. Simplified 87.8%

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

      1. associate-*r/ 76.4%

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

      2. *-commutative 76.4%

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

      3. fma-udef 57.4%

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

      4. cube-mult 57.4%

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

      5. distribute-rgt1-in 76.4%

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

      6. *-commutative 76.4%

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

      7. +-commutative 76.4%

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

      8. associate-+r+ 76.4%

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

      9. frac-times 91.1%

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

      10. *-commutative 91.1%

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

      11. clear-num 91.1%

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

      12. associate-/r* 99.6%

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

      13. frac-times 99.3%

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

      14. *-un-lft-identity 99.3%

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

      15. +-commutative 99.3%

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

      16. +-commutative 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in y around 0 85.5%

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

      1. +-commutative 85.5%

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

   8. Simplified 85.5%

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

      1. clear-num 86.0%

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

      2. associate-/r/ 85.5%

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

      3. associate-/r* 86.0%

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

      4. clear-num 86.1%

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

      5. +-commutative 86.1%

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

   10. Applied egg-rr 86.1%

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

   if 7.49999999999999945e29 < y

   1. Initial program 57.8%

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

      1. associate-*l* 57.8%

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

      2. +-commutative 57.8%

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

      3. +-commutative 57.8%

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

      4. +-commutative 57.8%

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

      5. associate-*l* 57.8%

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

      6. *-commutative 57.8%

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

      7. associate-*r/ 73.7%

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

      8. *-commutative 73.7%

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

      9. distribute-rgt1-in 68.0%

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

      10. fma-def 73.7%

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

      11. +-commutative 73.7%

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

      12. +-commutative 73.7%

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

      13. cube-unmult 73.7%

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

      14. +-commutative 73.7%

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

   3. Simplified 73.7%

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

      1. associate-*r/ 57.8%

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

      2. *-commutative 57.8%

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

      3. fma-udef 57.8%

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

      4. cube-mult 57.8%

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

      5. distribute-rgt1-in 57.8%

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

      6. *-commutative 57.8%

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

      7. +-commutative 57.8%

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

      8. associate-+r+ 57.8%

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

      9. frac-times 80.0%

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

      10. *-commutative 80.0%

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

      11. clear-num 80.0%

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

      12. associate-/r* 99.9%

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

      13. frac-times 99.8%

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

      14. *-un-lft-identity 99.8%

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

      15. +-commutative 99.8%

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

      16. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around -inf 80.6%

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

      1. mul-1-neg 80.6%

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

      2. unsub-neg 80.6%

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

      3. neg-mul-1 80.6%

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

      4. +-commutative 80.6%

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

      5. unsub-neg 80.6%

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

      6. distribute-lft-in 80.6%

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

      7. metadata-eval 80.6%

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

      8. neg-mul-1 80.6%

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

      9. unsub-neg 80.6%

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

   8. Simplified 80.6%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \]

Alternative 9: 99.2% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (/ (/ x (+ y x)) (* (+ y x) (/ (+ y (+ x 1.0)) y))))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x / (y + x)) / ((y + x) * ((y + (x + 1.0d0)) / y))
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return (x / (y + x)) / ((y + x) * ((y + (x + 1.0)) / y))

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(Float64(x / Float64(y + x)) / Float64(Float64(y + x) * Float64(Float64(y + Float64(x + 1.0)) / y)))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = (x / (y + x)) / ((y + x) * ((y + (x + 1.0)) / y));
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.3%

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

   1. associate-*l* 72.3%

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

   2. +-commutative 72.3%

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

   3. +-commutative 72.3%

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

   4. +-commutative 72.3%

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

   5. associate-*l* 72.3%

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

   6. *-commutative 72.3%

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

   7. associate-*r/ 84.7%

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

   8. *-commutative 84.7%

      \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   9. distribute-rgt1-in 67.1%

      \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   10. fma-def 84.7%

       \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

   11. +-commutative 84.7%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   12. +-commutative 84.7%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   13. cube-unmult 84.7%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

   14. +-commutative 84.7%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

3. Simplified 84.7%

   \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ 72.3%

      \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   2. *-commutative 72.3%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

   3. fma-udef 57.5%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   4. cube-mult 57.5%

      \[\leadsto \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   5. distribute-rgt1-in 72.3%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   6. *-commutative 72.3%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   7. +-commutative 72.3%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

   8. associate-+r+ 72.3%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

   9. frac-times 88.7%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   10. *-commutative 88.7%

       \[\leadsto \color{blue}{\frac{y}{y + \left(x + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   11. clear-num 88.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{y + \left(x + 1\right)}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

   12. associate-/r* 99.7%

       \[\leadsto \frac{1}{\frac{y + \left(x + 1\right)}{y}} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

   13. frac-times 99.4%

       \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{y + \left(x + 1\right)}{y} \cdot \left(x + y\right)}} \]

   14. *-un-lft-identity 99.4%

       \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{\frac{y + \left(x + 1\right)}{y} \cdot \left(x + y\right)} \]

   15. +-commutative 99.4%

       \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{\frac{y + \left(x + 1\right)}{y} \cdot \left(x + y\right)} \]

   16. +-commutative 99.4%

       \[\leadsto \frac{\frac{x}{y + x}}{\frac{y + \left(x + 1\right)}{y} \cdot \color{blue}{\left(y + x\right)}} \]

5. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{y + \left(x + 1\right)}{y} \cdot \left(y + x\right)}} \]

6. Final simplification 99.4%

   \[\leadsto \frac{\frac{x}{y + x}}{\left(y + x\right) \cdot \frac{y + \left(x + 1\right)}{y}} \]

Alternative 10: 81.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ (/ y x) x)
   (if (<= x -8.5e-98)
     (- (/ y x) y)
     (if (<= x 1.9e-32) (/ x (* y (+ y 1.0))) (/ (/ x y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) / x;
	} else if (x <= -8.5e-98) {
		tmp = (y / x) - y;
	} else if (x <= 1.9e-32) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (y / x) / x
    else if (x <= (-8.5d-98)) then
        tmp = (y / x) - y
    else if (x <= 1.9d-32) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) / x;
	} else if (x <= -8.5e-98) {
		tmp = (y / x) - y;
	} else if (x <= 1.9e-32) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (y / x) / x
	elif x <= -8.5e-98:
		tmp = (y / x) - y
	elif x <= 1.9e-32:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -8.5e-98)
		tmp = Float64(Float64(y / x) - y);
	elseif (x <= 1.9e-32)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (y / x) / x;
	elseif (x <= -8.5e-98)
		tmp = (y / x) - y;
	elseif (x <= 1.9e-32)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -8.5e-98], N[(N[(y / x), $MachinePrecision] - y), $MachinePrecision], If[LessEqual[x, 1.9e-32], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1

