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

Percentage Accurate: 69.7% → 99.8%

Time: 16.8s

Alternatives: 23

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: 69.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return ((y / (y + (1.0 + x))) * (x / (y + x))) / (y + 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 + (1.0d0 + x))) * (x / (y + x))) / (y + x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.0%

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

   1. associate-*r/ 84.4%

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

   2. *-commutative 84.4%

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

   3. distribute-rgt1-in 67.7%

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

   4. fma-def 84.5%

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

   5. cube-unmult 84.5%

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

3. Simplified 84.5%

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

   1. associate-*r/ 70.0%

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

   2. fma-udef 55.5%

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

   3. cube-mult 55.5%

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

   4. distribute-rgt1-in 70.0%

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

   5. associate-+r+ 70.0%

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

   6. *-commutative 70.0%

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

   7. frac-times 88.9%

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

   8. *-commutative 88.9%

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

   9. associate-/r* 99.8%

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

   10. associate-*r/ 99.8%

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

   11. +-commutative 99.8%

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

   12. associate-+l+ 99.8%

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

   13. +-commutative 99.8%

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

   14. +-commutative 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 90.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{+122}:\\ \;\;\;\;\frac{\frac{y}{y + \left(1 + x\right)}}{y + x}\\ \mathbf{elif}\;x \leq -2600000:\\ \;\;\;\;\frac{y}{x} \cdot t_0\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-158}:\\ \;\;\;\;t_0 \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* (+ y x) (+ y x)))))
   (if (<= x -3.6e+122)
     (/ (/ y (+ y (+ 1.0 x))) (+ y x))
     (if (<= x -2600000.0)
       (* (/ y x) t_0)
       (if (<= x -2.6e-158) (* t_0 (/ y (+ y 1.0))) (/ (/ x y) (+ y 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / ((y + x) * (y + x));
	double tmp;
	if (x <= -3.6e+122) {
		tmp = (y / (y + (1.0 + x))) / (y + x);
	} else if (x <= -2600000.0) {
		tmp = (y / x) * t_0;
	} else if (x <= -2.6e-158) {
		tmp = t_0 * (y / (y + 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) :: t_0
    real(8) :: tmp
    t_0 = x / ((y + x) * (y + x))
    if (x <= (-3.6d+122)) then
        tmp = (y / (y + (1.0d0 + x))) / (y + x)
    else if (x <= (-2600000.0d0)) then
        tmp = (y / x) * t_0
    else if (x <= (-2.6d-158)) then
        tmp = t_0 * (y / (y + 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 t_0 = x / ((y + x) * (y + x));
	double tmp;
	if (x <= -3.6e+122) {
		tmp = (y / (y + (1.0 + x))) / (y + x);
	} else if (x <= -2600000.0) {
		tmp = (y / x) * t_0;
	} else if (x <= -2.6e-158) {
		tmp = t_0 * (y / (y + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / ((y + x) * (y + x))
	tmp = 0
	if x <= -3.6e+122:
		tmp = (y / (y + (1.0 + x))) / (y + x)
	elif x <= -2600000.0:
		tmp = (y / x) * t_0
	elif x <= -2.6e-158:
		tmp = t_0 * (y / (y + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(Float64(y + x) * Float64(y + x)))
	tmp = 0.0
	if (x <= -3.6e+122)
		tmp = Float64(Float64(y / Float64(y + Float64(1.0 + x))) / Float64(y + x));
	elseif (x <= -2600000.0)
		tmp = Float64(Float64(y / x) * t_0);
	elseif (x <= -2.6e-158)
		tmp = Float64(t_0 * Float64(y / Float64(y + 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)
	t_0 = x / ((y + x) * (y + x));
	tmp = 0.0;
	if (x <= -3.6e+122)
		tmp = (y / (y + (1.0 + x))) / (y + x);
	elseif (x <= -2600000.0)
		tmp = (y / x) * t_0;
	elseif (x <= -2.6e-158)
		tmp = t_0 * (y / (y + 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_] := Block[{t$95$0 = N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.6e+122], N[(N[(y / N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2600000.0], N[(N[(y / x), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, -2.6e-158], N[(t$95$0 * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.6000000000000003e122

   1. Initial program 66.0%

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

      1. associate-*r/ 76.5%

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

      2. *-commutative 76.5%

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

      3. distribute-rgt1-in 13.8%

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

      4. fma-def 76.5%

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

      5. cube-unmult 76.5%

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

   3. Simplified 76.5%

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

      1. associate-*r/ 66.0%

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

      2. fma-udef 10.8%

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

      3. cube-mult 10.8%

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

      4. distribute-rgt1-in 66.0%

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

      5. associate-+r+ 66.0%

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

      6. *-commutative 66.0%

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

      7. frac-times 77.8%

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

      8. *-commutative 77.8%

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

      9. associate-/r* 99.8%

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

      10. frac-times 77.8%

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

      11. +-commutative 77.8%

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

      12. +-commutative 77.8%

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

      13. associate-+l+ 77.8%

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

      14. +-commutative 77.8%

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

   5. Applied egg-rr 77.8%

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

      1. associate-/r* 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*r/ 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-*r/ 73.1%

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

      7. +-commutative 73.1%

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

   7. Simplified 73.1%

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

   8. Taylor expanded in y around 0 82.5%

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

   if -3.6000000000000003e122 < x < -2.6e6

   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 96.3%

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

      2. /-rgt-identity 96.3%

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

      3. associate-/l/ 96.3%

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

      4. *-lft-identity 96.3%

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

      5. associate-+l+ 96.3%

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

   3. Simplified 96.3%

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

   4. Taylor expanded in x around inf 77.2%

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

   if -2.6e6 < x < -2.6e-158

   1. Initial program 88.9%

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

      1. times-frac 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. /-rgt-identity 99.7%

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

      3. associate-/l/ 99.7%

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

      4. *-lft-identity 99.7%

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

      5. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 97.5%

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

      1. +-commutative 97.5%

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

   6. Simplified 97.5%

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

   if -2.6e-158 < x

   1. Initial program 66.2%

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

      1. associate-*r/ 82.8%

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

      2. *-commutative 82.8%

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

      3. distribute-rgt1-in 72.0%

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

      4. fma-def 82.8%

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

      5. cube-unmult 82.8%

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

   3. Simplified 82.8%

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

      1. associate-*r/ 66.2%

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

      2. fma-udef 56.8%

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

      3. cube-mult 56.8%

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

      4. distribute-rgt1-in 66.2%

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

      5. associate-+r+ 66.2%

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

      6. *-commutative 66.2%

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

      7. frac-times 87.5%

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

      8. *-commutative 87.5%

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

      9. associate-/r* 99.8%

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

      10. associate-*r/ 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in x around 0 59.2%

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

       1. associate-/r* 59.3%

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

   12. Simplified 59.3%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+122}:\\ \;\;\;\;\frac{\frac{y}{y + \left(1 + x\right)}}{y + x}\\ \mathbf{elif}\;x \leq -2600000:\\ \;\;\;\;\frac{y}{x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-158}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 3: 91.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(1 + x\right)\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{y}{t_0}}{y + x}\\ \mathbf{elif}\;x \leq -3.45 \cdot 10^{-9}:\\ \;\;\;\;y \cdot \frac{\frac{1}{t_0}}{y + x}\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-158}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ 1.0 x))))
   (if (<= x -1.9e+170)
     (/ (/ y t_0) (+ y x))
     (if (<= x -3.45e-9)
       (* y (/ (/ 1.0 t_0) (+ y x)))
       (if (<= x -2.6e-158)
         (* (/ x (* (+ y x) (+ y x))) (/ y (+ y 1.0)))
         (/ (/ x y) (+ y 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (1.0 + x);
	double tmp;
	if (x <= -1.9e+170) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= -3.45e-9) {
		tmp = y * ((1.0 / t_0) / (y + x));
	} else if (x <= -2.6e-158) {
		tmp = (x / ((y + x) * (y + x))) * (y / (y + 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) :: t_0
    real(8) :: tmp
    t_0 = y + (1.0d0 + x)
    if (x <= (-1.9d+170)) then
        tmp = (y / t_0) / (y + x)
    else if (x <= (-3.45d-9)) then
        tmp = y * ((1.0d0 / t_0) / (y + x))
    else if (x <= (-2.6d-158)) then
        tmp = (x / ((y + x) * (y + x))) * (y / (y + 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 t_0 = y + (1.0 + x);
	double tmp;
	if (x <= -1.9e+170) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= -3.45e-9) {
		tmp = y * ((1.0 / t_0) / (y + x));
	} else if (x <= -2.6e-158) {
		tmp = (x / ((y + x) * (y + x))) * (y / (y + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (1.0 + x)
	tmp = 0
	if x <= -1.9e+170:
		tmp = (y / t_0) / (y + x)
	elif x <= -3.45e-9:
		tmp = y * ((1.0 / t_0) / (y + x))
	elif x <= -2.6e-158:
		tmp = (x / ((y + x) * (y + x))) * (y / (y + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(1.0 + x))
	tmp = 0.0
	if (x <= -1.9e+170)
		tmp = Float64(Float64(y / t_0) / Float64(y + x));
	elseif (x <= -3.45e-9)
		tmp = Float64(y * Float64(Float64(1.0 / t_0) / Float64(y + x)));
	elseif (x <= -2.6e-158)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + x))) * Float64(y / Float64(y + 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)
	t_0 = y + (1.0 + x);
	tmp = 0.0;
	if (x <= -1.9e+170)
		tmp = (y / t_0) / (y + x);
	elseif (x <= -3.45e-9)
		tmp = y * ((1.0 / t_0) / (y + x));
	elseif (x <= -2.6e-158)
		tmp = (x / ((y + x) * (y + x))) * (y / (y + 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_] := Block[{t$95$0 = N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.9e+170], N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.45e-9], N[(y * N[(N[(1.0 / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.6e-158], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.8999999999999999e170

   1. Initial program 70.1%

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

      1. associate-*r/ 80.2%

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

      2. *-commutative 80.2%

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

      3. distribute-rgt1-in 4.3%

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

      4. fma-def 80.2%

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

      5. cube-unmult 80.2%

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

   3. Simplified 80.2%

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

      1. associate-*r/ 70.1%

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

      2. fma-udef 0.0%

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

      3. cube-mult 0.0%

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

      4. distribute-rgt1-in 70.1%

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

      5. associate-+r+ 70.1%

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

      6. *-commutative 70.1%

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

      7. frac-times 80.2%

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

      8. *-commutative 80.2%

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

      9. associate-/r* 99.8%

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

      10. frac-times 80.2%

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

      11. +-commutative 80.2%

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

      12. +-commutative 80.2%

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

      13. associate-+l+ 80.2%

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

      14. +-commutative 80.2%

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

   5. Applied egg-rr 80.2%

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

      1. associate-/r* 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*r/ 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-*r/ 74.4%

