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

Percentage Accurate: 70.0% → 99.8%

Time: 15.0s

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: 70.0% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.9%

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

   1. times-frac 86.0%

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

   2. associate-+l+ 86.0%

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

3. Simplified 86.0%

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

   1. associate-/r* 99.8%

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

   2. div-inv 99.7%

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

5. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{x + y}\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-16}:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y} \cdot \frac{y}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.2e+160)
   (/ (/ y (+ y (+ x 1.0))) (+ x y))
   (if (<= x -1.15e-16)
     (* (/ y (+ x (+ y 1.0))) (/ x (* (+ x y) (+ x y))))
     (* (/ (/ x (+ x y)) (+ x y)) (/ y (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.2e+160) {
		tmp = (y / (y + (x + 1.0))) / (x + y);
	} else if (x <= -1.15e-16) {
		tmp = (y / (x + (y + 1.0))) * (x / ((x + y) * (x + y)));
	} else {
		tmp = ((x / (x + y)) / (x + y)) * (y / (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.2d+160)) then
        tmp = (y / (y + (x + 1.0d0))) / (x + y)
    else if (x <= (-1.15d-16)) then
        tmp = (y / (x + (y + 1.0d0))) * (x / ((x + y) * (x + y)))
    else
        tmp = ((x / (x + y)) / (x + y)) * (y / (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -3.2e+160) {
		tmp = (y / (y + (x + 1.0))) / (x + y);
	} else if (x <= -1.15e-16) {
		tmp = (y / (x + (y + 1.0))) * (x / ((x + y) * (x + y)));
	} else {
		tmp = ((x / (x + y)) / (x + y)) * (y / (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.2e+160:
		tmp = (y / (y + (x + 1.0))) / (x + y)
	elif x <= -1.15e-16:
		tmp = (y / (x + (y + 1.0))) * (x / ((x + y) * (x + y)))
	else:
		tmp = ((x / (x + y)) / (x + y)) * (y / (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.2e+160)
		tmp = Float64(Float64(y / Float64(y + Float64(x + 1.0))) / Float64(x + y));
	elseif (x <= -1.15e-16)
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) * Float64(x / Float64(Float64(x + y) * Float64(x + y))));
	else
		tmp = Float64(Float64(Float64(x / Float64(x + y)) / Float64(x + y)) * Float64(y / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.2e+160)
		tmp = (y / (y + (x + 1.0))) / (x + y);
	elseif (x <= -1.15e-16)
		tmp = (y / (x + (y + 1.0))) * (x / ((x + y) * (x + y)));
	else
		tmp = ((x / (x + y)) / (x + y)) * (y / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -3.2e+160], N[(N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.15e-16], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.1999999999999998e160

   1. Initial program 59.4%

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

      1. times-frac 82.1%

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

      2. associate-+l+ 82.1%

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

   3. Simplified 82.1%

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

      1. associate-/r* 99.9%

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

      2. div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around inf 86.1%

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

      1. associate-*l/ 86.1%

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

      2. *-un-lft-identity 86.1%

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

      3. associate-+r+ 86.1%

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

      4. +-commutative 86.1%

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

      5. associate-+l+ 86.1%

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

   10. Applied egg-rr 86.1%

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

   if -3.1999999999999998e160 < x < -1.15e-16

   1. Initial program 76.6%

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

      1. times-frac 91.6%

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

      2. associate-+l+ 91.6%

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

   3. Simplified 91.6%

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

   if -1.15e-16 < x

   1. Initial program 66.5%

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

      1. times-frac 85.7%

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

      2. associate-+l+ 85.7%

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

   3. Simplified 85.7%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.7%

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

      2. *-rgt-identity 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 84.1%

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

      1. +-commutative 84.1%

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

   10. Simplified 84.1%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{x + y}\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-16}:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y} \cdot \frac{y}{y + 1}\\ \end{array} \]

Alternative 3: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{x + y}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-21}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y} \cdot \frac{y}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.3e+160)
   (/ (/ y (+ y (+ x 1.0))) (+ x y))
   (if (<= x -2e-21)
     (* (/ y (* (+ x y) (+ x y))) (/ x (+ x (+ y 1.0))))
     (* (/ (/ x (+ x y)) (+ x y)) (/ y (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.3e+160) {
		tmp = (y / (y + (x + 1.0))) / (x + y);
	} else if (x <= -2e-21) {
		tmp = (y / ((x + y) * (x + y))) * (x / (x + (y + 1.0)));
	} else {
		tmp = ((x / (x + y)) / (x + y)) * (y / (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.3d+160)) then
        tmp = (y / (y + (x + 1.0d0))) / (x + y)
    else if (x <= (-2d-21)) then
        tmp = (y / ((x + y) * (x + y))) * (x / (x + (y + 1.0d0)))
    else
        tmp = ((x / (x + y)) / (x + y)) * (y / (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -3.3e+160) {
		tmp = (y / (y + (x + 1.0))) / (x + y);
	} else if (x <= -2e-21) {
		tmp = (y / ((x + y) * (x + y))) * (x / (x + (y + 1.0)));
	} else {
		tmp = ((x / (x + y)) / (x + y)) * (y / (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.3e+160:
		tmp = (y / (y + (x + 1.0))) / (x + y)
	elif x <= -2e-21:
		tmp = (y / ((x + y) * (x + y))) * (x / (x + (y + 1.0)))
	else:
		tmp = ((x / (x + y)) / (x + y)) * (y / (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.3e+160)
		tmp = Float64(Float64(y / Float64(y + Float64(x + 1.0))) / Float64(x + y));
	elseif (x <= -2e-21)
		tmp = Float64(Float64(y / Float64(Float64(x + y) * Float64(x + y))) * Float64(x / Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(Float64(x / Float64(x + y)) / Float64(x + y)) * Float64(y / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.3e+160)
		tmp = (y / (y + (x + 1.0))) / (x + y);
	elseif (x <= -2e-21)
		tmp = (y / ((x + y) * (x + y))) * (x / (x + (y + 1.0)));
	else
		tmp = ((x / (x + y)) / (x + y)) * (y / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -3.3e+160], N[(N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2e-21], N[(N[(y / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2999999999999997e160

   1. Initial program 59.4%

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

      1. times-frac 82.1%

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

      2. associate-+l+ 82.1%

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

   3. Simplified 82.1%

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

      1. associate-/r* 99.9%

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

      2. div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around inf 86.1%

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

      1. associate-*l/ 86.1%

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

      2. *-un-lft-identity 86.1%

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

      3. associate-+r+ 86.1%

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

      4. +-commutative 86.1%

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

      5. associate-+l+ 86.1%

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

   10. Applied egg-rr 86.1%

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

   if -3.2999999999999997e160 < x < -1.99999999999999982e-21

   1. Initial program 77.2%

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

      1. associate-/r* 82.5%

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

      2. +-commutative 82.5%

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

      3. +-commutative 82.5%

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

      4. +-commutative 82.5%

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

      5. associate-/l/ 77.2%

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

      6. times-frac 92.1%

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

      7. *-commutative 92.1%

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

      8. +-commutative 92.1%

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

      9. +-commutative 92.1%

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

      10. +-commutative 92.1%

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

      11. associate-+l+ 92.1%

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

   3. Simplified 92.1%

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

   if -1.99999999999999982e-21 < x

   1. Initial program 66.3%

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

      1. times-frac 85.7%

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

      2. associate-+l+ 85.7%

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

   3. Simplified 85.7%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.7%

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

      2. *-rgt-identity 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 84.0%

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

      1. +-commutative 84.0%

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

   10. Simplified 84.0%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{x + y}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-21}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y} \cdot \frac{y}{y + 1}\\ \end{array} \]

Alternative 4: 89.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{-168}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 150000000:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + \left(x + \left(x - -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.2e-168)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 150000000.0)
     (* (/ y (* (+ x y) (+ x y))) (/ x (+ x 1.0)))
     (/ (/ x (+ x y)) (+ y (+ x (- x -1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.2e-168) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 150000000.0) {
		tmp = (y / ((x + y) * (x + y))) * (x / (x + 1.0));
	} else {
		tmp = (x / (x + y)) / (y + (x + (x - -1.0)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.2d-168) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 150000000.0d0) then
        tmp = (y / ((x + y) * (x + y))) * (x / (x + 1.0d0))
    else
        tmp = (x / (x + y)) / (y + (x + (x - (-1.0d0))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.2e-168) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 150000000.0) {
		tmp = (y / ((x + y) * (x + y))) * (x / (x + 1.0));
	} else {
		tmp = (x / (x + y)) / (y + (x + (x - -1.0)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.2e-168:
		tmp = (y / x) / (x + 1.0)
	elif y <= 150000000.0:
		tmp = (y / ((x + y) * (x + y))) * (x / (x + 1.0))
	else:
		tmp = (x / (x + y)) / (y + (x + (x - -1.0)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.2e-168)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 150000000.0)
		tmp = Float64(Float64(y / Float64(Float64(x + y) * Float64(x + y))) * Float64(x / Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(y + Float64(x + Float64(x - -1.0))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.2e-168)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 150000000.0)
		tmp = (y / ((x + y) * (x + y))) * (x / (x + 1.0));
	else
		tmp = (x / (x + y)) / (y + (x + (x - -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.2e-168], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 150000000.0], N[(N[(y / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.2e-168