   1. Initial program 63.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 63.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 63.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 63.5%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 63.5%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 63.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 63.5%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. associate-*r/ 79.3%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 79.3%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 32.3%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 79.3%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 79.3%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 79.3%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 79.3%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 79.3%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 79.3%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 63.5%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 26.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 26.8%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 63.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 63.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. +-commutative 63.5%

         \[\leadsto \frac{y \cdot x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      7. associate-+r+ 63.5%

         \[\leadsto \frac{y \cdot x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      8. frac-times 80.9%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{y + \left(x + 1\right)}} \]

      9. *-commutative 80.9%

         \[\leadsto \color{blue}{\frac{x}{y + \left(x + 1\right)} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      10. clear-num 80.9%

          \[\leadsto \color{blue}{\frac{1}{\frac{y + \left(x + 1\right)}{x}}} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \]

      11. associate-/r* 99.7%

          \[\leadsto \frac{1}{\frac{y + \left(x + 1\right)}{x}} \cdot \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \]

      12. frac-times 99.8%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{y}{x + y}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)}} \]

      13. *-un-lft-identity 99.8%

          \[\leadsto \frac{\color{blue}{\frac{y}{x + y}}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)} \]

      14. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)} \]

      15. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{y + x}}{\frac{y + \left(x + 1\right)}{x} \cdot \color{blue}{\left(y + x\right)}} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(y + x\right)}} \]

   6. Taylor expanded in x around inf 80.9%

      \[\leadsto \frac{\frac{y}{y + x}}{\color{blue}{x}} \]

   7. Taylor expanded in y around 0 80.6%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x} \]

   if -1 < x < -8.4999999999999997e-98

   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. times-frac 99.6%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 99.6%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 99.6%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around 0 37.2%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 37.2%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 37.2%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   6. Simplified 37.2%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   7. Taylor expanded in x around 0 37.2%

      \[\leadsto \color{blue}{-1 \cdot y + \frac{y}{x}} \]
   8. Step-by-step derivation

      1. neg-mul-1 37.2%

         \[\leadsto \color{blue}{\left(-y\right)} + \frac{y}{x} \]

      2. +-commutative 37.2%

         \[\leadsto \color{blue}{\frac{y}{x} + \left(-y\right)} \]

      3. unsub-neg 37.2%

         \[\leadsto \color{blue}{\frac{y}{x} - y} \]

   9. Simplified 37.2%

      \[\leadsto \color{blue}{\frac{y}{x} - y} \]

   if -8.4999999999999997e-98 < x < 1.90000000000000004e-32

   1. Initial program 75.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 91.1%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 91.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 91.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 91.1%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around 0 82.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 82.1%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   6. Simplified 82.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   if 1.90000000000000004e-32 < x

   1. Initial program 72.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 72.3%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 72.3%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 72.3%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 72.3%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. *-commutative 72.3%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)\right)} \]

      6. associate-*l* 72.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      7. times-frac 88.7%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 88.7%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 88.7%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. associate-+l+ 88.7%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 88.7%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around inf 30.5%

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{y + \left(x + 1\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 30.6%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y + \left(x + 1\right)}}{y}} \]

      2. *-un-lft-identity 30.6%

         \[\leadsto \frac{\color{blue}{\frac{x}{y + \left(x + 1\right)}}}{y} \]

   6. Applied egg-rr 30.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + \left(x + 1\right)}}{y}} \]

   7. Taylor expanded in y around inf 29.0%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 11: 82.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.8e-170)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 1.35e+154) (/ x (* y (+ y (+ x 1.0)))) (/ (/ x y) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.8e-170) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 1.35e+154) {
		tmp = x / (y * (y + (x + 1.0)));
	} else {
		tmp = (x / y) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.8d-170) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 1.35d+154) then
        tmp = x / (y * (y + (x + 1.0d0)))
    else
        tmp = (x / y) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.8e-170) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 1.35e+154) {
		tmp = x / (y * (y + (x + 1.0)));
	} else {
		tmp = (x / y) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.8e-170:
		tmp = (y / x) / (x + 1.0)
	elif y <= 1.35e+154:
		tmp = x / (y * (y + (x + 1.0)))
	else:
		tmp = (x / y) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.8e-170)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 1.35e+154)
		tmp = Float64(x / Float64(y * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / y) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.8e-170)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 1.35e+154)
		tmp = x / (y * (y + (x + 1.0)));
	else
		tmp = (x / y) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.8e-170], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+154], N[(x / N[(y * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.8000000000000002e-170

   1. Initial program 73.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 89.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 89.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 89.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 89.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around 0 53.7%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 55.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 55.7%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   6. Simplified 55.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if 1.8000000000000002e-170 < y < 1.35000000000000003e154

   1. Initial program 76.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 76.3%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 76.3%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 76.3%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 76.3%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. *-commutative 76.3%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)\right)} \]

      6. associate-*l* 76.3%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      7. times-frac 91.6%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 91.6%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 91.6%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. associate-+l+ 91.6%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 91.6%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around inf 47.5%

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{y + \left(x + 1\right)} \]
   5. Step-by-step derivation

      1. frac-times 54.3%

         \[\leadsto \color{blue}{\frac{1 \cdot x}{y \cdot \left(y + \left(x + 1\right)\right)}} \]

      2. *-un-lft-identity 54.3%

         \[\leadsto \frac{\color{blue}{x}}{y \cdot \left(y + \left(x + 1\right)\right)} \]

   6. Applied egg-rr 54.3%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + \left(x + 1\right)\right)}} \]

   if 1.35000000000000003e154 < y

   1. Initial program 58.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 58.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 58.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 58.5%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 58.5%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 58.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 58.5%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. associate-*r/ 75.9%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 75.9%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 75.8%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 75.9%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 75.9%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 75.9%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 75.9%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 75.9%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 75.9%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 58.5%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. *-commutative 58.5%

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      3. fma-udef 58.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      4. cube-mult 58.5%

         \[\leadsto \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. distribute-rgt1-in 58.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      6. *-commutative 58.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. +-commutative 58.5%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      8. associate-+r+ 58.5%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      9. frac-times 75.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

      10. associate-*l/ 75.9%

          \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{y + \left(x + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      11. associate-/r* 99.8%

          \[\leadsto \color{blue}{\frac{\frac{x \cdot \frac{y}{y + \left(x + 1\right)}}{x + y}}{x + y}} \]

      12. clear-num 99.9%

          \[\leadsto \frac{\frac{x \cdot \color{blue}{\frac{1}{\frac{y + \left(x + 1\right)}{y}}}}{x + y}}{x + y} \]

      13. un-div-inv 99.9%

          \[\leadsto \frac{\frac{\color{blue}{\frac{x}{\frac{y + \left(x + 1\right)}{y}}}}{x + y}}{x + y} \]