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

      7. +-commutative 74.4%

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

   7. Simplified 74.4%

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

   8. Taylor expanded in y around 0 87.0%

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

   if -1.8999999999999999e170 < x < -3.44999999999999987e-9

   1. Initial program 68.9%

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

      1. associate-*r/ 80.1%

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

      2. *-commutative 80.1%

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

      3. distribute-rgt1-in 66.7%

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

      4. fma-def 80.1%

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

      5. cube-unmult 80.1%

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

   3. Simplified 80.1%

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

      1. associate-*r/ 68.8%

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

      2. fma-udef 60.9%

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

      3. cube-mult 61.0%

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

      4. distribute-rgt1-in 68.9%

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

      5. associate-+r+ 68.9%

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

      6. *-commutative 68.9%

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

      7. frac-times 90.2%

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

      8. *-commutative 90.2%

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

      9. associate-/r* 99.7%

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

      10. frac-times 90.4%

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

      11. +-commutative 90.4%

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

      12. +-commutative 90.4%

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

      13. associate-+l+ 90.4%

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

      14. +-commutative 90.4%

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

   5. Applied egg-rr 90.4%

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

      1. associate-/r* 99.7%

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

      2. associate-*l/ 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*r/ 99.7%

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

      5. *-commutative 99.7%

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

      6. associate-*r/ 82.0%

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

      7. +-commutative 82.0%

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

   7. Simplified 82.0%

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

   8. Taylor expanded in y around 0 59.7%

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

      1. +-commutative 59.7%

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

      2. div-inv 59.8%

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

      3. *-un-lft-identity 59.8%

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

      4. times-frac 77.8%

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

   10. Applied egg-rr 77.8%

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

   if -3.44999999999999987e-9 < x < -2.6e-158

   1. Initial program 88.6%

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

      1. times-frac 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. /-rgt-identity 99.6%

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

      3. associate-/l/ 99.6%

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

      4. *-lft-identity 99.6%

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

      5. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.4%

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

      1. +-commutative 99.4%

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

   6. Simplified 99.4%

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

   if -2.6e-158 < x

   1. Initial program 66.2%

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

      1. associate-*r/ 82.8%

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

      2. *-commutative 82.8%

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

      3. distribute-rgt1-in 72.0%

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

      4. fma-def 82.8%

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

      5. cube-unmult 82.8%

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

   3. Simplified 82.8%

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

      1. associate-*r/ 66.2%

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

      2. fma-udef 56.8%

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

      3. cube-mult 56.8%

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

      4. distribute-rgt1-in 66.2%

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

      5. associate-+r+ 66.2%

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

      6. *-commutative 66.2%

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

      7. frac-times 87.5%

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

      8. *-commutative 87.5%

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

      9. associate-/r* 99.8%

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

      10. associate-*r/ 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in x around 0 59.2%

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

       1. associate-/r* 59.3%

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

   12. Simplified 59.3%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{y}{y + \left(1 + x\right)}}{y + x}\\ \mathbf{elif}\;x \leq -3.45 \cdot 10^{-9}:\\ \;\;\;\;y \cdot \frac{\frac{1}{y + \left(1 + x\right)}}{y + x}\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-158}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 4: 96.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+122}:\\ \;\;\;\;\frac{\frac{y}{y + \left(1 + x\right)}}{y + x}\\ \mathbf{elif}\;x \leq -7.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.6e+122)
   (/ (/ y (+ y (+ 1.0 x))) (+ y x))
   (if (<= x -7.6e-17)
     (* (/ x (* (+ y x) (+ y x))) (/ y (+ x (+ y 1.0))))
     (/ (* (/ x (+ y x)) (/ y (+ y 1.0))) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.6e+122) {
		tmp = (y / (y + (1.0 + x))) / (y + x);
	} else if (x <= -7.6e-17) {
		tmp = (x / ((y + x) * (y + x))) * (y / (x + (y + 1.0)));
	} else {
		tmp = ((x / (y + x)) * (y / (y + 1.0))) / (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 (x <= (-3.6d+122)) then
        tmp = (y / (y + (1.0d0 + x))) / (y + x)
    else if (x <= (-7.6d-17)) then
        tmp = (x / ((y + x) * (y + x))) * (y / (x + (y + 1.0d0)))
    else
        tmp = ((x / (y + x)) * (y / (y + 1.0d0))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -3.6e+122) {
		tmp = (y / (y + (1.0 + x))) / (y + x);
	} else if (x <= -7.6e-17) {
		tmp = (x / ((y + x) * (y + x))) * (y / (x + (y + 1.0)));
	} else {
		tmp = ((x / (y + x)) * (y / (y + 1.0))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.6e+122:
		tmp = (y / (y + (1.0 + x))) / (y + x)
	elif x <= -7.6e-17:
		tmp = (x / ((y + x) * (y + x))) * (y / (x + (y + 1.0)))
	else:
		tmp = ((x / (y + x)) * (y / (y + 1.0))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.6e+122)
		tmp = Float64(Float64(y / Float64(y + Float64(1.0 + x))) / Float64(y + x));
	elseif (x <= -7.6e-17)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + x))) * Float64(y / Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(y + 1.0))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.6e+122)
		tmp = (y / (y + (1.0 + x))) / (y + x);
	elseif (x <= -7.6e-17)
		tmp = (x / ((y + x) * (y + x))) * (y / (x + (y + 1.0)));
	else
		tmp = ((x / (y + x)) * (y / (y + 1.0))) / (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[x, -3.6e+122], N[(N[(y / N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7.6e-17], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.6000000000000003e122

   1. Initial program 66.0%

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

      1. associate-*r/ 76.5%

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

      2. *-commutative 76.5%

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

      3. distribute-rgt1-in 13.8%

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

      4. fma-def 76.5%

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

      5. cube-unmult 76.5%

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

   3. Simplified 76.5%

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

      1. associate-*r/ 66.0%

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

      2. fma-udef 10.8%

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

      3. cube-mult 10.8%

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

      4. distribute-rgt1-in 66.0%

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

      5. associate-+r+ 66.0%

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

      6. *-commutative 66.0%

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

      7. frac-times 77.8%

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

      8. *-commutative 77.8%

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

      9. associate-/r* 99.8%

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

      10. frac-times 77.8%

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

      11. +-commutative 77.8%

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

      12. +-commutative 77.8%

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

      13. associate-+l+ 77.8%

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

      14. +-commutative 77.8%

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

   5. Applied egg-rr 77.8%

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

      1. associate-/r* 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*r/ 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-*r/ 73.1%

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

      7. +-commutative 73.1%

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

   7. Simplified 73.1%

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

   8. Taylor expanded in y around 0 82.5%

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

   if -3.6000000000000003e122 < x < -7.6000000000000002e-17

   1. Initial program 74.8%

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

      1. times-frac 96.6%

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

      2. /-rgt-identity 96.6%

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

      3. associate-/l/ 96.6%

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

      4. *-lft-identity 96.6%

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

      5. associate-+l+ 96.6%

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

   3. Simplified 96.6%

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

   if -7.6000000000000002e-17 < x

   1. Initial program 70.0%

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

      1. associate-*r/ 85.7%

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

      2. *-commutative 85.7%

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

      3. distribute-rgt1-in 75.2%

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

      4. fma-def 85.7%

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

      5. cube-unmult 85.7%

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

   3. Simplified 85.7%

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

      1. associate-*r/ 70.0%

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

      2. fma-udef 60.6%

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

      3. cube-mult 60.6%

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

      4. distribute-rgt1-in 70.0%

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

      5. associate-+r+ 70.0%

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

      6. *-commutative 70.0%

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

      7. frac-times 89.6%

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

      8. *-commutative 89.6%

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

      9. associate-/r* 99.8%

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

      10. associate-*r/ 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 84.3%

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

      1. +-commutative 84.3%

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

   8. Simplified 84.3%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+122}:\\ \;\;\;\;\frac{\frac{y}{y + \left(1 + x\right)}}{y + x}\\ \mathbf{elif}\;x \leq -7.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{y + 1}}{y + x}\\ \end{array} \]

Alternative 5: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+169}:\\ \;\;\;\;\frac{\frac{y}{y + \left(1 + x\right)} \cdot \left(1 - \frac{y}{x}\right)}{y + x}\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -4.2e+169)
   (/ (* (/ y (+ y (+ 1.0 x))) (- 1.0 (/ y x))) (+ y x))
   (if (<= x -7.8e-17)
     (* (/ x (* (+ y x) (+ y x))) (/ y (+ x (+ y 1.0))))
     (/ (* (/ x (+ y x)) (/ y (+ y 1.0))) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4.2e+169) {
		tmp = ((y / (y + (1.0 + x))) * (1.0 - (y / x))) / (y + x);
	} else if (x <= -7.8e-17) {
		tmp = (x / ((y + x) * (y + x))) * (y / (x + (y + 1.0)));
	} else {
		tmp = ((x / (y + x)) * (y / (y + 1.0))) / (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 (x <= (-4.2d+169)) then
        tmp = ((y / (y + (1.0d0 + x))) * (1.0d0 - (y / x))) / (y + x)
    else if (x <= (-7.8d-17)) then
        tmp = (x / ((y + x) * (y + x))) * (y / (x + (y + 1.0d0)))
    else
        tmp = ((x / (y + x)) * (y / (y + 1.0d0))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -4.2e+169) {
		tmp = ((y / (y + (1.0 + x))) * (1.0 - (y / x))) / (y + x);
	} else if (x <= -7.8e-17) {
		tmp = (x / ((y + x) * (y + x))) * (y / (x + (y + 1.0)));
	} else {
		tmp = ((x / (y + x)) * (y / (y + 1.0))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -4.2e+169:
		tmp = ((y / (y + (1.0 + x))) * (1.0 - (y / x))) / (y + x)
	elif x <= -7.8e-17:
		tmp = (x / ((y + x) * (y + x))) * (y / (x + (y + 1.0)))
	else:
		tmp = ((x / (y + x)) * (y / (y + 1.0))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -4.2e+169)
		tmp = Float64(Float64(Float64(y / Float64(y + Float64(1.0 + x))) * Float64(1.0 - Float64(y / x))) / Float64(y + x));
	elseif (x <= -7.8e-17)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + x))) * Float64(y / Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(y + 1.0))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.2e+169)
		tmp = ((y / (y + (1.0 + x))) * (1.0 - (y / x))) / (y + x);
	elseif (x <= -7.8e-17)
		tmp = (x / ((y + x) * (y + x))) * (y / (x + (y + 1.0)));
	else
		tmp = ((x / (y + x)) * (y / (y + 1.0))) / (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[x, -4.2e+169], N[(N[(N[(y / N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7.8e-17], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.2000000000000002e169