   1. Initial program 65.2%

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

      1. times-frac 85.4%

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

      2. associate-+l+ 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in y around 0 57.6%

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

      1. associate-/r* 56.1%

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

      2. +-commutative 56.1%

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

   6. Simplified 56.1%

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

   if 1.2e-168 < y < 1.5e8

   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. associate-/r* 91.5%

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

      2. +-commutative 91.5%

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

      3. +-commutative 91.5%

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

      4. +-commutative 91.5%

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

      5. associate-/l/ 88.9%

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

      6. times-frac 99.5%

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

      7. *-commutative 99.5%

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

      8. +-commutative 99.5%

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

      9. +-commutative 99.5%

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

      10. +-commutative 99.5%

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

      11. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 97.7%

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

      1. +-commutative 97.7%

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

   6. Simplified 97.7%

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

   if 1.5e8 < y

   1. Initial program 58.6%

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

      1. associate-/r* 63.2%

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

      2. +-commutative 63.2%

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

      3. +-commutative 63.2%

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

      4. +-commutative 63.2%

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

      5. associate-/r* 58.6%

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

      6. associate-*l/ 71.7%

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

      7. *-commutative 71.7%

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

      8. *-commutative 71.7%

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

      9. distribute-rgt1-in 68.6%

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

      10. fma-def 71.7%

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

      11. +-commutative 71.7%

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

      12. +-commutative 71.7%

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

      13. cube-unmult 71.8%

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

      14. +-commutative 71.8%

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

   3. Simplified 71.8%

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

      1. associate-*r/ 58.6%

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

      2. fma-udef 57.6%

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

      3. cube-mult 57.6%

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

      4. distribute-rgt1-in 58.6%

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

      5. associate-+r+ 58.6%

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

      6. *-commutative 58.6%

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

      7. *-commutative 58.6%

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

      8. frac-times 80.1%

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

      9. associate-/r* 99.8%

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

      10. clear-num 99.8%

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

      11. frac-times 99.2%

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

      12. *-un-lft-identity 99.2%

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

   5. Applied egg-rr 99.2%

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

   6. Taylor expanded in y around -inf 77.8%

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

      1. mul-1-neg 77.8%

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

      2. unsub-neg 77.8%

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

      3. neg-mul-1 77.8%

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

      4. +-commutative 77.8%

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

      5. unsub-neg 77.8%

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

      6. distribute-lft-in 77.8%

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

      7. metadata-eval 77.8%

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

      8. neg-mul-1 77.8%

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

      9. unsub-neg 77.8%

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

   8. Simplified 77.8%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{-168}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 150000000:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + \left(x + \left(x - -1\right)\right)}\\ \end{array} \]

Alternative 5: 94.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{x + y}\\ \mathbf{elif}\;x \leq -5.3 \cdot 10^{-20}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y} \cdot \frac{y}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -4.4e+161)
   (/ (/ y (+ y (+ x 1.0))) (+ x y))
   (if (<= x -5.3e-20)
     (* (/ y (* (+ x y) (+ x y))) (/ x (+ x 1.0)))
     (* (/ (/ x (+ x y)) (+ x y)) (/ y (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4.4e+161) {
		tmp = (y / (y + (x + 1.0))) / (x + y);
	} else if (x <= -5.3e-20) {
		tmp = (y / ((x + y) * (x + y))) * (x / (x + 1.0));
	} else {
		tmp = ((x / (x + y)) / (x + y)) * (y / (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.4d+161)) then
        tmp = (y / (y + (x + 1.0d0))) / (x + y)
    else if (x <= (-5.3d-20)) then
        tmp = (y / ((x + y) * (x + y))) * (x / (x + 1.0d0))
    else
        tmp = ((x / (x + y)) / (x + y)) * (y / (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -4.4e+161) {
		tmp = (y / (y + (x + 1.0))) / (x + y);
	} else if (x <= -5.3e-20) {
		tmp = (y / ((x + y) * (x + y))) * (x / (x + 1.0));
	} else {
		tmp = ((x / (x + y)) / (x + y)) * (y / (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -4.4e+161:
		tmp = (y / (y + (x + 1.0))) / (x + y)
	elif x <= -5.3e-20:
		tmp = (y / ((x + y) * (x + y))) * (x / (x + 1.0))
	else:
		tmp = ((x / (x + y)) / (x + y)) * (y / (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -4.4e+161)
		tmp = Float64(Float64(y / Float64(y + Float64(x + 1.0))) / Float64(x + y));
	elseif (x <= -5.3e-20)
		tmp = Float64(Float64(y / Float64(Float64(x + y) * Float64(x + y))) * Float64(x / Float64(x + 1.0)));
	else
		tmp = Float64(Float64(Float64(x / Float64(x + y)) / Float64(x + y)) * Float64(y / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.4e+161)
		tmp = (y / (y + (x + 1.0))) / (x + y);
	elseif (x <= -5.3e-20)
		tmp = (y / ((x + y) * (x + y))) * (x / (x + 1.0));
	else
		tmp = ((x / (x + y)) / (x + y)) * (y / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -4.4e+161], N[(N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.3e-20], N[(N[(y / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.4e161

   1. Initial program 61.2%

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

      1. times-frac 84.5%

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

      2. associate-+l+ 84.5%

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

   3. Simplified 84.5%

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

      1. associate-/r* 99.9%

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

      2. div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around inf 88.6%

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

      1. associate-*l/ 88.6%

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

      2. *-un-lft-identity 88.6%

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

      3. associate-+r+ 88.6%

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

      4. +-commutative 88.6%

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

      5. associate-+l+ 88.6%

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

   10. Applied egg-rr 88.6%

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

   if -4.4e161 < x < -5.3000000000000002e-20

   1. Initial program 75.1%

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

      1. associate-/r* 80.3%

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

      2. +-commutative 80.3%

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

      3. +-commutative 80.3%

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

      4. +-commutative 80.3%

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

      5. associate-/l/ 75.1%

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

      6. times-frac 89.7%

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

      7. *-commutative 89.7%

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

      8. +-commutative 89.7%

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

      9. +-commutative 89.7%

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

      10. +-commutative 89.7%

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

      11. associate-+l+ 89.7%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in y around 0 84.8%

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

      1. +-commutative 84.8%

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

   6. Simplified 84.8%

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

   if -5.3000000000000002e-20 < x

   1. Initial program 66.3%

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

      1. times-frac 85.7%

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

      2. associate-+l+ 85.7%

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

   3. Simplified 85.7%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.7%

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

      2. *-rgt-identity 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 84.0%

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

      1. +-commutative 84.0%

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

   10. Simplified 84.0%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{x + y}\\ \mathbf{elif}\;x \leq -5.3 \cdot 10^{-20}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y} \cdot \frac{y}{y + 1}\\ \end{array} \]

Alternative 6: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{-92}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 0.55:\\ \;\;\;\;x \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + \left(y + \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4.6e-92)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 0.55)
     (* x (/ y (* (+ x y) (+ x y))))
     (/ (/ x (+ x y)) (+ x (+ y (+ x 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.6e-92) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 0.55) {
		tmp = x * (y / ((x + y) * (x + y)));
	} else {
		tmp = (x / (x + y)) / (x + (y + (x + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.6d-92) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 0.55d0) then
        tmp = x * (y / ((x + y) * (x + y)))
    else
        tmp = (x / (x + y)) / (x + (y + (x + 1.0d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 4.6e-92) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 0.55) {
		tmp = x * (y / ((x + y) * (x + y)));
	} else {
		tmp = (x / (x + y)) / (x + (y + (x + 1.0)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 4.6e-92:
		tmp = (y / x) / (x + 1.0)
	elif y <= 0.55:
		tmp = x * (y / ((x + y) * (x + y)))
	else:
		tmp = (x / (x + y)) / (x + (y + (x + 1.0)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4.6e-92)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 0.55)
		tmp = Float64(x * Float64(y / Float64(Float64(x + y) * Float64(x + y))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(x + Float64(y + Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.6e-92)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 0.55)
		tmp = x * (y / ((x + y) * (x + y)));
	else
		tmp = (x / (x + y)) / (x + (y + (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 4.6e-92], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.55], N[(x * N[(y / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.60000000000000032e-92