      14. +-commutative 99.9%

          \[\leadsto \frac{\frac{\frac{x}{\frac{y + \left(x + 1\right)}{y}}}{\color{blue}{y + x}}}{x + y} \]

      15. +-commutative 99.9%

          \[\leadsto \frac{\frac{\frac{x}{\frac{y + \left(x + 1\right)}{y}}}{y + x}}{\color{blue}{y + x}} \]

   5. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{\frac{x}{\frac{y + \left(x + 1\right)}{y}}}{y + x}}{y + x}} \]

   6. Taylor expanded in y around inf 95.4%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y + x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \]

Alternative 12: 75.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.7 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-98}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ (/ y x) x)
   (if (<= x -5.7e-97)
     (- (/ y x) y)
     (if (<= x 2.25e-98) (/ x y) (/ (/ x y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) / x;
	} else if (x <= -5.7e-97) {
		tmp = (y / x) - y;
	} else if (x <= 2.25e-98) {
		tmp = x / y;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (y / x) / x
    else if (x <= (-5.7d-97)) then
        tmp = (y / x) - y
    else if (x <= 2.25d-98) then
        tmp = x / y
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) / x;
	} else if (x <= -5.7e-97) {
		tmp = (y / x) - y;
	} else if (x <= 2.25e-98) {
		tmp = x / y;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (y / x) / x
	elif x <= -5.7e-97:
		tmp = (y / x) - y
	elif x <= 2.25e-98:
		tmp = x / y
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -5.7e-97)
		tmp = Float64(Float64(y / x) - y);
	elseif (x <= 2.25e-98)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (y / x) / x;
	elseif (x <= -5.7e-97)
		tmp = (y / x) - y;
	elseif (x <= 2.25e-98)
		tmp = x / y;
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -5.7e-97], N[(N[(y / x), $MachinePrecision] - y), $MachinePrecision], If[LessEqual[x, 2.25e-98], N[(x / y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1

   1. Initial program 63.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 63.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 63.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 63.5%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 63.5%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 63.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 63.5%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. associate-*r/ 79.3%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 79.3%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 32.3%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 79.3%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 79.3%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 79.3%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 79.3%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 79.3%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 79.3%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 63.5%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 26.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 26.8%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 63.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 63.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. +-commutative 63.5%

         \[\leadsto \frac{y \cdot x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      7. associate-+r+ 63.5%

         \[\leadsto \frac{y \cdot x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      8. frac-times 80.9%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{y + \left(x + 1\right)}} \]

      9. *-commutative 80.9%

         \[\leadsto \color{blue}{\frac{x}{y + \left(x + 1\right)} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      10. clear-num 80.9%

          \[\leadsto \color{blue}{\frac{1}{\frac{y + \left(x + 1\right)}{x}}} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \]

      11. associate-/r* 99.7%

          \[\leadsto \frac{1}{\frac{y + \left(x + 1\right)}{x}} \cdot \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \]

      12. frac-times 99.8%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{y}{x + y}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)}} \]

      13. *-un-lft-identity 99.8%

          \[\leadsto \frac{\color{blue}{\frac{y}{x + y}}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)} \]

      14. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)} \]

      15. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{y + x}}{\frac{y + \left(x + 1\right)}{x} \cdot \color{blue}{\left(y + x\right)}} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(y + x\right)}} \]

   6. Taylor expanded in x around inf 80.9%

      \[\leadsto \frac{\frac{y}{y + x}}{\color{blue}{x}} \]

   7. Taylor expanded in y around 0 80.6%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x} \]

   if -1 < x < -5.7000000000000001e-97

   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. times-frac 99.6%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 99.6%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 99.6%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around 0 37.2%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 37.2%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 37.2%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   6. Simplified 37.2%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   7. Taylor expanded in x around 0 37.2%

      \[\leadsto \color{blue}{-1 \cdot y + \frac{y}{x}} \]
   8. Step-by-step derivation

      1. neg-mul-1 37.2%

         \[\leadsto \color{blue}{\left(-y\right)} + \frac{y}{x} \]

      2. +-commutative 37.2%

         \[\leadsto \color{blue}{\frac{y}{x} + \left(-y\right)} \]

      3. unsub-neg 37.2%

         \[\leadsto \color{blue}{\frac{y}{x} - y} \]

   9. Simplified 37.2%

      \[\leadsto \color{blue}{\frac{y}{x} - y} \]

   if -5.7000000000000001e-97 < x < 2.24999999999999998e-98

   1. Initial program 72.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 89.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 89.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 89.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 89.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around 0 87.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 87.1%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   6. Simplified 87.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   7. Taylor expanded in y around 0 71.6%

      \[\leadsto \color{blue}{\frac{x}{y}} \]

   if 2.24999999999999998e-98 < x

   1. Initial program 76.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 76.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 76.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 76.8%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 76.8%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. *-commutative 76.8%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)\right)} \]

      6. associate-*l* 76.9%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      7. times-frac 90.4%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 90.4%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 90.4%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. associate-+l+ 90.4%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 90.4%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around inf 33.7%

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{y + \left(x + 1\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 33.7%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y + \left(x + 1\right)}}{y}} \]

      2. *-un-lft-identity 33.7%

         \[\leadsto \frac{\color{blue}{\frac{x}{y + \left(x + 1\right)}}}{y} \]

   6. Applied egg-rr 33.7%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + \left(x + 1\right)}}{y}} \]

   7. Taylor expanded in y around inf 28.2%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.7 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-98}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 13: 77.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-170}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 1.66 \cdot 10^{+149}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.8e-170)
   (/ y (* x (+ x 1.0)))
   (if (<= y 6.5e+130)
     (/ x (* y (+ y 1.0)))
     (if (<= y 1.66e+149) (/ (/ y x) x) (/ (/ x y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.8e-170) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 6.5e+130) {
		tmp = x / (y * (y + 1.0));
	} else if (y <= 1.66e+149) {
		tmp = (y / x) / x;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.8d-170) then
        tmp = y / (x * (x + 1.0d0))
    else if (y <= 6.5d+130) then
        tmp = x / (y * (y + 1.0d0))
    else if (y <= 1.66d+149) then
        tmp = (y / x) / x
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.8e-170) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 6.5e+130) {
		tmp = x / (y * (y + 1.0));
	} else if (y <= 1.66e+149) {
		tmp = (y / x) / x;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.8e-170:
		tmp = y / (x * (x + 1.0))
	elif y <= 6.5e+130:
		tmp = x / (y * (y + 1.0))
	elif y <= 1.66e+149:
		tmp = (y / x) / x
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.8e-170)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (y <= 6.5e+130)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	elseif (y <= 1.66e+149)
		tmp = Float64(Float64(y / x) / x);
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.8e-170)
		tmp = y / (x * (x + 1.0));
	elseif (y <= 6.5e+130)
		tmp = x / (y * (y + 1.0));
	elseif (y <= 1.66e+149)
		tmp = (y / x) / x;
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.8e-170], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.5e+130], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.66e+149], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 1.8000000000000002e-170