   1. Initial program 70.1%

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

      1. associate-*r/ 80.2%

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

      2. *-commutative 80.2%

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

      3. distribute-rgt1-in 4.3%

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

      4. fma-def 80.2%

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

      5. cube-unmult 80.2%

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

   3. Simplified 80.2%

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

      1. associate-*r/ 70.1%

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

      2. fma-udef 0.0%

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

      3. cube-mult 0.0%

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

      4. distribute-rgt1-in 70.1%

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

      5. associate-+r+ 70.1%

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

      6. *-commutative 70.1%

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

      7. frac-times 80.2%

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

      8. *-commutative 80.2%

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

      9. associate-/r* 99.8%

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

      10. associate-*r/ 99.8%

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

      11. +-commutative 99.8%

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

      12. associate-+l+ 99.8%

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

      13. +-commutative 99.8%

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

      14. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around inf 87.7%

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

      1. mul-1-neg 87.7%

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

      2. unsub-neg 87.7%

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

   8. Simplified 87.7%

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

   if -4.2000000000000002e169 < x < -7.79999999999999979e-17

   1. Initial program 70.4%

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

      1. times-frac 90.7%

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

      2. /-rgt-identity 90.7%

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

      3. associate-/l/ 90.7%

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

      4. *-lft-identity 90.7%

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

      5. associate-+l+ 90.7%

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

   3. Simplified 90.7%

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

   if -7.79999999999999979e-17 < x

   1. Initial program 70.0%

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

      1. associate-*r/ 85.7%

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

      2. *-commutative 85.7%

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

      3. distribute-rgt1-in 75.2%

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

      4. fma-def 85.7%

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

      5. cube-unmult 85.7%

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

   3. Simplified 85.7%

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

      1. associate-*r/ 70.0%

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

      2. fma-udef 60.6%

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

      3. cube-mult 60.6%

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

      4. distribute-rgt1-in 70.0%

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

      5. associate-+r+ 70.0%

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

      6. *-commutative 70.0%

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

      7. frac-times 89.6%

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

      8. *-commutative 89.6%

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

      9. associate-/r* 99.8%

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

      10. associate-*r/ 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 84.3%

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

      1. +-commutative 84.3%

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

   8. Simplified 84.3%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+169}:\\ \;\;\;\;\frac{\frac{y}{y + \left(1 + x\right)} \cdot \left(1 - \frac{y}{x}\right)}{y + x}\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{y + 1}}{y + x}\\ \end{array} \]

Alternative 6: 94.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(1 + x\right)\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{+168}:\\ \;\;\;\;\frac{\frac{y}{t_0}}{y + x}\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-9}:\\ \;\;\;\;y \cdot \frac{\frac{1}{t_0}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ 1.0 x))))
   (if (<= x -2.9e+168)
     (/ (/ y t_0) (+ y x))
     (if (<= x -3.3e-9)
       (* y (/ (/ 1.0 t_0) (+ y x)))
       (/ (* (/ x (+ y x)) (/ y (+ y 1.0))) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (1.0 + x);
	double tmp;
	if (x <= -2.9e+168) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= -3.3e-9) {
		tmp = y * ((1.0 / t_0) / (y + x));
	} else {
		tmp = ((x / (y + x)) * (y / (y + 1.0))) / (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) :: t_0
    real(8) :: tmp
    t_0 = y + (1.0d0 + x)
    if (x <= (-2.9d+168)) then
        tmp = (y / t_0) / (y + x)
    else if (x <= (-3.3d-9)) then
        tmp = y * ((1.0d0 / t_0) / (y + x))
    else
        tmp = ((x / (y + x)) * (y / (y + 1.0d0))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (1.0 + x);
	double tmp;
	if (x <= -2.9e+168) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= -3.3e-9) {
		tmp = y * ((1.0 / t_0) / (y + x));
	} else {
		tmp = ((x / (y + x)) * (y / (y + 1.0))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (1.0 + x)
	tmp = 0
	if x <= -2.9e+168:
		tmp = (y / t_0) / (y + x)
	elif x <= -3.3e-9:
		tmp = y * ((1.0 / t_0) / (y + x))
	else:
		tmp = ((x / (y + x)) * (y / (y + 1.0))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(1.0 + x))
	tmp = 0.0
	if (x <= -2.9e+168)
		tmp = Float64(Float64(y / t_0) / Float64(y + x));
	elseif (x <= -3.3e-9)
		tmp = Float64(y * Float64(Float64(1.0 / t_0) / Float64(y + x)));
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(y + 1.0))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (1.0 + x);
	tmp = 0.0;
	if (x <= -2.9e+168)
		tmp = (y / t_0) / (y + x);
	elseif (x <= -3.3e-9)
		tmp = y * ((1.0 / t_0) / (y + x));
	else
		tmp = ((x / (y + x)) * (y / (y + 1.0))) / (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_] := Block[{t$95$0 = N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.9e+168], N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.3e-9], N[(y * N[(N[(1.0 / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.9e168

   1. Initial program 70.1%

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

      1. associate-*r/ 80.2%

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

      2. *-commutative 80.2%

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

      3. distribute-rgt1-in 4.3%

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

      4. fma-def 80.2%

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

      5. cube-unmult 80.2%

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

   3. Simplified 80.2%

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

      1. associate-*r/ 70.1%

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

      2. fma-udef 0.0%

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

      3. cube-mult 0.0%

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

      4. distribute-rgt1-in 70.1%

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

      5. associate-+r+ 70.1%

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

      6. *-commutative 70.1%

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

      7. frac-times 80.2%

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

      8. *-commutative 80.2%

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

      9. associate-/r* 99.8%

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

      10. frac-times 80.2%

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

      11. +-commutative 80.2%

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

      12. +-commutative 80.2%

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

      13. associate-+l+ 80.2%

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

      14. +-commutative 80.2%

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

   5. Applied egg-rr 80.2%

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

      1. associate-/r* 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*r/ 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-*r/ 74.4%

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

      7. +-commutative 74.4%

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

   7. Simplified 74.4%

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

   8. Taylor expanded in y around 0 87.0%

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

   if -2.9e168 < x < -3.30000000000000018e-9

   1. Initial program 68.9%

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

      1. associate-*r/ 80.1%

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

      2. *-commutative 80.1%

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

      3. distribute-rgt1-in 66.7%

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

      4. fma-def 80.1%

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

      5. cube-unmult 80.1%

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

   3. Simplified 80.1%

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

      1. associate-*r/ 68.8%

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

      2. fma-udef 60.9%

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

      3. cube-mult 61.0%

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

      4. distribute-rgt1-in 68.9%

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

      5. associate-+r+ 68.9%

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

      6. *-commutative 68.9%

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

      7. frac-times 90.2%

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

      8. *-commutative 90.2%

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

      9. associate-/r* 99.7%

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

      10. frac-times 90.4%

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

      11. +-commutative 90.4%

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

      12. +-commutative 90.4%

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

      13. associate-+l+ 90.4%

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

      14. +-commutative 90.4%

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

   5. Applied egg-rr 90.4%

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

      1. associate-/r* 99.7%

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

      2. associate-*l/ 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*r/ 99.7%

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

      5. *-commutative 99.7%

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

      6. associate-*r/ 82.0%

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

      7. +-commutative 82.0%

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

   7. Simplified 82.0%

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

   8. Taylor expanded in y around 0 59.7%

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

      1. +-commutative 59.7%

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

      2. div-inv 59.8%

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

      3. *-un-lft-identity 59.8%

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

      4. times-frac 77.8%

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

   10. Applied egg-rr 77.8%

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

   if -3.30000000000000018e-9 < x

   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-*r/ 85.8%

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

      2. *-commutative 85.8%

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

      3. distribute-rgt1-in 75.4%

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

      4. fma-def 85.8%

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

      5. cube-unmult 85.9%

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

   3. Simplified 85.9%

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

      1. associate-*r/ 70.3%

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

      2. fma-udef 61.0%

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

      3. cube-mult 61.0%

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

      4. distribute-rgt1-in 70.3%

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

      5. associate-+r+ 70.3%

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

      6. *-commutative 70.3%

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

      7. frac-times 89.7%

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

      8. *-commutative 89.7%

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

      9. associate-/r* 99.8%

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

      10. associate-*r/ 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 84.5%

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

      1. +-commutative 84.5%

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

   8. Simplified 84.5%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+168}:\\ \;\;\;\;\frac{\frac{y}{y + \left(1 + x\right)}}{y + x}\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-9}:\\ \;\;\;\;y \cdot \frac{\frac{1}{y + \left(1 + x\right)}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{y + 1}}{y + x}\\ \end{array} \]

Alternative 7: 82.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-168}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{\frac{x}{y}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -6.5e+30)
   (/ (/ y x) x)
   (if (<= y 2.3e-168)
     (/ y (* x (+ 1.0 x)))
     (if (<= y 8e+93) (/ x (+ y (* y y))) (/ 1.0 (/ y (/ x y)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -6.5e+30) {
		tmp = (y / x) / x;
	} else if (y <= 2.3e-168) {
		tmp = y / (x * (1.0 + x));
	} else if (y <= 8e+93) {
		tmp = x / (y + (y * y));
	} else {
		tmp = 1.0 / (y / (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 <= (-6.5d+30)) then
        tmp = (y / x) / x
    else if (y <= 2.3d-168) then
        tmp = y / (x * (1.0d0 + x))
    else if (y <= 8d+93) then
        tmp = x / (y + (y * y))
    else
        tmp = 1.0d0 / (y / (x / y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= -6.5e+30) {
		tmp = (y / x) / x;
	} else if (y <= 2.3e-168) {
		tmp = y / (x * (1.0 + x));
	} else if (y <= 8e+93) {
		tmp = x / (y + (y * y));
	} else {
		tmp = 1.0 / (y / (x / y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -6.5e+30:
		tmp = (y / x) / x
	elif y <= 2.3e-168:
		tmp = y / (x * (1.0 + x))
	elif y <= 8e+93:
		tmp = x / (y + (y * y))
	else:
		tmp = 1.0 / (y / (x / y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -6.5e+30)
		tmp = Float64(Float64(y / x) / x);
	elseif (y <= 2.3e-168)
		tmp = Float64(y / Float64(x * Float64(1.0 + x)));
	elseif (y <= 8e+93)
		tmp = Float64(x / Float64(y + Float64(y * y)));
	else
		tmp = Float64(1.0 / Float64(y / 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 <= -6.5e+30)
		tmp = (y / x) / x;
	elseif (y <= 2.3e-168)
		tmp = y / (x * (1.0 + x));
	elseif (y <= 8e+93)
		tmp = x / (y + (y * y));
	else
		tmp = 1.0 / (y / (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, -6.5e+30], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 2.3e-168], N[(y / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e+93], N[(x / N[(y + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -6.5e30