   1. Initial program 66.5%

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

      1. times-frac 86.1%

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

      2. associate-+l+ 86.1%

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

   3. Simplified 86.1%

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

   4. Taylor expanded in y around 0 58.7%

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

      1. associate-/r* 57.3%

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

      2. +-commutative 57.3%

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

   6. Simplified 57.3%

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

   if 4.60000000000000032e-92 < y < 0.55000000000000004

   1. Initial program 88.5%

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

      1. associate-/r* 92.1%

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

      2. +-commutative 92.1%

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

      3. +-commutative 92.1%

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

      4. +-commutative 92.1%

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

      5. associate-/l/ 88.5%

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

      6. times-frac 99.5%

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

      7. *-commutative 99.5%

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

      8. +-commutative 99.5%

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

      9. +-commutative 99.5%

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

      10. +-commutative 99.5%

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

      11. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 97.1%

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

      1. +-commutative 97.1%

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

   6. Simplified 97.1%

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

   7. Taylor expanded in x around 0 82.0%

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

   if 0.55000000000000004 < y

   1. Initial program 59.3%

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

      1. associate-/r* 63.8%

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

      2. +-commutative 63.8%

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

      3. +-commutative 63.8%

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

      4. +-commutative 63.8%

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

      5. associate-/r* 59.3%

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

      6. associate-*l/ 72.2%

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

      7. *-commutative 72.2%

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

      8. *-commutative 72.2%

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

      9. distribute-rgt1-in 69.1%

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

      10. fma-def 72.2%

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

      11. +-commutative 72.2%

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

      12. +-commutative 72.2%

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

      13. cube-unmult 72.3%

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

      14. +-commutative 72.3%

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

   3. Simplified 72.3%

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

      1. associate-*r/ 59.3%

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

      2. fma-udef 58.3%

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

      3. cube-mult 58.3%

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

      4. distribute-rgt1-in 59.3%

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

      5. associate-+r+ 59.3%

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

      6. *-commutative 59.3%

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

      7. *-commutative 59.3%

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

      8. frac-times 80.4%

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

      9. associate-/r* 99.8%

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

      10. clear-num 99.8%

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

      11. frac-times 99.2%

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

      12. *-un-lft-identity 99.2%

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

   5. Applied egg-rr 99.2%

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

   6. Taylor expanded in y around -inf 76.6%

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

      1. mul-1-neg 76.6%

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

      2. unsub-neg 76.6%

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

      3. neg-mul-1 76.6%

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

      4. +-commutative 76.6%

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

      5. unsub-neg 76.6%

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

      6. distribute-lft-in 76.6%

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

      7. metadata-eval 76.6%

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

      8. neg-mul-1 76.6%

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

      9. unsub-neg 76.6%

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

   8. Simplified 76.6%

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

      1. associate--r- 76.6%

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

   10. Applied egg-rr 76.6%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{-92}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 0.55:\\ \;\;\;\;x \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + \left(y + \left(x + 1\right)\right)}\\ \end{array} \]

Alternative 7: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{-92}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 1700000:\\ \;\;\;\;x \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + \left(x + \left(x - -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.7e-92)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 1700000.0)
     (* x (/ y (* (+ x y) (+ x y))))
     (/ (/ x (+ x y)) (+ y (+ x (- x -1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.7e-92) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 1700000.0) {
		tmp = x * (y / ((x + y) * (x + y)));
	} else {
		tmp = (x / (x + y)) / (y + (x + (x - -1.0)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.7d-92) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 1700000.0d0) then
        tmp = x * (y / ((x + y) * (x + y)))
    else
        tmp = (x / (x + y)) / (y + (x + (x - (-1.0d0))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.7e-92) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 1700000.0) {
		tmp = x * (y / ((x + y) * (x + y)));
	} else {
		tmp = (x / (x + y)) / (y + (x + (x - -1.0)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.7e-92:
		tmp = (y / x) / (x + 1.0)
	elif y <= 1700000.0:
		tmp = x * (y / ((x + y) * (x + y)))
	else:
		tmp = (x / (x + y)) / (y + (x + (x - -1.0)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.7e-92)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 1700000.0)
		tmp = Float64(x * Float64(y / Float64(Float64(x + y) * Float64(x + y))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(y + Float64(x + Float64(x - -1.0))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.7e-92)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 1700000.0)
		tmp = x * (y / ((x + y) * (x + y)));
	else
		tmp = (x / (x + y)) / (y + (x + (x - -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.7e-92], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1700000.0], N[(x * N[(y / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.7000000000000001e-92

   1. Initial program 66.5%

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

      1. times-frac 86.1%

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

      2. associate-+l+ 86.1%

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

   3. Simplified 86.1%

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

   4. Taylor expanded in y around 0 58.7%

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

      1. associate-/r* 57.3%

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

      2. +-commutative 57.3%

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

   6. Simplified 57.3%

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

   if 1.7000000000000001e-92 < y < 1.7e6

   1. Initial program 89.0%

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

      1. associate-/r* 92.4%

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

      2. +-commutative 92.4%

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

      3. +-commutative 92.4%

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

      4. +-commutative 92.4%

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

      5. associate-/l/ 89.0%

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

      6. times-frac 99.5%

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

      7. *-commutative 99.5%

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

      8. +-commutative 99.5%

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

      9. +-commutative 99.5%

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

      10. +-commutative 99.5%

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

      11. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 97.2%

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

      1. +-commutative 97.2%

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

   6. Simplified 97.2%

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

   7. Taylor expanded in x around 0 79.1%

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

   if 1.7e6 < y

   1. Initial program 58.6%

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

      1. associate-/r* 63.2%

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

      2. +-commutative 63.2%

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

      3. +-commutative 63.2%

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

      4. +-commutative 63.2%

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

      5. associate-/r* 58.6%

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

      6. associate-*l/ 71.7%

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

      7. *-commutative 71.7%

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

      8. *-commutative 71.7%

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

      9. distribute-rgt1-in 68.6%

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

      10. fma-def 71.7%

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

      11. +-commutative 71.7%

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

      12. +-commutative 71.7%

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

      13. cube-unmult 71.8%

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

      14. +-commutative 71.8%

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

   3. Simplified 71.8%

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

      1. associate-*r/ 58.6%

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

      2. fma-udef 57.6%

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

      3. cube-mult 57.6%

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

      4. distribute-rgt1-in 58.6%

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

      5. associate-+r+ 58.6%

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

      6. *-commutative 58.6%

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

      7. *-commutative 58.6%

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

      8. frac-times 80.1%

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

      9. associate-/r* 99.8%

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

      10. clear-num 99.8%

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

      11. frac-times 99.2%

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

      12. *-un-lft-identity 99.2%

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

   5. Applied egg-rr 99.2%

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

   6. Taylor expanded in y around -inf 77.8%

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

      1. mul-1-neg 77.8%

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

      2. unsub-neg 77.8%

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

      3. neg-mul-1 77.8%

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

      4. +-commutative 77.8%

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

      5. unsub-neg 77.8%

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

      6. distribute-lft-in 77.8%

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

      7. metadata-eval 77.8%

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

      8. neg-mul-1 77.8%

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

      9. unsub-neg 77.8%

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

   8. Simplified 77.8%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{-92}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 1700000:\\ \;\;\;\;x \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + \left(x + \left(x - -1\right)\right)}\\ \end{array} \]

Alternative 8: 84.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-92}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 110000000:\\ \;\;\;\;x \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.8e-92)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 110000000.0)
     (* x (/ y (* (+ x y) (+ x y))))
     (/ (/ x (+ x y)) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.8e-92) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 110000000.0) {
		tmp = x * (y / ((x + y) * (x + y)));
	} else {
		tmp = (x / (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 <= 1.8d-92) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 110000000.0d0) then
        tmp = x * (y / ((x + y) * (x + y)))
    else
        tmp = (x / (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 <= 1.8e-92) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 110000000.0) {
		tmp = x * (y / ((x + y) * (x + y)));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.8e-92:
		tmp = (y / x) / (x + 1.0)
	elif y <= 110000000.0:
		tmp = x * (y / ((x + y) * (x + y)))
	else:
		tmp = (x / (x + y)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.8e-92)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 110000000.0)
		tmp = Float64(x * Float64(y / Float64(Float64(x + y) * Float64(x + y))));
	else
		tmp = Float64(Float64(x / 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 <= 1.8e-92)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 110000000.0)
		tmp = x * (y / ((x + y) * (x + y)));
	else
		tmp = (x / (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, 1.8e-92], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 110000000.0], N[(x * N[(y / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.80000000000000008e-92