   1. Initial program 73.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 89.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 89.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 89.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 89.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around 0 53.7%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   if 1.8000000000000002e-170 < y < 6.5e130

   1. Initial program 83.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 92.1%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 92.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 92.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 92.1%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around 0 49.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 49.8%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   6. Simplified 49.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   if 6.5e130 < y < 1.6600000000000001e149

   1. Initial program 14.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 14.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 14.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 14.9%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 14.9%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 14.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 14.9%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. associate-*r/ 72.9%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 72.9%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 29.2%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 72.9%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 72.9%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 72.9%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 72.9%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 72.9%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 72.9%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 14.9%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 14.9%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 14.9%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 14.9%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 14.9%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. +-commutative 14.9%

         \[\leadsto \frac{y \cdot x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      7. associate-+r+ 14.9%

         \[\leadsto \frac{y \cdot x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      8. frac-times 86.6%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{y + \left(x + 1\right)}} \]

      9. *-commutative 86.6%

         \[\leadsto \color{blue}{\frac{x}{y + \left(x + 1\right)} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      10. clear-num 86.6%

          \[\leadsto \color{blue}{\frac{1}{\frac{y + \left(x + 1\right)}{x}}} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \]

      11. associate-/r* 100.0%

          \[\leadsto \frac{1}{\frac{y + \left(x + 1\right)}{x}} \cdot \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \]

      12. frac-times 99.8%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{y}{x + y}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)}} \]

      13. *-un-lft-identity 99.8%

          \[\leadsto \frac{\color{blue}{\frac{y}{x + y}}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)} \]

      14. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)} \]

      15. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{y + x}}{\frac{y + \left(x + 1\right)}{x} \cdot \color{blue}{\left(y + x\right)}} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(y + x\right)}} \]

   6. Taylor expanded in x around inf 73.4%

      \[\leadsto \frac{\frac{y}{y + x}}{\color{blue}{x}} \]

   7. Taylor expanded in y around 0 73.2%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x} \]

   if 1.6600000000000001e149 < y

   1. Initial program 58.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 58.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 58.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 58.5%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 58.5%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. *-commutative 58.5%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)\right)} \]

      6. associate-*l* 58.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      7. times-frac 75.9%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 75.9%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 75.9%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. associate-+l+ 75.9%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 75.9%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around inf 95.4%

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{y + \left(x + 1\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 95.4%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y + \left(x + 1\right)}}{y}} \]

      2. *-un-lft-identity 95.4%

         \[\leadsto \frac{\color{blue}{\frac{x}{y + \left(x + 1\right)}}}{y} \]

   6. Applied egg-rr 95.4%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + \left(x + 1\right)}}{y}} \]

   7. Taylor expanded in y around inf 95.3%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-170}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 1.66 \cdot 10^{+149}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 14: 81.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+171}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.65e+171)
   (/ (/ y x) x)
   (if (<= x -5.5e-98) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.65e+171) {
		tmp = (y / x) / x;
	} else if (x <= -5.5e-98) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.65d+171)) then
        tmp = (y / x) / x
    else if (x <= (-5.5d-98)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.65e+171) {
		tmp = (y / x) / x;
	} else if (x <= -5.5e-98) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.65e+171:
		tmp = (y / x) / x
	elif x <= -5.5e-98:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.65e+171)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -5.5e-98)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.65e+171)
		tmp = (y / x) / x;
	elseif (x <= -5.5e-98)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.65e+171], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -5.5e-98], 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 3 regimes

2. if x < -1.64999999999999996e171

   1. Initial program 57.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 57.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 57.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 57.9%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 57.9%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 57.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 57.9%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. associate-*r/ 77.4%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 77.4%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 3.0%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 77.4%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 77.4%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 77.4%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 77.4%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 77.4%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 77.4%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 57.9%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      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 57.9%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 57.9%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. +-commutative 57.9%

         \[\leadsto \frac{y \cdot x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      7. associate-+r+ 57.9%

         \[\leadsto \frac{y \cdot x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      8. frac-times 77.4%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{y + \left(x + 1\right)}} \]

      9. *-commutative 77.4%

         \[\leadsto \color{blue}{\frac{x}{y + \left(x + 1\right)} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      10. clear-num 77.4%

          \[\leadsto \color{blue}{\frac{1}{\frac{y + \left(x + 1\right)}{x}}} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \]

      11. associate-/r* 99.9%

          \[\leadsto \frac{1}{\frac{y + \left(x + 1\right)}{x}} \cdot \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \]

      12. frac-times 99.9%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{y}{x + y}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)}} \]

      13. *-un-lft-identity 99.9%

          \[\leadsto \frac{\color{blue}{\frac{y}{x + y}}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)} \]

      14. +-commutative 99.9%

          \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)} \]

      15. +-commutative 99.9%

          \[\leadsto \frac{\frac{y}{y + x}}{\frac{y + \left(x + 1\right)}{x} \cdot \color{blue}{\left(y + x\right)}} \]

   5. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(y + x\right)}} \]

   6. Taylor expanded in x around inf 87.8%

      \[\leadsto \frac{\frac{y}{y + x}}{\color{blue}{x}} \]

   7. Taylor expanded in y around 0 87.6%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x} \]

   if -1.64999999999999996e171 < x < -5.4999999999999997e-98

   1. Initial program 76.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 93.0%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 93.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 93.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 93.0%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around 0 58.4%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   if -5.4999999999999997e-98 < x

   1. Initial program 74.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 74.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 74.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 74.4%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 74.4%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 74.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 74.4%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. associate-*r/ 85.3%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 85.3%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 77.5%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 85.3%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 85.3%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 85.3%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 85.3%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 85.3%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 85.3%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 74.4%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 67.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 67.0%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 74.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 74.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. +-commutative 74.4%

         \[\leadsto \frac{y \cdot x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      7. associate-+r+ 74.4%

         \[\leadsto \frac{y \cdot x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      8. frac-times 90.2%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{y + \left(x + 1\right)}} \]

      9. *-commutative 90.2%

         \[\leadsto \color{blue}{\frac{x}{y + \left(x + 1\right)} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      10. clear-num 90.2%

          \[\leadsto \color{blue}{\frac{1}{\frac{y + \left(x + 1\right)}{x}}} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \]

      11. associate-/r* 99.7%

          \[\leadsto \frac{1}{\frac{y + \left(x + 1\right)}{x}} \cdot \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \]

      12. frac-times 99.3%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{y}{x + y}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)}} \]