   1. Initial program 57.5%

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

      1. times-frac 81.5%

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

      2. /-rgt-identity 81.5%

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

      3. associate-/l/ 81.5%

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

      4. *-lft-identity 81.5%

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

      5. associate-+l+ 81.5%

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

   3. Simplified 81.5%

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

   4. Taylor expanded in x around inf 17.2%

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

   5. Taylor expanded in x around inf 16.0%

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

      1. associate-*l/ 16.0%

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

      2. *-un-lft-identity 16.0%

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

   7. Applied egg-rr 16.0%

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

   if -6.5e30 < y < 2.29999999999999986e-168

   1. Initial program 73.4%

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

      1. times-frac 89.8%

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

      2. /-rgt-identity 89.8%

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

      3. associate-/l/ 89.8%

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

      4. *-lft-identity 89.8%

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

      5. associate-+l+ 89.8%

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

   3. Simplified 89.8%

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

   4. Taylor expanded in y around 0 78.4%

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

   if 2.29999999999999986e-168 < y < 8.00000000000000035e93

   1. Initial program 84.2%

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

      1. times-frac 96.8%

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

      2. /-rgt-identity 96.8%

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

      3. associate-/l/ 96.8%

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

      4. *-lft-identity 96.8%

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

      5. associate-+l+ 96.8%

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

   3. Simplified 96.8%

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

   4. Taylor expanded in x around 0 47.0%

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

      1. distribute-rgt-in 47.0%

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

      2. *-lft-identity 47.0%

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

   6. Simplified 47.0%

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

   if 8.00000000000000035e93 < y

   1. Initial program 63.9%

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

      1. associate-*r/ 84.3%

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

      2. *-commutative 84.3%

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

      3. distribute-rgt1-in 84.0%

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

      4. fma-def 84.3%

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

      5. cube-unmult 84.3%

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

   3. Simplified 84.3%

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

   4. Taylor expanded in y around inf 82.3%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 82.3%

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

   6. Simplified 82.3%

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

      1. clear-num 82.3%

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

      2. inv-pow 82.3%

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

   8. Applied egg-rr 82.3%

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

      1. unpow-1 82.3%

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

      2. associate-/l* 84.2%

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

   10. Simplified 84.2%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-168}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{\frac{x}{y}}}\\ \end{array} \]

Alternative 8: 82.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-168}:\\ \;\;\;\;\frac{\frac{y}{y + \left(1 + x\right)}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.3e-168)
   (/ (/ y (+ y (+ 1.0 x))) (+ y x))
   (/ (/ x (+ y 1.0)) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.3e-168) {
		tmp = (y / (y + (1.0 + x))) / (y + x);
	} else {
		tmp = (x / (y + 1.0)) / (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 <= 2.3d-168) then
        tmp = (y / (y + (1.0d0 + x))) / (y + x)
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.3e-168) {
		tmp = (y / (y + (1.0 + x))) / (y + x);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.3e-168:
		tmp = (y / (y + (1.0 + x))) / (y + x)
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.3e-168)
		tmp = Float64(Float64(y / Float64(y + Float64(1.0 + x))) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / 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 <= 2.3e-168)
		tmp = (y / (y + (1.0 + x))) / (y + x);
	else
		tmp = (x / (y + 1.0)) / (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, 2.3e-168], N[(N[(y / N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.29999999999999986e-168

   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. associate-*r/ 81.7%

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

      2. *-commutative 81.7%

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

      3. distribute-rgt1-in 56.4%

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

      4. fma-def 81.7%

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

      5. cube-unmult 81.7%

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

   3. Simplified 81.7%

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

      1. associate-*r/ 67.8%

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

      2. fma-udef 45.4%

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

      3. cube-mult 45.4%

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

      4. distribute-rgt1-in 67.8%

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

      5. associate-+r+ 67.8%

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

      6. *-commutative 67.8%

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

      7. frac-times 86.9%

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

      8. *-commutative 86.9%

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

      9. associate-/r* 99.7%

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

      10. frac-times 93.5%

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

      11. +-commutative 93.5%

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

      12. +-commutative 93.5%

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

      13. associate-+l+ 93.5%

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

      14. +-commutative 93.5%

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

   5. Applied egg-rr 93.5%

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

      1. associate-/r* 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*r/ 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-*r/ 73.5%

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

      7. +-commutative 73.5%

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

   7. Simplified 73.5%

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

   8. Taylor expanded in y around 0 57.5%

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

   if 2.29999999999999986e-168 < y

   1. Initial program 73.6%

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

      1. associate-*r/ 88.9%

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

      2. *-commutative 88.9%

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

      3. distribute-rgt1-in 85.7%

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

      4. fma-def 88.9%

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

      5. cube-unmult 88.9%

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

   3. Simplified 88.9%

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

      1. associate-*r/ 73.6%

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

      2. fma-udef 71.5%

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

      3. cube-mult 71.5%

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

      4. distribute-rgt1-in 73.6%

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

      5. associate-+r+ 73.6%

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

      6. *-commutative 73.6%

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

      7. frac-times 92.1%

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

      8. *-commutative 92.1%

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

      9. associate-/r* 99.8%

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

      10. associate-*r/ 99.8%

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

      11. +-commutative 99.8%

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

      12. associate-+l+ 99.8%

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

      13. +-commutative 99.8%

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

      14. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around 0 66.9%

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

      1. +-commutative 66.9%

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

   8. Simplified 66.9%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-168}:\\ \;\;\;\;\frac{\frac{y}{y + \left(1 + x\right)}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]

Alternative 9: 69.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -1.5 \cdot 10^{-92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.66 \cdot 10^{-96}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 16000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* x x))))
   (if (<= y -1.5e-92)
     t_0
     (if (<= y 1.66e-96)
       (/ y x)
       (if (<= y 16000000000.0) t_0 (/ x (* y y)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x * x);
	double tmp;
	if (y <= -1.5e-92) {
		tmp = t_0;
	} else if (y <= 1.66e-96) {
		tmp = y / x;
	} else if (y <= 16000000000.0) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = y / (x * x)
    if (y <= (-1.5d-92)) then
        tmp = t_0
    else if (y <= 1.66d-96) then
        tmp = y / x
    else if (y <= 16000000000.0d0) then
        tmp = t_0
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (x * x);
	double tmp;
	if (y <= -1.5e-92) {
		tmp = t_0;
	} else if (y <= 1.66e-96) {
		tmp = y / x;
	} else if (y <= 16000000000.0) {
		tmp = t_0;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (x * x)
	tmp = 0
	if y <= -1.5e-92:
		tmp = t_0
	elif y <= 1.66e-96:
		tmp = y / x
	elif y <= 16000000000.0:
		tmp = t_0
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(x * x))
	tmp = 0.0
	if (y <= -1.5e-92)
		tmp = t_0;
	elseif (y <= 1.66e-96)
		tmp = Float64(y / x);
	elseif (y <= 16000000000.0)
		tmp = t_0;
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (x * x);
	tmp = 0.0;
	if (y <= -1.5e-92)
		tmp = t_0;
	elseif (y <= 1.66e-96)
		tmp = y / x;
	elseif (y <= 16000000000.0)
		tmp = t_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_] := Block[{t$95$0 = N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.5e-92], t$95$0, If[LessEqual[y, 1.66e-96], N[(y / x), $MachinePrecision], If[LessEqual[y, 16000000000.0], t$95$0, N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.50000000000000007e-92 or 1.65999999999999993e-96 < y < 1.6e10

   1. Initial program 70.1%

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

      1. associate-*r/ 80.7%

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

      2. *-commutative 80.7%

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

      3. distribute-rgt1-in 51.2%

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

      4. fma-def 80.7%

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

      5. cube-unmult 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in x around inf 27.4%

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 27.4%

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

   6. Simplified 27.4%

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

   if -1.50000000000000007e-92 < y < 1.65999999999999993e-96

   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. times-frac 88.0%

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

      2. /-rgt-identity 88.0%

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

      3. associate-/l/ 88.0%

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

      4. *-lft-identity 88.0%

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

      5. associate-+l+ 88.0%

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

   3. Simplified 88.0%

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

   4. Taylor expanded in y around 0 81.4%

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

   5. Taylor expanded in x around 0 66.0%

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

   if 1.6e10 < y

   1. Initial program 69.5%

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

      1. associate-*r/ 87.3%

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

      2. *-commutative 87.3%

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

      3. distribute-rgt1-in 84.0%

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

      4. fma-def 87.3%

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

      5. cube-unmult 87.3%

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

   3. Simplified 87.3%

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

   4. Taylor expanded in y around inf 79.7%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 79.7%

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

   6. Simplified 79.7%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-92}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 1.66 \cdot 10^{-96}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 16000000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 10: 71.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -1 \cdot 10^{-84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-101}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 6500000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* x x))))
   (if (<= y -1e-84)
     t_0
     (if (<= y 6.5e-101) (/ y x) (if (<= y 6500000000.0) t_0 (/ (/ x y) y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x * x);
	double tmp;
	if (y <= -1e-84) {
		tmp = t_0;
	} else if (y <= 6.5e-101) {
		tmp = y / x;
	} else if (y <= 6500000000.0) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = y / (x * x)
    if (y <= (-1d-84)) then
        tmp = t_0
    else if (y <= 6.5d-101) then
        tmp = y / x
    else if (y <= 6500000000.0d0) then
        tmp = t_0
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (x * x);
	double tmp;
	if (y <= -1e-84) {
		tmp = t_0;
	} else if (y <= 6.5e-101) {
		tmp = y / x;
	} else if (y <= 6500000000.0) {
		tmp = t_0;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (x * x)
	tmp = 0
	if y <= -1e-84:
		tmp = t_0
	elif y <= 6.5e-101:
		tmp = y / x
	elif y <= 6500000000.0:
		tmp = t_0
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(x * x))
	tmp = 0.0
	if (y <= -1e-84)
		tmp = t_0;
	elseif (y <= 6.5e-101)
		tmp = Float64(y / x);
	elseif (y <= 6500000000.0)
		tmp = t_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)
	t_0 = y / (x * x);
	tmp = 0.0;
	if (y <= -1e-84)
		tmp = t_0;
	elseif (y <= 6.5e-101)
		tmp = y / x;
	elseif (y <= 6500000000.0)
		tmp = t_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_] := Block[{t$95$0 = N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1e-84], t$95$0, If[LessEqual[y, 6.5e-101], N[(y / x), $MachinePrecision], If[LessEqual[y, 6500000000.0], t$95$0, N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1e-84 or 6.4999999999999996e-101 < y < 6.5e9