   1. Initial program 66.5%

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

      1. times-frac 86.1%

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

      2. associate-+l+ 86.1%

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

   3. Simplified 86.1%

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

   4. Taylor expanded in y around 0 58.7%

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

      1. associate-/r* 57.3%

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

      2. +-commutative 57.3%

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

   6. Simplified 57.3%

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

   if 1.80000000000000008e-92 < y < 1.1e8

   1. Initial program 89.0%

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

      1. associate-/r* 92.4%

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

      2. +-commutative 92.4%

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

      3. +-commutative 92.4%

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

      4. +-commutative 92.4%

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

      5. associate-/l/ 89.0%

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

      6. times-frac 99.5%

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

      7. *-commutative 99.5%

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

      8. +-commutative 99.5%

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

      9. +-commutative 99.5%

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

      10. +-commutative 99.5%

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

      11. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 97.2%

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

      1. +-commutative 97.2%

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

   6. Simplified 97.2%

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

   7. Taylor expanded in x around 0 79.1%

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

   if 1.1e8 < y

   1. Initial program 58.6%

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

      1. associate-/r* 63.2%

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

      2. +-commutative 63.2%

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

      3. +-commutative 63.2%

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

      4. +-commutative 63.2%

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

      5. associate-/r* 58.6%

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

      6. associate-*l/ 71.7%

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

      7. *-commutative 71.7%

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

      8. *-commutative 71.7%

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

      9. distribute-rgt1-in 68.6%

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

      10. fma-def 71.7%

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

      11. +-commutative 71.7%

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

      12. +-commutative 71.7%

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

      13. cube-unmult 71.8%

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

      14. +-commutative 71.8%

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

   3. Simplified 71.8%

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

      1. associate-*r/ 58.6%

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

      2. fma-udef 57.6%

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

      3. cube-mult 57.6%

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

      4. distribute-rgt1-in 58.6%

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

      5. associate-+r+ 58.6%

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

      6. *-commutative 58.6%

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

      7. *-commutative 58.6%

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

      8. frac-times 80.1%

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

      9. associate-/r* 99.8%

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

      10. clear-num 99.8%

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

      11. frac-times 99.2%

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

      12. *-un-lft-identity 99.2%

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

   5. Applied egg-rr 99.2%

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

   6. Taylor expanded in x around 0 76.9%

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

      1. +-commutative 76.9%

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

   8. Simplified 76.9%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-92}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 110000000:\\ \;\;\;\;x \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]

Alternative 9: 71.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{-137}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-161}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-86}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 96000000:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* x x))))
   (if (<= y -5.8e-137)
     t_0
     (if (<= y 3.8e-161)
       (/ y x)
       (if (<= y 3.3e-86)
         t_0
         (if (<= y 96000000.0) (- (/ x y) x) (/ x (* y y))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x * x);
	double tmp;
	if (y <= -5.8e-137) {
		tmp = t_0;
	} else if (y <= 3.8e-161) {
		tmp = y / x;
	} else if (y <= 3.3e-86) {
		tmp = t_0;
	} else if (y <= 96000000.0) {
		tmp = (x / y) - x;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (x * x)
    if (y <= (-5.8d-137)) then
        tmp = t_0
    else if (y <= 3.8d-161) then
        tmp = y / x
    else if (y <= 3.3d-86) then
        tmp = t_0
    else if (y <= 96000000.0d0) then
        tmp = (x / y) - x
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (x * x);
	double tmp;
	if (y <= -5.8e-137) {
		tmp = t_0;
	} else if (y <= 3.8e-161) {
		tmp = y / x;
	} else if (y <= 3.3e-86) {
		tmp = t_0;
	} else if (y <= 96000000.0) {
		tmp = (x / y) - x;
	} 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 <= -5.8e-137:
		tmp = t_0
	elif y <= 3.8e-161:
		tmp = y / x
	elif y <= 3.3e-86:
		tmp = t_0
	elif y <= 96000000.0:
		tmp = (x / y) - x
	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 <= -5.8e-137)
		tmp = t_0;
	elseif (y <= 3.8e-161)
		tmp = Float64(y / x);
	elseif (y <= 3.3e-86)
		tmp = t_0;
	elseif (y <= 96000000.0)
		tmp = Float64(Float64(x / 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)
	t_0 = y / (x * x);
	tmp = 0.0;
	if (y <= -5.8e-137)
		tmp = t_0;
	elseif (y <= 3.8e-161)
		tmp = y / x;
	elseif (y <= 3.3e-86)
		tmp = t_0;
	elseif (y <= 96000000.0)
		tmp = (x / y) - x;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.8e-137], t$95$0, If[LessEqual[y, 3.8e-161], N[(y / x), $MachinePrecision], If[LessEqual[y, 3.3e-86], t$95$0, If[LessEqual[y, 96000000.0], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.7999999999999997e-137 or 3.8000000000000001e-161 < y < 3.29999999999999987e-86

   1. Initial program 68.5%

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

      1. associate-/r* 75.2%

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

      2. +-commutative 75.2%

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

      3. +-commutative 75.2%

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

      4. +-commutative 75.2%

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

      5. associate-/r* 68.5%

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

      6. associate-*l/ 83.4%

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

      7. *-commutative 83.4%

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

      8. *-commutative 83.4%

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

      9. distribute-rgt1-in 52.5%

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

      10. fma-def 83.4%

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

      11. +-commutative 83.4%

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

      12. +-commutative 83.4%

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

      13. cube-unmult 83.4%

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

      14. +-commutative 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in x around inf 39.2%

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

      1. unpow2 39.2%

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

   6. Simplified 39.2%

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

   if -5.7999999999999997e-137 < y < 3.8000000000000001e-161

   1. Initial program 64.6%

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

      1. associate-/r* 64.5%

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

      2. +-commutative 64.5%

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

      3. +-commutative 64.5%

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

      4. +-commutative 64.5%

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

      5. associate-/r* 64.6%

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

      6. associate-*l/ 75.9%

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

      7. *-commutative 75.9%

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

      8. *-commutative 75.9%

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

      9. distribute-rgt1-in 62.0%

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

      10. fma-def 75.9%

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

      11. +-commutative 75.9%

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

      12. +-commutative 75.9%

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

      13. cube-unmult 75.9%

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

      14. +-commutative 75.9%

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

   3. Simplified 75.9%

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

      1. associate-*r/ 64.6%

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

      2. fma-udef 50.7%

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

      3. cube-mult 50.7%

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

      4. distribute-rgt1-in 64.6%

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

      5. associate-+r+ 64.6%

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

      6. *-commutative 64.6%

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

      7. *-commutative 64.6%

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

      8. frac-times 75.9%

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

      9. associate-/r* 99.7%

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

      10. clear-num 99.6%

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

      11. frac-times 99.7%

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

      12. *-un-lft-identity 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in y around 0 86.5%

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

      1. +-commutative 86.5%

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

   8. Simplified 86.5%

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

   9. Taylor expanded in x around 0 71.1%

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

   if 3.29999999999999987e-86 < y < 9.6e7

   1. Initial program 87.6%

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

      1. times-frac 99.1%

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

      2. associate-+l+ 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around 0 58.1%

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

      1. distribute-rgt-in 58.1%

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

      2. *-lft-identity 58.1%

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

   6. Simplified 58.1%

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

   7. Taylor expanded in y around 0 58.0%

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

      1. neg-mul-1 58.0%

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

      2. +-commutative 58.0%

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

      3. unsub-neg 58.0%

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

   9. Simplified 58.0%

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

   if 9.6e7 < y

   1. Initial program 58.6%

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

      1. associate-/r* 63.2%

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

      2. +-commutative 63.2%

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

      3. +-commutative 63.2%

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

      4. +-commutative 63.2%

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

      5. associate-/r* 58.6%

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

      6. associate-*l/ 71.7%

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

      7. *-commutative 71.7%

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

      8. *-commutative 71.7%

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

      9. distribute-rgt1-in 68.6%

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

      10. fma-def 71.7%

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

      11. +-commutative 71.7%

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

      12. +-commutative 71.7%

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

      13. cube-unmult 71.8%

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

      14. +-commutative 71.8%

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

   3. Simplified 71.8%

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

   4. Taylor expanded in y around inf 69.5%

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

      1. unpow2 69.5%

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

   6. Simplified 69.5%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-137}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-161}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 96000000:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 10: 73.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -7.1 \cdot 10^{-137}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.85 \cdot 10^{-161}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-87}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 96000000:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* x x))))
   (if (<= y -7.1e-137)
     t_0
     (if (<= y 2.85e-161)
       (/ y x)
       (if (<= y 4.4e-87)
         t_0
         (if (<= y 96000000.0) (- (/ x y) x) (/ (/ x y) y)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x * x);
	double tmp;
	if (y <= -7.1e-137) {
		tmp = t_0;
	} else if (y <= 2.85e-161) {
		tmp = y / x;
	} else if (y <= 4.4e-87) {
		tmp = t_0;
	} else if (y <= 96000000.0) {
		tmp = (x / y) - x;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (x * x)
    if (y <= (-7.1d-137)) then
        tmp = t_0
    else if (y <= 2.85d-161) then
        tmp = y / x
    else if (y <= 4.4d-87) then
        tmp = t_0
    else if (y <= 96000000.0d0) then
        tmp = (x / y) - x
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (x * x);
	double tmp;
	if (y <= -7.1e-137) {
		tmp = t_0;
	} else if (y <= 2.85e-161) {
		tmp = y / x;
	} else if (y <= 4.4e-87) {
		tmp = t_0;
	} else if (y <= 96000000.0) {
		tmp = (x / y) - x;
	} 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 <= -7.1e-137:
		tmp = t_0
	elif y <= 2.85e-161:
		tmp = y / x
	elif y <= 4.4e-87:
		tmp = t_0
	elif y <= 96000000.0:
		tmp = (x / y) - x
	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 <= -7.1e-137)
		tmp = t_0;
	elseif (y <= 2.85e-161)
		tmp = Float64(y / x);
	elseif (y <= 4.4e-87)
		tmp = t_0;
	elseif (y <= 96000000.0)
		tmp = Float64(Float64(x / y) - x);
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (x * x);
	tmp = 0.0;
	if (y <= -7.1e-137)
		tmp = t_0;
	elseif (y <= 2.85e-161)
		tmp = y / x;
	elseif (y <= 4.4e-87)
		tmp = t_0;
	elseif (y <= 96000000.0)
		tmp = (x / y) - x;
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.1e-137], t$95$0, If[LessEqual[y, 2.85e-161], N[(y / x), $MachinePrecision], If[LessEqual[y, 4.4e-87], t$95$0, If[LessEqual[y, 96000000.0], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -7.0999999999999997e-137 or 2.85000000000000011e-161 < y < 4.39999999999999976e-87