      13. *-un-lft-identity 99.3%

          \[\leadsto \frac{\color{blue}{\frac{y}{x + y}}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)} \]

      14. +-commutative 99.3%

          \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)} \]

      15. +-commutative 99.3%

          \[\leadsto \frac{\frac{y}{y + x}}{\frac{y + \left(x + 1\right)}{x} \cdot \color{blue}{\left(y + x\right)}} \]

   5. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(y + x\right)}} \]
   6. Step-by-step derivation

      1. clear-num 99.2%

         \[\leadsto \frac{\color{blue}{\frac{1}{\frac{y + x}{y}}}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(y + x\right)} \]

      2. inv-pow 99.2%

         \[\leadsto \frac{\color{blue}{{\left(\frac{y + x}{y}\right)}^{-1}}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(y + x\right)} \]

   7. Applied egg-rr 99.2%

      \[\leadsto \frac{\color{blue}{{\left(\frac{y + x}{y}\right)}^{-1}}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(y + x\right)} \]
   8. Step-by-step derivation

      1. unpow-1 99.2%

         \[\leadsto \frac{\color{blue}{\frac{1}{\frac{y + x}{y}}}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(y + x\right)} \]

   9. Simplified 99.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{y + x}{y}}}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(y + x\right)} \]

   10. Taylor expanded in x around 0 59.3%

       \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   11. Step-by-step derivation

       1. associate-/r* 61.3%

          \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

       2. +-commutative 61.3%

          \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y + 1}} \]

   12. Simplified 61.3%

       \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+171}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 15: 82.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+166}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.9e+166)
   (/ (/ y x) x)
   (if (<= x -3.6e-96) (/ y (* x (+ x 1.0))) (/ (/ x (+ y 1.0)) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.9e+166) {
		tmp = (y / x) / x;
	} else if (x <= -3.6e-96) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.9d+166)) then
        tmp = (y / x) / x
    else if (x <= (-3.6d-96)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.9e+166) {
		tmp = (y / x) / x;
	} else if (x <= -3.6e-96) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.9e+166:
		tmp = (y / x) / x
	elif x <= -3.6e-96:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.9e+166)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -3.6e-96)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.9e+166)
		tmp = (y / x) / x;
	elseif (x <= -3.6e-96)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.9e+166], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -3.6e-96], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.9000000000000001e166

   1. Initial program 57.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 57.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 57.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 57.9%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 57.9%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 57.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 57.9%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. associate-*r/ 77.4%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 77.4%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 3.0%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 77.4%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 77.4%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 77.4%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 77.4%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 77.4%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 77.4%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 57.9%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      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 57.9%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 57.9%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. +-commutative 57.9%

         \[\leadsto \frac{y \cdot x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      7. associate-+r+ 57.9%

         \[\leadsto \frac{y \cdot x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      8. frac-times 77.4%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{y + \left(x + 1\right)}} \]

      9. *-commutative 77.4%

         \[\leadsto \color{blue}{\frac{x}{y + \left(x + 1\right)} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      10. clear-num 77.4%

          \[\leadsto \color{blue}{\frac{1}{\frac{y + \left(x + 1\right)}{x}}} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \]

      11. associate-/r* 99.9%

          \[\leadsto \frac{1}{\frac{y + \left(x + 1\right)}{x}} \cdot \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \]

      12. frac-times 99.9%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{y}{x + y}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)}} \]

      13. *-un-lft-identity 99.9%

          \[\leadsto \frac{\color{blue}{\frac{y}{x + y}}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)} \]

      14. +-commutative 99.9%

          \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)} \]

      15. +-commutative 99.9%

          \[\leadsto \frac{\frac{y}{y + x}}{\frac{y + \left(x + 1\right)}{x} \cdot \color{blue}{\left(y + x\right)}} \]

   5. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(y + x\right)}} \]

   6. Taylor expanded in x around inf 87.8%

      \[\leadsto \frac{\frac{y}{y + x}}{\color{blue}{x}} \]

   7. Taylor expanded in y around 0 87.6%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x} \]

   if -2.9000000000000001e166 < x < -3.60000000000000008e-96

   1. Initial program 76.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 93.0%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 93.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 93.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 93.0%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around 0 58.4%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   if -3.60000000000000008e-96 < x

   1. Initial program 74.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 74.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 74.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 74.4%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 74.4%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. *-commutative 74.4%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)\right)} \]

      6. associate-*l* 74.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      7. times-frac 90.2%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 90.2%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 90.2%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. associate-+l+ 90.2%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 90.2%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around inf 61.5%

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{y + \left(x + 1\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 61.6%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y + \left(x + 1\right)}}{y}} \]

      2. *-un-lft-identity 61.6%

         \[\leadsto \frac{\color{blue}{\frac{x}{y + \left(x + 1\right)}}}{y} \]

   6. Applied egg-rr 61.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + \left(x + 1\right)}}{y}} \]

   7. Taylor expanded in x around 0 61.3%

      \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{y} \]
   8. Step-by-step derivation

      1. +-commutative 61.3%

         \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{y} \]

   9. Simplified 61.3%

      \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+166}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]

Alternative 16: 81.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.8e-170) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ y x)) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.8e-170) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + x)) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.8d-170) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (y + x)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.8e-170) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + x)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.8e-170:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (y + x)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.8e-170)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.8e-170)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (y + x)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.8e-170], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.8000000000000002e-170

   1. Initial program 73.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 89.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 89.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 89.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 89.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around 0 53.7%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 55.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 55.7%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   6. Simplified 55.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if 1.8000000000000002e-170 < y

   1. Initial program 71.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 71.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 71.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 71.0%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 71.0%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 71.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 71.0%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. associate-*r/ 83.5%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 83.5%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 75.6%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 83.5%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 83.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 83.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 83.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 83.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 83.5%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 71.0%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. *-commutative 71.0%

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      3. fma-udef 66.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      4. cube-mult 66.2%

         \[\leadsto \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. distribute-rgt1-in 71.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      6. *-commutative 71.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. +-commutative 71.0%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      8. associate-+r+ 71.0%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      9. frac-times 86.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

      10. *-commutative 86.9%

          \[\leadsto \color{blue}{\frac{y}{y + \left(x + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      11. clear-num 86.9%

          \[\leadsto \color{blue}{\frac{1}{\frac{y + \left(x + 1\right)}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      12. associate-/r* 99.7%

          \[\leadsto \frac{1}{\frac{y + \left(x + 1\right)}{y}} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      13. frac-times 99.8%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{y + \left(x + 1\right)}{y} \cdot \left(x + y\right)}} \]

      14. *-un-lft-identity 99.8%

          \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{\frac{y + \left(x + 1\right)}{y} \cdot \left(x + y\right)} \]