   1. Initial program 69.5%

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

      1. associate-*r/ 80.3%

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

      2. *-commutative 80.3%

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

      3. distribute-rgt1-in 50.2%

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

      4. fma-def 80.3%

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

      5. cube-unmult 80.3%

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

   3. Simplified 80.3%

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

   4. Taylor expanded in x around inf 26.9%

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 26.9%

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

   6. Simplified 26.9%

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

   if -1e-84 < y < 6.4999999999999996e-101

   1. Initial program 70.9%

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

      1. times-frac 88.2%

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

      2. /-rgt-identity 88.2%

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

      3. associate-/l/ 88.2%

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

      4. *-lft-identity 88.2%

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

      5. associate-+l+ 88.2%

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

   3. Simplified 88.2%

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

   4. Taylor expanded in y around 0 80.9%

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

   5. Taylor expanded in x around 0 64.8%

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

   if 6.5e9 < y

   1. Initial program 69.5%

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

      1. associate-*r/ 87.3%

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

      2. *-commutative 87.3%

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

      3. distribute-rgt1-in 84.0%

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

      4. fma-def 87.3%

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

      5. cube-unmult 87.3%

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

   3. Simplified 87.3%

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

   4. Taylor expanded in y around inf 79.7%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 79.7%

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

   6. Simplified 79.7%

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

      1. associate-/r* 81.2%

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

      2. div-inv 81.1%

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

   8. Applied egg-rr 81.1%

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

      1. associate-*r/ 81.2%

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

      2. *-rgt-identity 81.2%

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

   10. Simplified 81.2%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-84}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-101}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 6500000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 11: 73.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{\frac{y}{x}}{x}\\ \mathbf{if}\;y \leq -1.2 \cdot 10^{-93}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.72 \cdot 10^{-102}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1500000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ y x) x)))
   (if (<= y -1.2e-93)
     t_0
     (if (<= y 1.72e-102)
       (/ y x)
       (if (<= y 1500000000.0) t_0 (/ (/ x y) y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y / x) / x;
	double tmp;
	if (y <= -1.2e-93) {
		tmp = t_0;
	} else if (y <= 1.72e-102) {
		tmp = y / x;
	} else if (y <= 1500000000.0) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (y / x) / x
    if (y <= (-1.2d-93)) then
        tmp = t_0
    else if (y <= 1.72d-102) then
        tmp = y / x
    else if (y <= 1500000000.0d0) then
        tmp = t_0
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y / x) / x;
	double tmp;
	if (y <= -1.2e-93) {
		tmp = t_0;
	} else if (y <= 1.72e-102) {
		tmp = y / x;
	} else if (y <= 1500000000.0) {
		tmp = t_0;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y / x) / x
	tmp = 0
	if y <= -1.2e-93:
		tmp = t_0
	elif y <= 1.72e-102:
		tmp = y / x
	elif y <= 1500000000.0:
		tmp = t_0
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y / x) / x)
	tmp = 0.0
	if (y <= -1.2e-93)
		tmp = t_0;
	elseif (y <= 1.72e-102)
		tmp = Float64(y / x);
	elseif (y <= 1500000000.0)
		tmp = t_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)
	t_0 = (y / x) / x;
	tmp = 0.0;
	if (y <= -1.2e-93)
		tmp = t_0;
	elseif (y <= 1.72e-102)
		tmp = y / x;
	elseif (y <= 1500000000.0)
		tmp = t_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_] := Block[{t$95$0 = N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[y, -1.2e-93], t$95$0, If[LessEqual[y, 1.72e-102], N[(y / x), $MachinePrecision], If[LessEqual[y, 1500000000.0], t$95$0, N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2000000000000001e-93 or 1.72000000000000009e-102 < y < 1.5e9

   1. Initial program 70.1%

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

      1. times-frac 89.0%

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

      2. /-rgt-identity 89.0%

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

      3. associate-/l/ 89.0%

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

      4. *-lft-identity 89.0%

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

      5. associate-+l+ 89.0%

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

   3. Simplified 89.0%

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

   4. Taylor expanded in x around inf 34.1%

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

   5. Taylor expanded in x around inf 30.8%

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

      1. associate-*l/ 30.8%

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

      2. *-un-lft-identity 30.8%

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

   7. Applied egg-rr 30.8%

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

   if -1.2000000000000001e-93 < y < 1.72000000000000009e-102

   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. times-frac 88.0%

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

      2. /-rgt-identity 88.0%

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

      3. associate-/l/ 88.0%

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

      4. *-lft-identity 88.0%

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

      5. associate-+l+ 88.0%

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

   3. Simplified 88.0%

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

   4. Taylor expanded in y around 0 81.4%

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

   5. Taylor expanded in x around 0 66.0%

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

   if 1.5e9 < y

   1. Initial program 69.5%

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

      1. associate-*r/ 87.3%

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

      2. *-commutative 87.3%

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

      3. distribute-rgt1-in 84.0%

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

      4. fma-def 87.3%

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

      5. cube-unmult 87.3%

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

   3. Simplified 87.3%

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

   4. Taylor expanded in y around inf 79.7%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 79.7%

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

   6. Simplified 79.7%

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

      1. associate-/r* 81.2%

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

      2. div-inv 81.1%

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

   8. Applied egg-rr 81.1%

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

      1. associate-*r/ 81.2%

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

      2. *-rgt-identity 81.2%

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

   10. Simplified 81.2%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-93}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 1.72 \cdot 10^{-102}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1500000000:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 12: 80.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-101}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \end{array} \end{array} \]

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 -1.75e-101) (- (/ y x) y) (/ x (+ y (* 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 <= -1.75e-101) {
		tmp = (y / x) - y;
	} else {
		tmp = x / (y + (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 <= (-1.75d-101)) then
        tmp = (y / x) - y
    else
        tmp = x / (y + (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 <= -1.75e-101) {
		tmp = (y / x) - y;
	} else {
		tmp = x / (y + (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 <= -1.75e-101:
		tmp = (y / x) - y
	else:
		tmp = x / (y + (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 <= -1.75e-101)
		tmp = Float64(Float64(y / x) - y);
	else
		tmp = Float64(x / Float64(y + Float64(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 <= -1.75e-101)
		tmp = (y / x) - y;
	else
		tmp = x / (y + (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, -1.75e-101], N[(N[(y / x), $MachinePrecision] - y), $MachinePrecision], N[(x / N[(y + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 68.8%

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

      1. times-frac 86.3%

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

      2. /-rgt-identity 86.3%

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

      3. associate-/l/ 86.3%

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

      4. *-lft-identity 86.3%

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

      5. associate-+l+ 86.3%

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

   3. Simplified 86.3%

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

   4. Taylor expanded in x around inf 68.8%

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

   5. Taylor expanded in x around inf 68.3%

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

      1. associate-*l/ 68.3%

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

      2. *-un-lft-identity 68.3%

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

   7. Applied egg-rr 68.3%

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

   if -1 < x < -1.74999999999999997e-101

   1. Initial program 88.0%

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

      1. times-frac 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. /-rgt-identity 99.6%

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

      3. associate-/l/ 99.6%

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

      4. *-lft-identity 99.6%

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

      5. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 53.3%

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

   5. Taylor expanded in x around 0 50.9%

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

      1. neg-mul-1 50.9%

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

      2. unsub-neg 50.9%

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

   7. Simplified 50.9%

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

   if -1.74999999999999997e-101 < x

   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 88.3%

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

      2. /-rgt-identity 88.3%

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

      3. associate-/l/ 88.3%

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

      4. *-lft-identity 88.3%

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

      5. associate-+l+ 88.3%

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

   3. Simplified 88.3%

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

   4. Taylor expanded in x around 0 61.3%

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

      1. distribute-rgt-in 61.3%

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

      2. *-lft-identity 61.3%

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

   6. Simplified 61.3%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-101}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \end{array} \]

Alternative 13: 82.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -6.1 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-168}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -6.1e+30)
   (/ (/ y x) x)
   (if (<= y 2.3e-168) (/ y (* x (+ 1.0 x))) (/ (/ x y) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -6.1e+30) {
		tmp = (y / x) / x;
	} else if (y <= 2.3e-168) {
		tmp = y / (x * (1.0 + x));
	} 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 (y <= (-6.1d+30)) then
        tmp = (y / x) / x
    else if (y <= 2.3d-168) then
        tmp = y / (x * (1.0d0 + x))
    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 (y <= -6.1e+30) {
		tmp = (y / x) / x;
	} else if (y <= 2.3e-168) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -6.1e+30:
		tmp = (y / x) / x
	elif y <= 2.3e-168:
		tmp = y / (x * (1.0 + x))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -6.1e+30)
		tmp = Float64(Float64(y / x) / x);
	elseif (y <= 2.3e-168)
		tmp = Float64(y / Float64(x * Float64(1.0 + x)));
	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 (y <= -6.1e+30)
		tmp = (y / x) / x;
	elseif (y <= 2.3e-168)
		tmp = y / (x * (1.0 + x));
	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[y, -6.1e+30], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 2.3e-168], N[(y / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.1000000000000001e30