   1. Initial program 68.5%

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

      1. associate-/r* 75.2%

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

      2. +-commutative 75.2%

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

      3. +-commutative 75.2%

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

      4. +-commutative 75.2%

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

      5. associate-/r* 68.5%

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

      6. associate-*l/ 83.4%

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

      7. *-commutative 83.4%

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

      8. *-commutative 83.4%

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

      9. distribute-rgt1-in 52.5%

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

      10. fma-def 83.4%

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

      11. +-commutative 83.4%

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

      12. +-commutative 83.4%

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

      13. cube-unmult 83.4%

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

      14. +-commutative 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in x around inf 39.2%

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

      1. unpow2 39.2%

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

   6. Simplified 39.2%

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

   if -7.0999999999999997e-137 < y < 2.85000000000000011e-161

   1. Initial program 64.6%

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

      1. associate-/r* 64.5%

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

      2. +-commutative 64.5%

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

      3. +-commutative 64.5%

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

      4. +-commutative 64.5%

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

      5. associate-/r* 64.6%

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

      6. associate-*l/ 75.9%

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

      7. *-commutative 75.9%

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

      8. *-commutative 75.9%

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

      9. distribute-rgt1-in 62.0%

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

      10. fma-def 75.9%

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

      11. +-commutative 75.9%

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

      12. +-commutative 75.9%

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

      13. cube-unmult 75.9%

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

      14. +-commutative 75.9%

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

   3. Simplified 75.9%

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

      1. associate-*r/ 64.6%

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

      2. fma-udef 50.7%

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

      3. cube-mult 50.7%

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

      4. distribute-rgt1-in 64.6%

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

      5. associate-+r+ 64.6%

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

      6. *-commutative 64.6%

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

      7. *-commutative 64.6%

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

      8. frac-times 75.9%

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

      9. associate-/r* 99.7%

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

      10. clear-num 99.6%

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

      11. frac-times 99.7%

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

      12. *-un-lft-identity 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in y around 0 86.5%

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

      1. +-commutative 86.5%

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

   8. Simplified 86.5%

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

   9. Taylor expanded in x around 0 71.1%

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

   if 4.39999999999999976e-87 < y < 9.6e7

   1. Initial program 87.6%

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

      1. times-frac 99.1%

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

      2. associate-+l+ 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around 0 58.1%

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

      1. distribute-rgt-in 58.1%

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

      2. *-lft-identity 58.1%

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

   6. Simplified 58.1%

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

   7. Taylor expanded in y around 0 58.0%

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

      1. neg-mul-1 58.0%

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

      2. +-commutative 58.0%

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

      3. unsub-neg 58.0%

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

   9. Simplified 58.0%

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

   if 9.6e7 < y

   1. Initial program 58.6%

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

      1. associate-/r* 63.2%

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

      2. +-commutative 63.2%

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

      3. +-commutative 63.2%

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

      4. +-commutative 63.2%

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

      5. associate-/r* 58.6%

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

      6. associate-*l/ 71.7%

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

      7. *-commutative 71.7%

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

      8. *-commutative 71.7%

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

      9. distribute-rgt1-in 68.6%

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

      10. fma-def 71.7%

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

      11. +-commutative 71.7%

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

      12. +-commutative 71.7%

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

      13. cube-unmult 71.8%

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

      14. +-commutative 71.8%

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

   3. Simplified 71.8%

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

   4. Taylor expanded in y around inf 69.5%

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

      1. unpow2 69.5%

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

   6. Simplified 69.5%

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

      1. *-un-lft-identity 69.5%

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

      2. times-frac 75.9%

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

   8. Applied egg-rr 75.9%

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

      1. associate-*l/ 75.9%

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

      2. *-lft-identity 75.9%

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

   10. Simplified 75.9%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.1 \cdot 10^{-137}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 2.85 \cdot 10^{-161}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-87}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 96000000:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 11: 75.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-137}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{-161}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 96000000:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -7e-137)
   (/ (/ y x) x)
   (if (<= y 2.15e-161)
     (/ y x)
     (if (<= y 3.1e-86)
       (/ y (* x x))
       (if (<= y 96000000.0) (- (/ x y) x) (/ (/ x y) y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -7e-137) {
		tmp = (y / x) / x;
	} else if (y <= 2.15e-161) {
		tmp = y / x;
	} else if (y <= 3.1e-86) {
		tmp = y / (x * x);
	} else if (y <= 96000000.0) {
		tmp = (x / y) - x;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-7d-137)) then
        tmp = (y / x) / x
    else if (y <= 2.15d-161) then
        tmp = y / x
    else if (y <= 3.1d-86) then
        tmp = y / (x * x)
    else if (y <= 96000000.0d0) then
        tmp = (x / y) - x
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= -7e-137) {
		tmp = (y / x) / x;
	} else if (y <= 2.15e-161) {
		tmp = y / x;
	} else if (y <= 3.1e-86) {
		tmp = y / (x * x);
	} else if (y <= 96000000.0) {
		tmp = (x / y) - x;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -7e-137:
		tmp = (y / x) / x
	elif y <= 2.15e-161:
		tmp = y / x
	elif y <= 3.1e-86:
		tmp = y / (x * x)
	elif y <= 96000000.0:
		tmp = (x / y) - x
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -7e-137)
		tmp = Float64(Float64(y / x) / x);
	elseif (y <= 2.15e-161)
		tmp = Float64(y / x);
	elseif (y <= 3.1e-86)
		tmp = Float64(y / Float64(x * x));
	elseif (y <= 96000000.0)
		tmp = Float64(Float64(x / y) - x);
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7e-137)
		tmp = (y / x) / x;
	elseif (y <= 2.15e-161)
		tmp = y / x;
	elseif (y <= 3.1e-86)
		tmp = y / (x * x);
	elseif (y <= 96000000.0)
		tmp = (x / y) - x;
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -7e-137], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 2.15e-161], N[(y / x), $MachinePrecision], If[LessEqual[y, 3.1e-86], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 96000000.0], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -7.0000000000000002e-137

   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* 73.5%

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

      2. +-commutative 73.5%

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

      3. +-commutative 73.5%

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

      4. +-commutative 73.5%

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

      5. associate-/r* 66.0%

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

      6. associate-*l/ 81.6%

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

      7. *-commutative 81.6%

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

      8. *-commutative 81.6%

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

      9. distribute-rgt1-in 48.3%

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

      10. fma-def 81.6%

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

      11. +-commutative 81.6%

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

      12. +-commutative 81.6%

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

      13. cube-unmult 81.6%

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

      14. +-commutative 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in x around inf 37.3%

      \[\leadsto y \cdot \color{blue}{\frac{1}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 37.3%

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

   6. Simplified 37.3%

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

      1. un-div-inv 37.3%

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

      2. associate-/r* 34.8%

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

   8. Applied egg-rr 34.8%

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

   if -7.0000000000000002e-137 < y < 2.14999999999999983e-161

   1. Initial program 64.6%

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

      1. associate-/r* 64.5%

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

      2. +-commutative 64.5%

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

      3. +-commutative 64.5%

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

      4. +-commutative 64.5%

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

      5. associate-/r* 64.6%

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

      6. associate-*l/ 75.9%

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

      7. *-commutative 75.9%

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

      8. *-commutative 75.9%

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

      9. distribute-rgt1-in 62.0%

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

      10. fma-def 75.9%

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

      11. +-commutative 75.9%

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

      12. +-commutative 75.9%

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

      13. cube-unmult 75.9%

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

      14. +-commutative 75.9%

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

   3. Simplified 75.9%

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

      1. associate-*r/ 64.6%

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

      2. fma-udef 50.7%

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

      3. cube-mult 50.7%

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

      4. distribute-rgt1-in 64.6%

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

      5. associate-+r+ 64.6%

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

      6. *-commutative 64.6%

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

      7. *-commutative 64.6%

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

      8. frac-times 75.9%

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

      9. associate-/r* 99.7%

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

      10. clear-num 99.6%

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

      11. frac-times 99.7%

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

      12. *-un-lft-identity 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in y around 0 86.5%