      15. +-commutative 99.8%

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{\frac{y + \left(x + 1\right)}{y} \cdot \left(x + y\right)} \]

      16. +-commutative 99.8%

          \[\leadsto \frac{\frac{x}{y + x}}{\frac{y + \left(x + 1\right)}{y} \cdot \color{blue}{\left(y + x\right)}} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{y + \left(x + 1\right)}{y} \cdot \left(y + x\right)}} \]

   6. Taylor expanded in x around 0 62.1%

      \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{1 + y}} \]
   7. Step-by-step derivation

      1. +-commutative 62.1%

         \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + 1}} \]

   8. Simplified 62.1%

      \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \]

Alternative 17: 81.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.8e-170) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ y (+ x 1.0))) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.8e-170) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + (x + 1.0))) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.8d-170) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (y + (x + 1.0d0))) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.8e-170) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + (x + 1.0))) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.8e-170:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (y + (x + 1.0))) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.8e-170)
		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

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.8e-170)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (y + (x + 1.0))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.8e-170], 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 y < 1.8000000000000002e-170

   1. Initial program 73.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 89.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 89.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 89.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 89.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around 0 53.7%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 55.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 55.7%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   6. Simplified 55.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if 1.8000000000000002e-170 < y

   1. Initial program 71.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 71.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 71.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 71.0%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 71.0%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. *-commutative 71.0%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)\right)} \]

      6. associate-*l* 71.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      7. times-frac 86.9%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 86.9%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 86.9%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. associate-+l+ 86.9%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 86.9%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around inf 61.9%

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{y + \left(x + 1\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 62.0%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y + \left(x + 1\right)}}{y}} \]

      2. *-un-lft-identity 62.0%

         \[\leadsto \frac{\color{blue}{\frac{x}{y + \left(x + 1\right)}}}{y} \]

   6. Applied egg-rr 62.0%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + \left(x + 1\right)}}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \]

Alternative 18: 80.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-176}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 7.5e-176) (/ (/ y (+ y x)) (+ x 1.0)) (/ (/ x (+ y (+ x 1.0))) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 7.5e-176) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = (x / (y + (x + 1.0))) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 7.5d-176) then
        tmp = (y / (y + x)) / (x + 1.0d0)
    else
        tmp = (x / (y + (x + 1.0d0))) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 7.5e-176) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = (x / (y + (x + 1.0))) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 7.5e-176:
		tmp = (y / (y + x)) / (x + 1.0)
	else:
		tmp = (x / (y + (x + 1.0))) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 7.5e-176)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7.5e-176)
		tmp = (y / (y + x)) / (x + 1.0);
	else
		tmp = (x / (y + (x + 1.0))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 7.5e-176], N[(N[(y / N[(y + x), $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 y < 7.5e-176

   1. Initial program 73.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 73.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 73.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 73.7%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 73.7%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 73.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 73.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. associate-*r/ 86.0%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 86.0%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 61.8%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 86.0%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 86.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 86.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 86.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 86.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 86.0%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 73.7%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 52.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 52.0%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 73.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 73.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. +-commutative 73.7%

         \[\leadsto \frac{y \cdot x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      7. associate-+r+ 73.7%

         \[\leadsto \frac{y \cdot x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      8. frac-times 90.6%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{y + \left(x + 1\right)}} \]

      9. *-commutative 90.6%

         \[\leadsto \color{blue}{\frac{x}{y + \left(x + 1\right)} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      10. clear-num 90.6%

          \[\leadsto \color{blue}{\frac{1}{\frac{y + \left(x + 1\right)}{x}}} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \]

      11. associate-/r* 99.7%

          \[\leadsto \frac{1}{\frac{y + \left(x + 1\right)}{x}} \cdot \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \]

      12. frac-times 99.4%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{y}{x + y}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)}} \]

      13. *-un-lft-identity 99.4%

          \[\leadsto \frac{\color{blue}{\frac{y}{x + y}}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)} \]

      14. +-commutative 99.4%

          \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)} \]

      15. +-commutative 99.4%

          \[\leadsto \frac{\frac{y}{y + x}}{\frac{y + \left(x + 1\right)}{x} \cdot \color{blue}{\left(y + x\right)}} \]

   5. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(y + x\right)}} \]

   6. Taylor expanded in y around 0 56.5%

      \[\leadsto \frac{\frac{y}{y + x}}{\color{blue}{1 + x}} \]
   7. Step-by-step derivation

      1. +-commutative 56.5%

         \[\leadsto \frac{\frac{y}{y + x}}{\color{blue}{x + 1}} \]

   8. Simplified 56.5%

      \[\leadsto \frac{\frac{y}{y + x}}{\color{blue}{x + 1}} \]

   if 7.5e-176 < y

   1. Initial program 70.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 70.3%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 70.3%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 70.3%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 70.3%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. *-commutative 70.3%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)\right)} \]

      6. associate-*l* 70.3%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      7. times-frac 86.1%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 86.1%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 86.1%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. associate-+l+ 86.1%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 86.1%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around inf 62.3%

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{y + \left(x + 1\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 62.4%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y + \left(x + 1\right)}}{y}} \]

      2. *-un-lft-identity 62.4%

         \[\leadsto \frac{\color{blue}{\frac{x}{y + \left(x + 1\right)}}}{y} \]

   6. Applied egg-rr 62.4%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + \left(x + 1\right)}}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-176}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \]

Alternative 19: 56.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{-170}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;y \leq 0.76:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.7e-170)
   (- (/ y x) y)
   (if (<= y 0.76) (- (/ x y) x) (/ (/ x y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.7e-170) {
		tmp = (y / x) - y;
	} else if (y <= 0.76) {
		tmp = (x / y) - x;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.7d-170) then
        tmp = (y / x) - y
    else if (y <= 0.76d0) then
        tmp = (x / y) - x
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.7e-170) {
		tmp = (y / x) - y;
	} else if (y <= 0.76) {
		tmp = (x / y) - x;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.7e-170:
		tmp = (y / x) - y
	elif y <= 0.76:
		tmp = (x / y) - x
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.7e-170)
		tmp = Float64(Float64(y / x) - y);
	elseif (y <= 0.76)
		tmp = Float64(Float64(x / y) - x);
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.7e-170)
		tmp = (y / x) - y;
	elseif (y <= 0.76)
		tmp = (x / y) - x;
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.7e-170], N[(N[(y / x), $MachinePrecision] - y), $MachinePrecision], If[LessEqual[y, 0.76], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.70000000000000006e-170

   1. Initial program 73.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 89.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 89.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 89.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 89.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around 0 53.7%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 55.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 55.7%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   6. Simplified 55.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   7. Taylor expanded in x around 0 18.4%

      \[\leadsto \color{blue}{-1 \cdot y + \frac{y}{x}} \]
   8. Step-by-step derivation