   1. Initial program 57.5%

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

      1. times-frac 81.5%

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

      2. /-rgt-identity 81.5%

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

      3. associate-/l/ 81.5%

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

      4. *-lft-identity 81.5%

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

      5. associate-+l+ 81.5%

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

   3. Simplified 81.5%

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

   4. Taylor expanded in x around inf 17.2%

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

   5. Taylor expanded in x around inf 16.0%

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

      1. associate-*l/ 16.0%

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

      2. *-un-lft-identity 16.0%

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

   7. Applied egg-rr 16.0%

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

   if -6.1000000000000001e30 < y < 2.29999999999999986e-168

   1. Initial program 73.4%

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

      1. times-frac 89.8%

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

      2. /-rgt-identity 89.8%

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

      3. associate-/l/ 89.8%

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

      4. *-lft-identity 89.8%

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

      5. associate-+l+ 89.8%

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

   3. Simplified 89.8%

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

   4. Taylor expanded in y around 0 78.4%

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

   if 2.29999999999999986e-168 < y

   1. Initial program 73.6%

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

      1. associate-*r/ 88.9%

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

      2. *-commutative 88.9%

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

      3. distribute-rgt1-in 85.7%

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

      4. fma-def 88.9%

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

      5. cube-unmult 88.9%

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

   3. Simplified 88.9%

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

      1. associate-*r/ 73.6%

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

      2. fma-udef 71.5%

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

      3. cube-mult 71.5%

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

      4. distribute-rgt1-in 73.6%

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

      5. associate-+r+ 73.6%

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

      6. *-commutative 73.6%

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

      7. frac-times 92.1%

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

      8. *-commutative 92.1%

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

      9. associate-/r* 99.8%

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

      10. associate-*r/ 99.8%

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

      11. +-commutative 99.8%

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

      12. associate-+l+ 99.8%

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

      13. +-commutative 99.8%

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

      14. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in x around 0 65.5%

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

       1. associate-/r* 66.5%

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

   12. Simplified 66.5%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.1 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-168}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 14: 82.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-168}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -6e+30)
   (/ (/ y x) (+ y x))
   (if (<= y 2.3e-168) (/ y (* x (+ 1.0 x))) (/ (/ x y) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -6e+30) {
		tmp = (y / x) / (y + x);
	} else if (y <= 2.3e-168) {
		tmp = y / (x * (1.0 + x));
	} 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 (y <= (-6d+30)) then
        tmp = (y / x) / (y + x)
    else if (y <= 2.3d-168) then
        tmp = y / (x * (1.0d0 + x))
    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 (y <= -6e+30) {
		tmp = (y / x) / (y + x);
	} else if (y <= 2.3e-168) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -6e+30:
		tmp = (y / x) / (y + x)
	elif y <= 2.3e-168:
		tmp = y / (x * (1.0 + x))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -6e+30)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (y <= 2.3e-168)
		tmp = Float64(y / Float64(x * Float64(1.0 + x)));
	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 (y <= -6e+30)
		tmp = (y / x) / (y + x);
	elseif (y <= 2.3e-168)
		tmp = y / (x * (1.0 + x));
	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[y, -6e+30], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.3e-168], N[(y / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.99999999999999956e30

   1. Initial program 57.5%

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

      1. associate-*r/ 71.4%

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

      2. *-commutative 71.4%

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

      3. distribute-rgt1-in 28.3%

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

      4. fma-def 71.4%

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

      5. cube-unmult 71.5%

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

   3. Simplified 71.5%

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

      1. associate-*r/ 57.5%

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

      2. fma-udef 22.6%

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

      3. cube-mult 22.5%

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

      4. distribute-rgt1-in 57.5%

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

      5. associate-+r+ 57.5%

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

      6. *-commutative 57.5%

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

      7. frac-times 81.5%

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

      8. *-commutative 81.5%

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

      9. associate-/r* 99.8%

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

      10. associate-*r/ 99.8%

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

      11. +-commutative 99.8%

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

      12. associate-+l+ 99.8%

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

      13. +-commutative 99.8%

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

      14. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around inf 17.1%

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

   if -5.99999999999999956e30 < y < 2.29999999999999986e-168

   1. Initial program 73.4%

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

      1. times-frac 89.8%

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

      2. /-rgt-identity 89.8%

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

      3. associate-/l/ 89.8%

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

      4. *-lft-identity 89.8%

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

      5. associate-+l+ 89.8%

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

   3. Simplified 89.8%

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

   4. Taylor expanded in y around 0 78.4%

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

   if 2.29999999999999986e-168 < y

   1. Initial program 73.6%

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

      1. associate-*r/ 88.9%

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

      2. *-commutative 88.9%

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

      3. distribute-rgt1-in 85.7%

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

      4. fma-def 88.9%

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

      5. cube-unmult 88.9%

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

   3. Simplified 88.9%

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

      1. associate-*r/ 73.6%

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

      2. fma-udef 71.5%

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

      3. cube-mult 71.5%

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

      4. distribute-rgt1-in 73.6%

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

      5. associate-+r+ 73.6%

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

      6. *-commutative 73.6%

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

      7. frac-times 92.1%

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

      8. *-commutative 92.1%

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

      9. associate-/r* 99.8%

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

      10. associate-*r/ 99.8%

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

      11. +-commutative 99.8%

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

      12. associate-+l+ 99.8%

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

      13. +-commutative 99.8%

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

      14. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in x around 0 65.5%

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

       1. associate-/r* 66.5%

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

   12. Simplified 66.5%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-168}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 15: 82.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-168}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.3e-168) (/ (/ y (+ 1.0 x)) x) (/ (/ x (+ y 1.0)) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.3e-168) {
		tmp = (y / (1.0 + x)) / x;
	} else {
		tmp = (x / (y + 1.0)) / (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 <= 2.3d-168) then
        tmp = (y / (1.0d0 + x)) / x
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.3e-168) {
		tmp = (y / (1.0 + x)) / x;
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.3e-168:
		tmp = (y / (1.0 + x)) / x
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.3e-168)
		tmp = Float64(Float64(y / Float64(1.0 + x)) / x);
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / 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 <= 2.3e-168)
		tmp = (y / (1.0 + x)) / x;
	else
		tmp = (x / (y + 1.0)) / (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, 2.3e-168], N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.29999999999999986e-168

   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. associate-*r/ 81.7%

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

      2. *-commutative 81.7%

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

      3. distribute-rgt1-in 56.4%

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

      4. fma-def 81.7%

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

      5. cube-unmult 81.7%

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

   3. Simplified 81.7%

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

      1. associate-*r/ 67.8%

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

      2. fma-udef 45.4%

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

      3. cube-mult 45.4%

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

      4. distribute-rgt1-in 67.8%

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

      5. associate-+r+ 67.8%

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

      6. *-commutative 67.8%

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

      7. frac-times 86.9%

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

      8. *-commutative 86.9%

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

      9. associate-/r* 99.7%

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

      10. associate-*r/ 99.8%

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

      11. +-commutative 99.8%

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

      12. associate-+l+ 99.8%

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

      13. +-commutative 99.8%

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

      14. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around 0 54.5%

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

      1. associate-/r* 56.5%

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

   8. Simplified 56.5%

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

   if 2.29999999999999986e-168 < y

   1. Initial program 73.6%

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

      1. associate-*r/ 88.9%

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

      2. *-commutative 88.9%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      3. distribute-rgt1-in 85.7%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. fma-def 88.9%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(x + y, x + y, \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)\right)}} \]

      5. cube-unmult 88.9%

         \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(x + y\right)}^{3}}\right)} \]

   3. Simplified 88.9%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 73.6%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 71.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 71.5%

         \[\leadsto \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 73.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. associate-+r+ 73.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 73.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. frac-times 92.1%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

      8. *-commutative 92.1%

         \[\leadsto \color{blue}{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      9. associate-/r* 99.8%

         \[\leadsto \frac{y}{x + \left(y + 1\right)} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      10. associate-*r/ 99.8%

          \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{x + y}}{x + y}} \]

      11. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(y + 1\right) + x}} \cdot \frac{x}{x + y}}{x + y} \]

      12. associate-+l+ 99.8%

          \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      13. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{\color{blue}{y + x}}}{x + y} \]

      14. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{y + x}}{\color{blue}{y + x}} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{y + x}}{y + x}} \]

   6. Taylor expanded in x around 0 66.9%

      \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{y + x} \]
   7. Step-by-step derivation

      1. +-commutative 66.9%

         \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{y + x} \]

   8. Simplified 66.9%

      \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{y + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-168}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]

Alternative 16: 82.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-168}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.3e-168) (/ (/ y (+ 1.0 x)) (+ y x)) (/ (/ x (+ y 1.0)) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.3e-168) {
		tmp = (y / (1.0 + x)) / (y + x);
	} else {
		tmp = (x / (y + 1.0)) / (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 <= 2.3d-168) then
        tmp = (y / (1.0d0 + x)) / (y + x)
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.3e-168) {
		tmp = (y / (1.0 + x)) / (y + x);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.3e-168:
		tmp = (y / (1.0 + x)) / (y + x)
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.3e-168)
		tmp = Float64(Float64(y / Float64(1.0 + x)) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / 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 <= 2.3e-168)
		tmp = (y / (1.0 + x)) / (y + x);
	else
		tmp = (x / (y + 1.0)) / (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, 2.3e-168], N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.29999999999999986e-168

   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. associate-*r/ 81.7%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. *-commutative 81.7%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      3. distribute-rgt1-in 56.4%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. fma-def 81.7%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(x + y, x + y, \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)\right)}} \]

      5. cube-unmult 81.7%

         \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(x + y\right)}^{3}}\right)} \]

   3. Simplified 81.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 67.8%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 45.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 45.4%

         \[\leadsto \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 67.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. associate-+r+ 67.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 67.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. frac-times 86.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

      8. *-commutative 86.9%

         \[\leadsto \color{blue}{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      9. associate-/r* 99.7%

         \[\leadsto \frac{y}{x + \left(y + 1\right)} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      10. associate-*r/ 99.8%

          \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{x + y}}{x + y}} \]

      11. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(y + 1\right) + x}} \cdot \frac{x}{x + y}}{x + y} \]

      12. associate-+l+ 99.8%

          \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      13. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{\color{blue}{y + x}}}{x + y} \]

      14. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{y + x}}{\color{blue}{y + x}} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{y + x}}{y + x}} \]

   6. Taylor expanded in y around 0 57.1%

      \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{y + x} \]

   if 2.29999999999999986e-168 < y

   1. Initial program 73.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*r/ 88.9%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. *-commutative 88.9%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      3. distribute-rgt1-in 85.7%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. fma-def 88.9%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(x + y, x + y, \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)\right)}} \]

      5. cube-unmult 88.9%

         \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(x + y\right)}^{3}}\right)} \]

   3. Simplified 88.9%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 73.6%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 71.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 71.5%

         \[\leadsto \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 73.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. associate-+r+ 73.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 73.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. frac-times 92.1%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

      8. *-commutative 92.1%

         \[\leadsto \color{blue}{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      9. associate-/r* 99.8%

         \[\leadsto \frac{y}{x + \left(y + 1\right)} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      10. associate-*r/ 99.8%

          \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{x + y}}{x + y}} \]