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

      1. +-commutative 86.5%

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

   8. Simplified 86.5%

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

   9. Taylor expanded in x around 0 71.1%

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

   if 2.14999999999999983e-161 < y < 3.09999999999999989e-86

   1. Initial program 90.5%

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

      1. associate-/r* 90.7%

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

      2. +-commutative 90.7%

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

      3. +-commutative 90.7%

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

      4. +-commutative 90.7%

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

      5. associate-/r* 90.5%

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

      6. associate-*l/ 99.2%

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

      7. *-commutative 99.2%

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

      8. *-commutative 99.2%

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

      9. distribute-rgt1-in 90.0%

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

      10. fma-def 99.1%

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

      11. +-commutative 99.1%

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

      12. +-commutative 99.1%

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

      13. cube-unmult 99.2%

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

      14. +-commutative 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around inf 55.4%

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

      1. unpow2 55.4%

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

   6. Simplified 55.4%

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

   if 3.09999999999999989e-86 < y < 9.6e7

   1. Initial program 87.6%

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

      1. times-frac 99.1%

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

      2. associate-+l+ 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around 0 58.1%

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

      1. distribute-rgt-in 58.1%

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

      2. *-lft-identity 58.1%

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

   6. Simplified 58.1%

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

   7. Taylor expanded in y around 0 58.0%

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

      1. neg-mul-1 58.0%

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

      2. +-commutative 58.0%

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

      3. unsub-neg 58.0%

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

   9. Simplified 58.0%

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

   if 9.6e7 < y

   1. Initial program 58.6%

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

      1. associate-/r* 63.2%

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

      2. +-commutative 63.2%

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

      3. +-commutative 63.2%

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

      4. +-commutative 63.2%

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

      5. associate-/r* 58.6%

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

      6. associate-*l/ 71.7%

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

      7. *-commutative 71.7%

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

      8. *-commutative 71.7%

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

      9. distribute-rgt1-in 68.6%

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

      10. fma-def 71.7%

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

      11. +-commutative 71.7%

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

      12. +-commutative 71.7%

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

      13. cube-unmult 71.8%

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

      14. +-commutative 71.8%

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

   3. Simplified 71.8%

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

   4. Taylor expanded in y around inf 69.5%

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

      1. unpow2 69.5%

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

   6. Simplified 69.5%

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

      1. *-un-lft-identity 69.5%

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

      2. times-frac 75.9%

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

   8. Applied egg-rr 75.9%

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

      1. associate-*l/ 75.9%

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

      2. *-lft-identity 75.9%

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

   10. Simplified 75.9%

       \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-137}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{-161}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 96000000:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 12: 81.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ (/ y x) x)
   (if (<= x -7.5e-95)
     (- (/ y x) y)
     (if (<= x 3.3e+15) (/ x (* y (+ y 1.0))) (/ (/ x y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) / x;
	} else if (x <= -7.5e-95) {
		tmp = (y / x) - y;
	} else if (x <= 3.3e+15) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (y / x) / x
    else if (x <= (-7.5d-95)) then
        tmp = (y / x) - y
    else if (x <= 3.3d+15) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) / x;
	} else if (x <= -7.5e-95) {
		tmp = (y / x) - y;
	} else if (x <= 3.3e+15) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (y / x) / x
	elif x <= -7.5e-95:
		tmp = (y / x) - y
	elif x <= 3.3e+15:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -7.5e-95)
		tmp = Float64(Float64(y / x) - y);
	elseif (x <= 3.3e+15)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (y / x) / x;
	elseif (x <= -7.5e-95)
		tmp = (y / x) - y;
	elseif (x <= 3.3e+15)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -7.5e-95], N[(N[(y / x), $MachinePrecision] - y), $MachinePrecision], If[LessEqual[x, 3.3e+15], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1

   1. Initial program 65.0%

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

      1. associate-/r* 68.0%

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

      2. +-commutative 68.0%

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

      3. +-commutative 68.0%

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

      4. +-commutative 68.0%

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

      5. associate-/r* 65.0%

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

      6. associate-*l/ 82.6%

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

      7. *-commutative 82.6%

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

      8. *-commutative 82.6%

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

      9. distribute-rgt1-in 36.2%

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

      10. fma-def 82.5%

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

      11. +-commutative 82.5%

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

      12. +-commutative 82.5%

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

      13. cube-unmult 82.6%

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

      14. +-commutative 82.6%

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

   3. Simplified 82.6%

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

   4. Taylor expanded in x around inf 77.4%

      \[\leadsto y \cdot \color{blue}{\frac{1}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 77.4%

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

   6. Simplified 77.4%

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

      1. un-div-inv 77.6%

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

      2. associate-/r* 77.8%

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

   8. Applied egg-rr 77.8%

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

   if -1 < x < -7.5000000000000003e-95

   1. Initial program 88.3%

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

      1. associate-/r* 93.9%

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

      2. +-commutative 93.9%

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

      3. +-commutative 93.9%

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

      4. +-commutative 93.9%

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

      5. associate-/r* 88.3%

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

      6. associate-*l/ 94.1%

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

      7. *-commutative 94.1%

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

      8. *-commutative 94.1%

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

      9. distribute-rgt1-in 81.4%

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

      10. fma-def 94.0%

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

      11. +-commutative 94.0%

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

      12. +-commutative 94.0%

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

      13. cube-unmult 94.0%

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

      14. +-commutative 94.0%

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

   3. Simplified 94.0%

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

      1. associate-*r/ 88.2%

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

      2. fma-udef 75.7%

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

      3. cube-mult 75.7%

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

      4. distribute-rgt1-in 88.3%

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

      5. associate-+r+ 88.3%

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

      6. *-commutative 88.3%

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

      7. *-commutative 88.3%

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

      8. frac-times 99.5%

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

      9. associate-/r* 99.6%

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

      10. clear-num 99.6%

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

      11. frac-times 99.8%

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

      12. *-un-lft-identity 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around 0 45.4%

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

      1. +-commutative 45.4%

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

   8. Simplified 45.4%

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

   9. Taylor expanded in x around 0 33.1%

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

       1. neg-mul-1 33.1%

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

       2. +-commutative 33.1%

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

       3. unsub-neg 33.1%

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

   11. Simplified 33.1%

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

   if -7.5000000000000003e-95 < x < 3.3e15

   1. Initial program 76.0%

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

      1. times-frac 86.7%

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

      2. associate-+l+ 86.7%

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

   3. Simplified 86.7%

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

   4. Taylor expanded in x around 0 77.0%

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

      1. +-commutative 77.0%

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

   6. Simplified 77.0%

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

   if 3.3e15 < x

   1. Initial program 44.6%

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

      1. associate-/r* 54.0%

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

      2. +-commutative 54.0%

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

      3. +-commutative 54.0%

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

      4. +-commutative 54.0%

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

      5. associate-/r* 44.6%

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

      6. associate-*l/ 72.1%

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

      7. *-commutative 72.1%

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

      8. *-commutative 72.1%

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

      9. distribute-rgt1-in 68.3%

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

      10. fma-def 72.1%

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

      11. +-commutative 72.1%

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

      12. +-commutative 72.1%

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

      13. cube-unmult 72.1%

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

      14. +-commutative 72.1%

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

   3. Simplified 72.1%

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

   4. Taylor expanded in y around inf 28.5%

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

      1. unpow2 28.5%

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

   6. Simplified 28.5%

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

      1. *-un-lft-identity 28.5%

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

      2. times-frac 32.5%

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

   8. Applied egg-rr 32.5%

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

      1. associate-*l/ 32.5%

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

      2. *-lft-identity 32.5%

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

   10. Simplified 32.5%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 13: 81.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ (/ y x) x)
   (if (<= x -7.5e-95)
     (- (/ y x) y)
     (if (<= x 2.05e+18) (/ x (+ y (* y y))) (/ (/ x y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) / x;
	} else if (x <= -7.5e-95) {
		tmp = (y / x) - y;
	} else if (x <= 2.05e+18) {
		tmp = x / (y + (y * y));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (y / x) / x
    else if (x <= (-7.5d-95)) then
        tmp = (y / x) - y
    else if (x <= 2.05d+18) then
        tmp = x / (y + (y * y))
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) / x;
	} else if (x <= -7.5e-95) {
		tmp = (y / x) - y;
	} else if (x <= 2.05e+18) {
		tmp = x / (y + (y * y));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (y / x) / x
	elif x <= -7.5e-95:
		tmp = (y / x) - y
	elif x <= 2.05e+18:
		tmp = x / (y + (y * y))
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -7.5e-95)
		tmp = Float64(Float64(y / x) - y);
	elseif (x <= 2.05e+18)
		tmp = Float64(x / Float64(y + Float64(y * y)));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (y / x) / x;
	elseif (x <= -7.5e-95)
		tmp = (y / x) - y;
	elseif (x <= 2.05e+18)
		tmp = x / (y + (y * y));
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -7.5e-95], N[(N[(y / x), $MachinePrecision] - y), $MachinePrecision], If[LessEqual[x, 2.05e+18], N[(x / N[(y + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1