      1. neg-mul-1 18.4%

         \[\leadsto \color{blue}{\left(-y\right)} + \frac{y}{x} \]

      2. +-commutative 18.4%

         \[\leadsto \color{blue}{\frac{y}{x} + \left(-y\right)} \]

      3. unsub-neg 18.4%

         \[\leadsto \color{blue}{\frac{y}{x} - y} \]

   9. Simplified 18.4%

      \[\leadsto \color{blue}{\frac{y}{x} - y} \]

   if 1.70000000000000006e-170 < y < 0.76000000000000001

   1. Initial program 87.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 96.7%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 96.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 96.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 96.7%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around 0 46.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 46.7%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   6. Simplified 46.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   7. Taylor expanded in y around 0 46.3%

      \[\leadsto \color{blue}{-1 \cdot x + \frac{x}{y}} \]
   8. Step-by-step derivation

      1. neg-mul-1 46.3%

         \[\leadsto \color{blue}{\left(-x\right)} + \frac{x}{y} \]

      2. +-commutative 46.3%

         \[\leadsto \color{blue}{\frac{x}{y} + \left(-x\right)} \]

      3. unsub-neg 46.3%

         \[\leadsto \color{blue}{\frac{x}{y} - x} \]

   9. Simplified 46.3%

      \[\leadsto \color{blue}{\frac{x}{y} - x} \]

   if 0.76000000000000001 < y

   1. Initial program 59.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 59.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 59.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 59.2%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 59.2%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. *-commutative 59.2%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)\right)} \]

      6. associate-*l* 59.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      7. times-frac 79.8%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 79.8%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 79.8%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. associate-+l+ 79.8%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 79.8%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around inf 72.6%

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{y + \left(x + 1\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 72.7%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y + \left(x + 1\right)}}{y}} \]

      2. *-un-lft-identity 72.7%

         \[\leadsto \frac{\color{blue}{\frac{x}{y + \left(x + 1\right)}}}{y} \]

   6. Applied egg-rr 72.7%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + \left(x + 1\right)}}{y}} \]

   7. Taylor expanded in y around inf 72.3%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{-170}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;y \leq 0.76:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 20: 82.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-97}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.35e-97) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ y 1.0)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.35e-97) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.35d-97)) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.35e-97) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.35e-97:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.35e-97)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.35e-97)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.35e-97], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.34999999999999993e-97

   1. Initial program 67.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 85.6%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 85.6%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around 0 67.4%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 71.0%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 71.0%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   6. Simplified 71.0%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if -1.34999999999999993e-97 < x

   1. Initial program 74.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 74.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 74.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 74.4%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 74.4%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. *-commutative 74.4%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)\right)} \]

      6. associate-*l* 74.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      7. times-frac 90.2%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 90.2%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 90.2%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. associate-+l+ 90.2%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 90.2%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around inf 61.5%

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{y + \left(x + 1\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 61.6%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y + \left(x + 1\right)}}{y}} \]

      2. *-un-lft-identity 61.6%

         \[\leadsto \frac{\color{blue}{\frac{x}{y + \left(x + 1\right)}}}{y} \]

   6. Applied egg-rr 61.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + \left(x + 1\right)}}{y}} \]

   7. Taylor expanded in x around 0 61.3%

      \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{y} \]
   8. Step-by-step derivation

      1. +-commutative 61.3%

         \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{y} \]

   9. Simplified 61.3%

      \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-97}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]

Alternative 21: 34.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-170}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 1.8e-170) (- (/ y x) y) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.8e-170) {
		tmp = (y / x) - y;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.8d-170) then
        tmp = (y / x) - y
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.8e-170) {
		tmp = (y / x) - y;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.8e-170:
		tmp = (y / x) - y
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.8e-170)
		tmp = Float64(Float64(y / x) - y);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.8e-170)
		tmp = (y / x) - y;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.8e-170], N[(N[(y / x), $MachinePrecision] - y), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.8000000000000002e-170

   1. Initial program 73.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 89.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 89.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 89.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 89.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around 0 53.7%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 55.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 55.7%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   6. Simplified 55.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   7. Taylor expanded in x around 0 18.4%

      \[\leadsto \color{blue}{-1 \cdot y + \frac{y}{x}} \]
   8. Step-by-step derivation

      1. neg-mul-1 18.4%

         \[\leadsto \color{blue}{\left(-y\right)} + \frac{y}{x} \]

      2. +-commutative 18.4%

         \[\leadsto \color{blue}{\frac{y}{x} + \left(-y\right)} \]

      3. unsub-neg 18.4%

         \[\leadsto \color{blue}{\frac{y}{x} - y} \]

   9. Simplified 18.4%

      \[\leadsto \color{blue}{\frac{y}{x} - y} \]

   if 1.8000000000000002e-170 < y

   1. Initial program 71.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 86.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 86.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 86.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 86.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around 0 55.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 55.8%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   6. Simplified 55.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   7. Taylor expanded in y around 0 32.9%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-170}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 22: 4.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{0.5}{y} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ 0.5 y))

C

assert(x < y);
double code(double x, double y) {
	return 0.5 / y;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.5d0 / y
end function

Java

assert x < y;
public static double code(double x, double y) {
	return 0.5 / y;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return 0.5 / y

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(0.5 / y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 0.5 / y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(0.5 / y), $MachinePrecision]

Derivation

1. Initial program 72.3%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 72.3%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

   2. +-commutative 72.3%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

   3. +-commutative 72.3%

      \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

   4. +-commutative 72.3%

      \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

   5. associate-*l* 72.3%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

   6. *-commutative 72.3%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

   7. associate-*r/ 84.7%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

   8. *-commutative 84.7%

      \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   9. distribute-rgt1-in 67.1%

      \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   10. fma-def 84.7%

       \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

   11. +-commutative 84.7%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   12. +-commutative 84.7%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   13. cube-unmult 84.7%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

   14. +-commutative 84.7%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

3. Simplified 84.7%

   \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ 72.3%

      \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   2. fma-udef 57.5%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   3. cube-mult 57.5%

      \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   4. distribute-rgt1-in 72.3%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   5. *-commutative 72.3%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   6. +-commutative 72.3%

      \[\leadsto \frac{y \cdot x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

   7. associate-+r+ 72.3%

      \[\leadsto \frac{y \cdot x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

   8. frac-times 88.8%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{y + \left(x + 1\right)}} \]

   9. *-commutative 88.8%

      \[\leadsto \color{blue}{\frac{x}{y + \left(x + 1\right)} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   10. clear-num 88.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{y + \left(x + 1\right)}{x}}} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \]

   11. associate-/r* 99.7%

       \[\leadsto \frac{1}{\frac{y + \left(x + 1\right)}{x}} \cdot \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \]