      11. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(y + 1\right) + x}} \cdot \frac{x}{x + y}}{x + y} \]

      12. associate-+l+ 99.8%

          \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      13. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{\color{blue}{y + x}}}{x + y} \]

      14. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{y + x}}{\color{blue}{y + x}} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{y + x}}{y + x}} \]

   6. Taylor expanded in x around 0 66.9%

      \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{y + x} \]
   7. Step-by-step derivation

      1. +-commutative 66.9%

         \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{y + x} \]

   8. Simplified 66.9%

      \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{y + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-168}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]

Alternative 17: 82.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-168}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.3e-168) (/ (/ y (+ 1.0 x)) x) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.3e-168) {
		tmp = (y / (1.0 + x)) / x;
	} 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 (y <= 2.3d-168) then
        tmp = (y / (1.0d0 + x)) / x
    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 (y <= 2.3e-168) {
		tmp = (y / (1.0 + x)) / x;
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.3e-168:
		tmp = (y / (1.0 + x)) / x
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.3e-168)
		tmp = Float64(Float64(y / Float64(1.0 + x)) / x);
	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 (y <= 2.3e-168)
		tmp = (y / (1.0 + x)) / x;
	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[y, 2.3e-168], N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.29999999999999986e-168

   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. associate-*r/ 81.7%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. *-commutative 81.7%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      3. distribute-rgt1-in 56.4%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. fma-def 81.7%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(x + y, x + y, \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)\right)}} \]

      5. cube-unmult 81.7%

         \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(x + y\right)}^{3}}\right)} \]

   3. Simplified 81.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 67.8%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 45.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 45.4%

         \[\leadsto \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 67.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. associate-+r+ 67.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 67.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. frac-times 86.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

      8. *-commutative 86.9%

         \[\leadsto \color{blue}{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      9. associate-/r* 99.7%

         \[\leadsto \frac{y}{x + \left(y + 1\right)} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      10. associate-*r/ 99.8%

          \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{x + y}}{x + y}} \]

      11. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(y + 1\right) + x}} \cdot \frac{x}{x + y}}{x + y} \]

      12. associate-+l+ 99.8%

          \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      13. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{\color{blue}{y + x}}}{x + y} \]

      14. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{y + x}}{\color{blue}{y + x}} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{y + x}}{y + x}} \]

   6. Taylor expanded in y around 0 54.5%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]
   7. Step-by-step derivation

      1. associate-/r* 56.5%

         \[\leadsto \color{blue}{\frac{\frac{y}{1 + x}}{x}} \]

   8. Simplified 56.5%

      \[\leadsto \color{blue}{\frac{\frac{y}{1 + x}}{x}} \]

   if 2.29999999999999986e-168 < y

   1. Initial program 73.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*r/ 88.9%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. *-commutative 88.9%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      3. distribute-rgt1-in 85.7%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. fma-def 88.9%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(x + y, x + y, \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)\right)}} \]

      5. cube-unmult 88.9%

         \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(x + y\right)}^{3}}\right)} \]

   3. Simplified 88.9%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 73.6%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 71.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 71.5%

         \[\leadsto \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 73.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. associate-+r+ 73.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 73.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. frac-times 92.1%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

      8. *-commutative 92.1%

         \[\leadsto \color{blue}{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      9. associate-/r* 99.8%

         \[\leadsto \frac{y}{x + \left(y + 1\right)} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      10. associate-*r/ 99.8%

          \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{x + y}}{x + y}} \]

      11. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(y + 1\right) + x}} \cdot \frac{x}{x + y}}{x + y} \]

      12. associate-+l+ 99.8%

          \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      13. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{\color{blue}{y + x}}}{x + y} \]

      14. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{y + x}}{\color{blue}{y + x}} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{y + x}}{y + x}} \]
   6. Step-by-step derivation

      1. clear-num 99.8%

         \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{1}{\frac{y + x}{x}}}}{y + x} \]

      2. inv-pow 99.8%

         \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{{\left(\frac{y + x}{x}\right)}^{-1}}}{y + x} \]

   7. Applied egg-rr 99.8%

      \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{{\left(\frac{y + x}{x}\right)}^{-1}}}{y + x} \]
   8. Step-by-step derivation

      1. unpow-1 99.8%

         \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{1}{\frac{y + x}{x}}}}{y + x} \]

   9. Simplified 99.8%

      \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{1}{\frac{y + x}{x}}}}{y + x} \]

   10. Taylor expanded in x around 0 65.5%

       \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   11. Step-by-step derivation

       1. associate-/r* 66.5%

          \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

   12. Simplified 66.5%

       \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-168}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 18: 27.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-47}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 8e-47) (/ y x) (/ 1.0 (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 8e-47) {
		tmp = y / x;
	} else {
		tmp = 1.0 / (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) :: tmp
    if (y <= 8d-47) then
        tmp = y / x
    else
        tmp = 1.0d0 / (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 <= 8e-47) {
		tmp = y / x;
	} else {
		tmp = 1.0 / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 8e-47:
		tmp = y / x
	else:
		tmp = 1.0 / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 8e-47)
		tmp = Float64(y / x);
	else
		tmp = Float64(1.0 / 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 <= 8e-47)
		tmp = y / x;
	else
		tmp = 1.0 / (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_] := If[LessEqual[y, 8e-47], N[(y / x), $MachinePrecision], N[(1.0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.9999999999999998e-47

   1. Initial program 69.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 88.0%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 88.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. associate-/l/ 88.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lft-identity 88.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      5. associate-+l+ 88.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 88.0%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 56.3%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]

   5. Taylor expanded in x around 0 37.8%

      \[\leadsto \frac{y}{\color{blue}{x}} \]

   if 7.9999999999999998e-47 < y

   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. associate-*r/ 88.0%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. *-commutative 88.0%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      3. distribute-rgt1-in 85.1%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. fma-def 88.0%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(x + y, x + y, \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)\right)}} \]

      5. cube-unmult 88.0%

         \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(x + y\right)}^{3}}\right)} \]

   3. Simplified 88.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 72.2%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 70.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 70.8%

         \[\leadsto \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 72.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. associate-+r+ 72.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 72.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. frac-times 91.2%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

      8. *-commutative 91.2%

         \[\leadsto \color{blue}{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      9. associate-/r* 99.8%

         \[\leadsto \frac{y}{x + \left(y + 1\right)} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      10. frac-times 91.2%

          \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{x + y}}{\left(x + \left(y + 1\right)\right) \cdot \left(x + y\right)}} \]

      11. +-commutative 91.2%

          \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{y + x}}}{\left(x + \left(y + 1\right)\right) \cdot \left(x + y\right)} \]

      12. +-commutative 91.2%

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\color{blue}{\left(\left(y + 1\right) + x\right)} \cdot \left(x + y\right)} \]

      13. associate-+l+ 91.2%

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\color{blue}{\left(y + \left(1 + x\right)\right)} \cdot \left(x + y\right)} \]

      14. +-commutative 91.2%

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\left(y + \left(1 + x\right)\right) \cdot \color{blue}{\left(y + x\right)}} \]

   5. Applied egg-rr 91.2%

      \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{y + x}}{\left(y + \left(1 + x\right)\right) \cdot \left(y + x\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 99.9%

         \[\leadsto \color{blue}{\frac{\frac{y \cdot \frac{x}{y + x}}{y + \left(1 + x\right)}}{y + x}} \]

      2. associate-*l/ 99.8%

         \[\leadsto \frac{\color{blue}{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{y + x}}}{y + x} \]

      3. *-commutative 99.8%

         \[\leadsto \frac{\color{blue}{\frac{x}{y + x} \cdot \frac{y}{y + \left(1 + x\right)}}}{y + x} \]

      4. associate-*r/ 99.9%

         \[\leadsto \frac{\color{blue}{\frac{\frac{x}{y + x} \cdot y}{y + \left(1 + x\right)}}}{y + x} \]

      5. *-commutative 99.9%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot \frac{x}{y + x}}}{y + \left(1 + x\right)}}{y + x} \]

      6. associate-*r/ 78.3%

         \[\leadsto \frac{\frac{\color{blue}{\frac{y \cdot x}{y + x}}}{y + \left(1 + x\right)}}{y + x} \]

      7. +-commutative 78.3%

         \[\leadsto \frac{\frac{\frac{y \cdot x}{y + x}}{y + \color{blue}{\left(x + 1\right)}}}{y + x} \]

   7. Simplified 78.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{y \cdot x}{y + x}}{y + \left(x + 1\right)}}{y + x}} \]

   8. Taylor expanded in y around 0 25.5%

      \[\leadsto \frac{\frac{\color{blue}{y}}{y + \left(x + 1\right)}}{y + x} \]

   9. Taylor expanded in x around 0 5.7%

      \[\leadsto \color{blue}{\frac{1}{1 + y}} \]
   10. Step-by-step derivation

       1. +-commutative 5.7%

          \[\leadsto \frac{1}{\color{blue}{y + 1}} \]

   11. Simplified 5.7%

       \[\leadsto \color{blue}{\frac{1}{y + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-47}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y + 1}\\ \end{array} \]

Alternative 19: 60.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-17}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 1.15e-17) (/ y x) (/ x (* y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.15e-17) {
		tmp = 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 <= 1.15d-17) then
        tmp = 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 <= 1.15e-17) {
		tmp = y / x;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.15e-17:
		tmp = y / x
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.15e-17)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.15e-17)
		tmp = 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, 1.15e-17], N[(y / x), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.15000000000000004e-17

   1. Initial program 69.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 88.1%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 88.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. associate-/l/ 88.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lft-identity 88.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      5. associate-+l+ 88.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 88.1%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 56.8%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]

   5. Taylor expanded in x around 0 37.4%

      \[\leadsto \frac{y}{\color{blue}{x}} \]

   if 1.15000000000000004e-17 < y

   1. Initial program 71.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*r/ 87.7%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. *-commutative 87.7%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      3. distribute-rgt1-in 84.7%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. fma-def 87.7%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(x + y, x + y, \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)\right)}} \]

      5. cube-unmult 87.7%

         \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(x + y\right)}^{3}}\right)} \]

   3. Simplified 87.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   4. Taylor expanded in y around inf 73.8%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 73.8%

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   6. Simplified 73.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-17}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 20: 27.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-47}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 8e-47) (/ y x) (/ 1.0 y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 8e-47) {
		tmp = y / x;
	} else {
		tmp = 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 <= 8d-47) then
        tmp = y / x
    else
        tmp = 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 <= 8e-47) {
		tmp = y / x;
	} else {
		tmp = 1.0 / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 8e-47:
		tmp = y / x
	else:
		tmp = 1.0 / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 8e-47)
		tmp = Float64(y / x);
	else
		tmp = Float64(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 <= 8e-47)
		tmp = y / x;
	else
		tmp = 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, 8e-47], N[(y / x), $MachinePrecision], N[(1.0 / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.9999999999999998e-47