   1. Initial program 65.0%

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

      1. associate-/r* 68.0%

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

      2. +-commutative 68.0%

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

      3. +-commutative 68.0%

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

      4. +-commutative 68.0%

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

      5. associate-/r* 65.0%

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

      6. associate-*l/ 82.6%

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

      7. *-commutative 82.6%

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

      8. *-commutative 82.6%

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

      9. distribute-rgt1-in 36.2%

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

      10. fma-def 82.5%

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

      11. +-commutative 82.5%

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

      12. +-commutative 82.5%

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

      13. cube-unmult 82.6%

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

      14. +-commutative 82.6%

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

   3. Simplified 82.6%

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

   4. Taylor expanded in x around inf 77.4%

      \[\leadsto y \cdot \color{blue}{\frac{1}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 77.4%

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

   6. Simplified 77.4%

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

      1. un-div-inv 77.6%

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

      2. associate-/r* 77.8%

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

   8. Applied egg-rr 77.8%

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

   if -1 < x < -7.5000000000000003e-95

   1. Initial program 88.3%

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

      1. associate-/r* 93.9%

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

      2. +-commutative 93.9%

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

      3. +-commutative 93.9%

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

      4. +-commutative 93.9%

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

      5. associate-/r* 88.3%

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

      6. associate-*l/ 94.1%

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

      7. *-commutative 94.1%

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

      8. *-commutative 94.1%

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

      9. distribute-rgt1-in 81.4%

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

      10. fma-def 94.0%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 94.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 94.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 94.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 94.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 94.0%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 88.2%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 75.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 75.7%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 88.3%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. associate-+r+ 88.3%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 88.3%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. *-commutative 88.3%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      8. frac-times 99.5%

         \[\leadsto \color{blue}{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      9. associate-/r* 99.6%

         \[\leadsto \frac{y}{x + \left(y + 1\right)} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      10. clear-num 99.6%

          \[\leadsto \color{blue}{\frac{1}{\frac{x + \left(y + 1\right)}{y}}} \cdot \frac{\frac{x}{x + y}}{x + y} \]

      11. frac-times 99.8%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}} \]

      12. *-un-lft-identity 99.8%

          \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}} \]

   6. Taylor expanded in y around 0 45.4%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 45.4%

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   8. Simplified 45.4%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]

   9. Taylor expanded in x around 0 33.1%

      \[\leadsto \color{blue}{-1 \cdot y + \frac{y}{x}} \]
   10. Step-by-step derivation

       1. neg-mul-1 33.1%

          \[\leadsto \color{blue}{\left(-y\right)} + \frac{y}{x} \]

       2. +-commutative 33.1%

          \[\leadsto \color{blue}{\frac{y}{x} + \left(-y\right)} \]

       3. unsub-neg 33.1%

          \[\leadsto \color{blue}{\frac{y}{x} - y} \]

   11. Simplified 33.1%

       \[\leadsto \color{blue}{\frac{y}{x} - y} \]

   if -7.5000000000000003e-95 < x < 2.05e18

   1. Initial program 76.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 86.7%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+l+ 86.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 86.7%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around 0 77.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. distribute-rgt-in 77.0%

         \[\leadsto \frac{x}{\color{blue}{1 \cdot y + y \cdot y}} \]

      2. *-lft-identity 77.0%

         \[\leadsto \frac{x}{\color{blue}{y} + y \cdot y} \]

   6. Simplified 77.0%

      \[\leadsto \color{blue}{\frac{x}{y + y \cdot y}} \]

   if 2.05e18 < x

   1. Initial program 44.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 54.0%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. +-commutative 54.0%

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 54.0%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      4. +-commutative 54.0%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

      5. associate-/r* 44.6%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. associate-*l/ 72.1%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

      7. *-commutative 72.1%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 72.1%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 68.3%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 72.1%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 72.1%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 72.1%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 72.1%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 72.1%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 72.1%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   4. Taylor expanded in y around inf 28.5%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 28.5%

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   6. Simplified 28.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 28.5%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{y \cdot y} \]

      2. times-frac 32.5%

         \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y}} \]

   8. Applied egg-rr 32.5%

      \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y}} \]
   9. Step-by-step derivation

      1. associate-*l/ 32.5%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y}}{y}} \]

      2. *-lft-identity 32.5%

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y} \]

   10. Simplified 32.5%

       \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 14: 82.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-84}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+29}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -2.8e+118)
   (/ (/ y x) x)
   (if (<= y 3.2e-84)
     (/ y (* x (+ x 1.0)))
     (if (<= y 4e+29) (/ x (+ y (* y y))) (/ (/ x y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -2.8e+118) {
		tmp = (y / x) / x;
	} else if (y <= 3.2e-84) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 4e+29) {
		tmp = x / (y + (y * y));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.8d+118)) then
        tmp = (y / x) / x
    else if (y <= 3.2d-84) then
        tmp = y / (x * (x + 1.0d0))
    else if (y <= 4d+29) then
        tmp = x / (y + (y * y))
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= -2.8e+118) {
		tmp = (y / x) / x;
	} else if (y <= 3.2e-84) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 4e+29) {
		tmp = x / (y + (y * y));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -2.8e+118:
		tmp = (y / x) / x
	elif y <= 3.2e-84:
		tmp = y / (x * (x + 1.0))
	elif y <= 4e+29:
		tmp = x / (y + (y * y))
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -2.8e+118)
		tmp = Float64(Float64(y / x) / x);
	elseif (y <= 3.2e-84)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (y <= 4e+29)
		tmp = Float64(x / Float64(y + Float64(y * y)));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.8e+118)
		tmp = (y / x) / x;
	elseif (y <= 3.2e-84)
		tmp = y / (x * (x + 1.0));
	elseif (y <= 4e+29)
		tmp = x / (y + (y * y));
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -2.8e+118], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 3.2e-84], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4e+29], N[(x / N[(y + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.79999999999999986e118

   1. Initial program 42.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 53.7%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. +-commutative 53.7%

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 53.7%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      4. +-commutative 53.7%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

      5. associate-/r* 42.9%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. associate-*l/ 70.9%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

      7. *-commutative 70.9%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 70.9%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 15.9%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 70.9%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 70.9%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 70.9%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 70.9%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 70.9%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 70.9%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   4. Taylor expanded in x around inf 27.1%

      \[\leadsto y \cdot \color{blue}{\frac{1}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 27.1%

         \[\leadsto y \cdot \frac{1}{\color{blue}{x \cdot x}} \]

   6. Simplified 27.1%

      \[\leadsto y \cdot \color{blue}{\frac{1}{x \cdot x}} \]
   7. Step-by-step derivation

      1. un-div-inv 27.1%

         \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

      2. associate-/r* 21.2%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   8. Applied egg-rr 21.2%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   if -2.79999999999999986e118 < y < 3.1999999999999999e-84

   1. Initial program 74.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 87.8%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+l+ 87.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 87.8%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 70.1%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   if 3.1999999999999999e-84 < y < 3.99999999999999966e29

   1. Initial program 86.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.1%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+l+ 99.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around 0 57.2%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. distribute-rgt-in 57.3%

         \[\leadsto \frac{x}{\color{blue}{1 \cdot y + y \cdot y}} \]

      2. *-lft-identity 57.3%

         \[\leadsto \frac{x}{\color{blue}{y} + y \cdot y} \]

   6. Simplified 57.3%

      \[\leadsto \color{blue}{\frac{x}{y + y \cdot y}} \]

   if 3.99999999999999966e29 < y

   1. Initial program 57.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 60.6%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. +-commutative 60.6%

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 60.6%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      4. +-commutative 60.6%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

      5. associate-/r* 57.3%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. associate-*l/ 72.0%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

      7. *-commutative 72.0%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 72.0%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 68.6%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 72.0%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 72.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 72.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 72.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 72.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 72.0%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   4. Taylor expanded in y around inf 71.5%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 71.5%

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   6. Simplified 71.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 71.5%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{y \cdot y} \]

      2. times-frac 78.3%

         \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y}} \]

   8. Applied egg-rr 78.3%

      \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y}} \]
   9. Step-by-step derivation

      1. associate-*l/ 78.4%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y}}{y}} \]

      2. *-lft-identity 78.4%

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y} \]