   12. frac-times 99.5%

       \[\leadsto \color{blue}{\frac{1 \cdot \frac{y}{x + y}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)}} \]

   13. *-un-lft-identity 99.5%

       \[\leadsto \frac{\color{blue}{\frac{y}{x + y}}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)} \]

   14. +-commutative 99.5%

       \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)} \]

   15. +-commutative 99.5%

       \[\leadsto \frac{\frac{y}{y + x}}{\frac{y + \left(x + 1\right)}{x} \cdot \color{blue}{\left(y + x\right)}} \]

5. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(y + x\right)}} \]

6. Taylor expanded in x around -inf 50.4%

   \[\leadsto \frac{\frac{y}{y + x}}{\color{blue}{x + -1 \cdot \left(-1 \cdot y + -1 \cdot \left(1 + y\right)\right)}} \]
7. Step-by-step derivation

   1. mul-1-neg 50.4%

      \[\leadsto \frac{\frac{y}{y + x}}{x + \color{blue}{\left(-\left(-1 \cdot y + -1 \cdot \left(1 + y\right)\right)\right)}} \]

   2. unsub-neg 50.4%

      \[\leadsto \frac{\frac{y}{y + x}}{\color{blue}{x - \left(-1 \cdot y + -1 \cdot \left(1 + y\right)\right)}} \]

   3. neg-mul-1 50.4%

      \[\leadsto \frac{\frac{y}{y + x}}{x - \left(\color{blue}{\left(-y\right)} + -1 \cdot \left(1 + y\right)\right)} \]

   4. +-commutative 50.4%

      \[\leadsto \frac{\frac{y}{y + x}}{x - \color{blue}{\left(-1 \cdot \left(1 + y\right) + \left(-y\right)\right)}} \]

   5. unsub-neg 50.4%

      \[\leadsto \frac{\frac{y}{y + x}}{x - \color{blue}{\left(-1 \cdot \left(1 + y\right) - y\right)}} \]

   6. distribute-lft-in 50.4%

      \[\leadsto \frac{\frac{y}{y + x}}{x - \left(\color{blue}{\left(-1 \cdot 1 + -1 \cdot y\right)} - y\right)} \]

   7. metadata-eval 50.4%

      \[\leadsto \frac{\frac{y}{y + x}}{x - \left(\left(\color{blue}{-1} + -1 \cdot y\right) - y\right)} \]

   8. neg-mul-1 50.4%

      \[\leadsto \frac{\frac{y}{y + x}}{x - \left(\left(-1 + \color{blue}{\left(-y\right)}\right) - y\right)} \]

   9. unsub-neg 50.4%

      \[\leadsto \frac{\frac{y}{y + x}}{x - \left(\color{blue}{\left(-1 - y\right)} - y\right)} \]

8. Simplified 50.4%

   \[\leadsto \frac{\frac{y}{y + x}}{\color{blue}{x - \left(\left(-1 - y\right) - y\right)}} \]

9. Taylor expanded in y around inf 4.0%

   \[\leadsto \color{blue}{\frac{0.5}{y}} \]

10. Final simplification 4.0%

    \[\leadsto \frac{0.5}{y} \]

Alternative 23: 4.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{1}{x} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ 1.0 x))

C

assert(x < y);
double code(double x, double y) {
	return 1.0 / x;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / x
end function

Java

assert x < y;
public static double code(double x, double y) {
	return 1.0 / x;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return 1.0 / x

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(1.0 / x)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 1.0 / x;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(1.0 / x), $MachinePrecision]

Derivation

1. Initial program 72.3%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 72.3%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

   2. +-commutative 72.3%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

   3. +-commutative 72.3%

      \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

   4. +-commutative 72.3%

      \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

   5. associate-*l* 72.3%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

   6. *-commutative 72.3%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

   7. associate-*r/ 84.7%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

   8. *-commutative 84.7%

      \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   9. distribute-rgt1-in 67.1%

      \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   10. fma-def 84.7%

       \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

   11. +-commutative 84.7%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   12. +-commutative 84.7%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   13. cube-unmult 84.7%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

   14. +-commutative 84.7%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

3. Simplified 84.7%

   \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ 72.3%

      \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   2. fma-udef 57.5%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   3. cube-mult 57.5%

      \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   4. distribute-rgt1-in 72.3%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   5. *-commutative 72.3%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   6. +-commutative 72.3%

      \[\leadsto \frac{y \cdot x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

   7. associate-+r+ 72.3%

      \[\leadsto \frac{y \cdot x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

   8. frac-times 88.8%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{y + \left(x + 1\right)}} \]

   9. *-commutative 88.8%

      \[\leadsto \color{blue}{\frac{x}{y + \left(x + 1\right)} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   10. clear-num 88.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{y + \left(x + 1\right)}{x}}} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \]

   11. associate-/r* 99.7%

       \[\leadsto \frac{1}{\frac{y + \left(x + 1\right)}{x}} \cdot \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \]

   12. frac-times 99.5%

       \[\leadsto \color{blue}{\frac{1 \cdot \frac{y}{x + y}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)}} \]

   13. *-un-lft-identity 99.5%

       \[\leadsto \frac{\color{blue}{\frac{y}{x + y}}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)} \]

   14. +-commutative 99.5%

       \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(x + y\right)} \]

   15. +-commutative 99.5%

       \[\leadsto \frac{\frac{y}{y + x}}{\frac{y + \left(x + 1\right)}{x} \cdot \color{blue}{\left(y + x\right)}} \]

5. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\frac{y + \left(x + 1\right)}{x} \cdot \left(y + x\right)}} \]

6. Taylor expanded in x around inf 38.3%

   \[\leadsto \frac{\frac{y}{y + x}}{\color{blue}{x}} \]

7. Taylor expanded in y around inf 4.3%

   \[\leadsto \color{blue}{\frac{1}{x}} \]

8. Final simplification 4.3%

   \[\leadsto \frac{1}{x} \]

Alternative 24: 26.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{x}{y} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ x y))

C

assert(x < y);
double code(double x, double y) {
	return x / y;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / y
end function

Java

assert x < y;
public static double code(double x, double y) {
	return x / y;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return x / y

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x / y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x / y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 72.3%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. times-frac 88.7%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. +-commutative 88.7%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   3. associate-+l+ 88.7%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

3. Simplified 88.7%

   \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

4. Taylor expanded in x around 0 48.2%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
5. Step-by-step derivation

   1. +-commutative 48.2%

      \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

6. Simplified 48.2%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

7. Taylor expanded in y around 0 29.6%

   \[\leadsto \color{blue}{\frac{x}{y}} \]

8. Final simplification 29.6%

   \[\leadsto \frac{x}{y} \]

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 2023336 
(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))))