   1. Initial program 69.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 88.0%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 88.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. associate-/l/ 88.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lft-identity 88.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      5. associate-+l+ 88.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 88.0%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 56.3%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]

   5. Taylor expanded in x around 0 37.8%

      \[\leadsto \frac{y}{\color{blue}{x}} \]

   if 7.9999999999999998e-47 < y

   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. associate-*r/ 88.0%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. *-commutative 88.0%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      3. distribute-rgt1-in 85.1%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. fma-def 88.0%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(x + y, x + y, \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)\right)}} \]

      5. cube-unmult 88.0%

         \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(x + y\right)}^{3}}\right)} \]

   3. Simplified 88.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 72.2%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 70.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 70.8%

         \[\leadsto \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 72.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. associate-+r+ 72.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 72.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. frac-times 91.2%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

      8. *-commutative 91.2%

         \[\leadsto \color{blue}{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      9. associate-/r* 99.8%

         \[\leadsto \frac{y}{x + \left(y + 1\right)} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      10. frac-times 91.2%

          \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{x + y}}{\left(x + \left(y + 1\right)\right) \cdot \left(x + y\right)}} \]

      11. +-commutative 91.2%

          \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{y + x}}}{\left(x + \left(y + 1\right)\right) \cdot \left(x + y\right)} \]

      12. +-commutative 91.2%

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\color{blue}{\left(\left(y + 1\right) + x\right)} \cdot \left(x + y\right)} \]

      13. associate-+l+ 91.2%

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\color{blue}{\left(y + \left(1 + x\right)\right)} \cdot \left(x + y\right)} \]

      14. +-commutative 91.2%

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\left(y + \left(1 + x\right)\right) \cdot \color{blue}{\left(y + x\right)}} \]

   5. Applied egg-rr 91.2%

      \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{y + x}}{\left(y + \left(1 + x\right)\right) \cdot \left(y + x\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 99.9%

         \[\leadsto \color{blue}{\frac{\frac{y \cdot \frac{x}{y + x}}{y + \left(1 + x\right)}}{y + x}} \]

      2. associate-*l/ 99.8%

         \[\leadsto \frac{\color{blue}{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{y + x}}}{y + x} \]

      3. *-commutative 99.8%

         \[\leadsto \frac{\color{blue}{\frac{x}{y + x} \cdot \frac{y}{y + \left(1 + x\right)}}}{y + x} \]

      4. associate-*r/ 99.9%

         \[\leadsto \frac{\color{blue}{\frac{\frac{x}{y + x} \cdot y}{y + \left(1 + x\right)}}}{y + x} \]

      5. *-commutative 99.9%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot \frac{x}{y + x}}}{y + \left(1 + x\right)}}{y + x} \]

      6. associate-*r/ 78.3%

         \[\leadsto \frac{\frac{\color{blue}{\frac{y \cdot x}{y + x}}}{y + \left(1 + x\right)}}{y + x} \]

      7. +-commutative 78.3%

         \[\leadsto \frac{\frac{\frac{y \cdot x}{y + x}}{y + \color{blue}{\left(x + 1\right)}}}{y + x} \]

   7. Simplified 78.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{y \cdot x}{y + x}}{y + \left(x + 1\right)}}{y + x}} \]

   8. Taylor expanded in y around 0 25.5%

      \[\leadsto \frac{\frac{\color{blue}{y}}{y + \left(x + 1\right)}}{y + x} \]

   9. Taylor expanded in y around inf 5.6%

      \[\leadsto \color{blue}{\frac{1}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-47}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y}\\ \end{array} \]

Alternative 21: 3.8% 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 70.0%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-*r/ 84.4%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   2. *-commutative 84.4%

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   3. distribute-rgt1-in 67.7%

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   4. fma-def 84.5%

      \[\leadsto x \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(x + y, x + y, \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)\right)}} \]

   5. cube-unmult 84.5%

      \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(x + y\right)}^{3}}\right)} \]

3. Simplified 84.5%

   \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ 70.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   2. fma-udef 55.5%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   3. cube-mult 55.5%

      \[\leadsto \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   4. distribute-rgt1-in 70.0%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   5. associate-+r+ 70.0%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

   6. *-commutative 70.0%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

   7. frac-times 88.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   8. *-commutative 88.9%

      \[\leadsto \color{blue}{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   9. associate-/r* 99.8%

      \[\leadsto \frac{y}{x + \left(y + 1\right)} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

   10. associate-*r/ 99.8%

       \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{x + y}}{x + y}} \]

   11. +-commutative 99.8%

       \[\leadsto \frac{\frac{y}{\color{blue}{\left(y + 1\right) + x}} \cdot \frac{x}{x + y}}{x + y} \]

   12. associate-+l+ 99.8%

       \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

   13. +-commutative 99.8%

       \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{\color{blue}{y + x}}}{x + y} \]

   14. +-commutative 99.8%

       \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{y + x}}{\color{blue}{y + x}} \]

5. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{y + x}}{y + x}} \]

6. Taylor expanded in x around inf 47.9%

   \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\left(1 + -1 \cdot \frac{y}{x}\right)}}{y + x} \]
7. Step-by-step derivation

   1. mul-1-neg 47.9%

      \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \left(1 + \color{blue}{\left(-\frac{y}{x}\right)}\right)}{y + x} \]

   2. unsub-neg 47.9%

      \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\left(1 - \frac{y}{x}\right)}}{y + x} \]

8. Simplified 47.9%

   \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\left(1 - \frac{y}{x}\right)}}{y + x} \]

9. Taylor expanded in y around inf 3.7%

   \[\leadsto \color{blue}{\frac{-1}{x}} \]

10. Final simplification 3.7%

    \[\leadsto \frac{-1}{x} \]

Alternative 22: 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 70.0%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. times-frac 88.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. /-rgt-identity 88.9%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

   3. associate-/l/ 88.9%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

   4. *-lft-identity 88.9%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

   5. associate-+l+ 88.9%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

3. Simplified 88.9%

   \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

4. Taylor expanded in x around inf 48.4%

   \[\leadsto \color{blue}{\frac{1}{x}} \cdot \frac{y}{x + \left(y + 1\right)} \]

5. Taylor expanded in y around inf 4.1%

   \[\leadsto \color{blue}{\frac{1}{x}} \]

6. Final simplification 4.1%

   \[\leadsto \frac{1}{x} \]

Alternative 23: 4.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{1}{y} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ 1.0 y))

C

assert(x < y);
double code(double x, double y) {
	return 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 = 1.0d0 / y
end function

Java

assert x < y;
public static double code(double x, double y) {
	return 1.0 / y;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return 1.0 / y

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(1.0 / y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 1.0 / y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(1.0 / y), $MachinePrecision]

Derivation

1. Initial program 70.0%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-*r/ 84.4%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   2. *-commutative 84.4%

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   3. distribute-rgt1-in 67.7%

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   4. fma-def 84.5%

      \[\leadsto x \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(x + y, x + y, \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)\right)}} \]

   5. cube-unmult 84.5%

      \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(x + y\right)}^{3}}\right)} \]

3. Simplified 84.5%

   \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ 70.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   2. fma-udef 55.5%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   3. cube-mult 55.5%

      \[\leadsto \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   4. distribute-rgt1-in 70.0%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   5. associate-+r+ 70.0%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

   6. *-commutative 70.0%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

   7. frac-times 88.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   8. *-commutative 88.9%

      \[\leadsto \color{blue}{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   9. associate-/r* 99.8%

      \[\leadsto \frac{y}{x + \left(y + 1\right)} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

   10. frac-times 93.5%

       \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{x + y}}{\left(x + \left(y + 1\right)\right) \cdot \left(x + y\right)}} \]

   11. +-commutative 93.5%

       \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{y + x}}}{\left(x + \left(y + 1\right)\right) \cdot \left(x + y\right)} \]

   12. +-commutative 93.5%

       \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\color{blue}{\left(\left(y + 1\right) + x\right)} \cdot \left(x + y\right)} \]

   13. associate-+l+ 93.5%

       \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\color{blue}{\left(y + \left(1 + x\right)\right)} \cdot \left(x + y\right)} \]

   14. +-commutative 93.5%

       \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\left(y + \left(1 + x\right)\right) \cdot \color{blue}{\left(y + x\right)}} \]

5. Applied egg-rr 93.5%

   \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{y + x}}{\left(y + \left(1 + x\right)\right) \cdot \left(y + x\right)}} \]
6. Step-by-step derivation

   1. associate-/r* 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y \cdot \frac{x}{y + x}}{y + \left(1 + x\right)}}{y + x}} \]

   2. associate-*l/ 99.8%

      \[\leadsto \frac{\color{blue}{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{y + x}}}{y + x} \]

   3. *-commutative 99.8%

      \[\leadsto \frac{\color{blue}{\frac{x}{y + x} \cdot \frac{y}{y + \left(1 + x\right)}}}{y + x} \]

   4. associate-*r/ 99.8%

      \[\leadsto \frac{\color{blue}{\frac{\frac{x}{y + x} \cdot y}{y + \left(1 + x\right)}}}{y + x} \]

   5. *-commutative 99.8%

      \[\leadsto \frac{\frac{\color{blue}{y \cdot \frac{x}{y + x}}}{y + \left(1 + x\right)}}{y + x} \]

   6. associate-*r/ 75.3%

      \[\leadsto \frac{\frac{\color{blue}{\frac{y \cdot x}{y + x}}}{y + \left(1 + x\right)}}{y + x} \]

   7. +-commutative 75.3%

      \[\leadsto \frac{\frac{\frac{y \cdot x}{y + x}}{y + \color{blue}{\left(x + 1\right)}}}{y + x} \]

7. Simplified 75.3%

   \[\leadsto \color{blue}{\frac{\frac{\frac{y \cdot x}{y + x}}{y + \left(x + 1\right)}}{y + x}} \]

8. Taylor expanded in y around 0 49.4%

   \[\leadsto \frac{\frac{\color{blue}{y}}{y + \left(x + 1\right)}}{y + x} \]

9. Taylor expanded in y around inf 4.1%

   \[\leadsto \color{blue}{\frac{1}{y}} \]

10. Final simplification 4.1%

    \[\leadsto \frac{1}{y} \]

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 2023257 
(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))))