   10. Simplified 78.4%

       \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-84}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+29}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 15: 82.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-84}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+28}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -3.1e+119)
   (/ (/ y x) x)
   (if (<= y 3.2e-84)
     (/ y (* x (+ x 1.0)))
     (if (<= y 5.8e+28) (/ x (+ y (* y y))) (/ (/ x (+ x y)) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -3.1e+119) {
		tmp = (y / x) / x;
	} else if (y <= 3.2e-84) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 5.8e+28) {
		tmp = x / (y + (y * y));
	} else {
		tmp = (x / (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 <= (-3.1d+119)) then
        tmp = (y / x) / x
    else if (y <= 3.2d-84) then
        tmp = y / (x * (x + 1.0d0))
    else if (y <= 5.8d+28) then
        tmp = x / (y + (y * y))
    else
        tmp = (x / (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 <= -3.1e+119) {
		tmp = (y / x) / x;
	} else if (y <= 3.2e-84) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 5.8e+28) {
		tmp = x / (y + (y * y));
	} else {
		tmp = (x / (x + y)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -3.1e+119:
		tmp = (y / x) / x
	elif y <= 3.2e-84:
		tmp = y / (x * (x + 1.0))
	elif y <= 5.8e+28:
		tmp = x / (y + (y * y))
	else:
		tmp = (x / (x + y)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -3.1e+119)
		tmp = Float64(Float64(y / x) / x);
	elseif (y <= 3.2e-84)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (y <= 5.8e+28)
		tmp = Float64(x / Float64(y + Float64(y * y)));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.1e+119)
		tmp = (y / x) / x;
	elseif (y <= 3.2e-84)
		tmp = y / (x * (x + 1.0));
	elseif (y <= 5.8e+28)
		tmp = x / (y + (y * y));
	else
		tmp = (x / (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, -3.1e+119], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 3.2e-84], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.8e+28], N[(x / N[(y + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.09999999999999995e119

   1. Initial program 42.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 53.7%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. +-commutative 53.7%

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 53.7%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      4. +-commutative 53.7%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

      5. associate-/r* 42.9%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. associate-*l/ 70.9%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

      7. *-commutative 70.9%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 70.9%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 15.9%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 70.9%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 70.9%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 70.9%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 70.9%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 70.9%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 70.9%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   4. Taylor expanded in x around inf 27.1%

      \[\leadsto y \cdot \color{blue}{\frac{1}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 27.1%

         \[\leadsto y \cdot \frac{1}{\color{blue}{x \cdot x}} \]

   6. Simplified 27.1%

      \[\leadsto y \cdot \color{blue}{\frac{1}{x \cdot x}} \]
   7. Step-by-step derivation

      1. un-div-inv 27.1%

         \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

      2. associate-/r* 21.2%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   8. Applied egg-rr 21.2%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   if -3.09999999999999995e119 < y < 3.1999999999999999e-84

   1. Initial program 74.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 87.8%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+l+ 87.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 87.8%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 70.1%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   if 3.1999999999999999e-84 < y < 5.8000000000000002e28

   1. Initial program 86.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.1%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+l+ 99.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around 0 57.2%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. distribute-rgt-in 57.3%

         \[\leadsto \frac{x}{\color{blue}{1 \cdot y + y \cdot y}} \]

      2. *-lft-identity 57.3%

         \[\leadsto \frac{x}{\color{blue}{y} + y \cdot y} \]

   6. Simplified 57.3%

      \[\leadsto \color{blue}{\frac{x}{y + y \cdot y}} \]

   if 5.8000000000000002e28 < y

   1. Initial program 57.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 60.6%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. +-commutative 60.6%

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 60.6%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      4. +-commutative 60.6%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

      5. associate-/r* 57.3%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. associate-*l/ 72.0%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

      7. *-commutative 72.0%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 72.0%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 68.6%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 72.0%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 72.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 72.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 72.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 72.0%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 72.0%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 57.3%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 56.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 56.2%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 57.3%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. associate-+r+ 57.3%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 57.3%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. *-commutative 57.3%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      8. frac-times 78.7%

         \[\leadsto \color{blue}{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      9. associate-/r* 99.9%

         \[\leadsto \frac{y}{x + \left(y + 1\right)} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      10. clear-num 99.9%

          \[\leadsto \color{blue}{\frac{1}{\frac{x + \left(y + 1\right)}{y}}} \cdot \frac{\frac{x}{x + y}}{x + y} \]

      11. frac-times 99.2%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}} \]

      12. *-un-lft-identity 99.2%

          \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)} \]

   5. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}} \]

   6. Taylor expanded in y around inf 78.7%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-84}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+28}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y}\\ \end{array} \]

Alternative 16: 65.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;y \leq -2.1 \cdot 10^{+141}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-153}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 96000000:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

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 y))))
   (if (<= y -2.1e+141)
     t_0
     (if (<= y 1.75e-153) (/ y x) (if (<= y 96000000.0) (- (/ x y) x) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (y <= -2.1e+141) {
		tmp = t_0;
	} else if (y <= 1.75e-153) {
		tmp = y / x;
	} else if (y <= 96000000.0) {
		tmp = (x / y) - x;
	} else {
		tmp = t_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 * y)
    if (y <= (-2.1d+141)) then
        tmp = t_0
    else if (y <= 1.75d-153) then
        tmp = y / x
    else if (y <= 96000000.0d0) then
        tmp = (x / y) - x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (y <= -2.1e+141) {
		tmp = t_0;
	} else if (y <= 1.75e-153) {
		tmp = y / x;
	} else if (y <= 96000000.0) {
		tmp = (x / y) - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y * y)
	tmp = 0
	if y <= -2.1e+141:
		tmp = t_0
	elif y <= 1.75e-153:
		tmp = y / x
	elif y <= 96000000.0:
		tmp = (x / y) - x
	else:
		tmp = t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y * y))
	tmp = 0.0
	if (y <= -2.1e+141)
		tmp = t_0;
	elseif (y <= 1.75e-153)
		tmp = Float64(y / x);
	elseif (y <= 96000000.0)
		tmp = Float64(Float64(x / y) - x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y * y);
	tmp = 0.0;
	if (y <= -2.1e+141)
		tmp = t_0;
	elseif (y <= 1.75e-153)
		tmp = y / x;
	elseif (y <= 96000000.0)
		tmp = (x / y) - x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.1e+141], t$95$0, If[LessEqual[y, 1.75e-153], N[(y / x), $MachinePrecision], If[LessEqual[y, 96000000.0], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.0999999999999998e141 or 9.6e7 < y

   1. Initial program 51.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 58.8%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. +-commutative 58.8%

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 58.8%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      4. +-commutative 58.8%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

      5. associate-/r* 51.5%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. associate-*l/ 71.1%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

      7. *-commutative 71.1%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 71.1%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 45.8%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 71.1%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 71.1%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 71.1%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 71.1%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 71.1%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 71.1%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   4. Taylor expanded in y around inf 72.7%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 72.7%

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   6. Simplified 72.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]

   if -2.0999999999999998e141 < y < 1.7499999999999999e-153

   1. Initial program 73.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 75.9%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. +-commutative 75.9%

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 75.9%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      4. +-commutative 75.9%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

      5. associate-/r* 73.7%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. associate-*l/ 82.5%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

      7. *-commutative 82.5%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 82.5%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 67.6%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 82.5%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 82.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 82.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 82.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 82.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 82.5%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 73.7%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 59.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 59.7%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 73.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. associate-+r+ 73.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 73.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. *-commutative 73.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      8. frac-times 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. clear-num 99.6%

          \[\leadsto \color{blue}{\frac{1}{\frac{x + \left(y + 1\right)}{y}}} \cdot \frac{\frac{x}{x + y}}{x + y} \]

      11. frac-times 99.7%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}} \]

      12. *-un-lft-identity 99.7%

          \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)} \]

   5. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}} \]

   6. Taylor expanded in y around 0 68.3%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 68.3%

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   8. Simplified 68.3%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]

   9. Taylor expanded in x around 0 44.3%

      \[\leadsto \color{blue}{\frac{y}{x}} \]

   if 1.7499999999999999e-153 < y < 9.6e7

   1. Initial program 88.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.1%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+l+ 99.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around 0 45.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. distribute-rgt-in 45.7%

         \[\leadsto \frac{x}{\color{blue}{1 \cdot y + y \cdot y}} \]

      2. *-lft-identity 45.7%

         \[\leadsto \frac{x}{\color{blue}{y} + y \cdot y} \]

   6. Simplified 45.7%

      \[\leadsto \color{blue}{\frac{x}{y + y \cdot y}} \]

   7. Taylor expanded in y around 0 45.6%

      \[\leadsto \color{blue}{-1 \cdot x + \frac{x}{y}} \]
   8. Step-by-step derivation

      1. neg-mul-1 45.6%

         \[\leadsto \color{blue}{\left(-x\right)} + \frac{x}{y} \]

      2. +-commutative 45.6%

         \[\leadsto \color{blue}{\frac{x}{y} + \left(-x\right)} \]

      3. unsub-neg 45.6%

         \[\leadsto \color{blue}{\frac{x}{y} - x} \]

   9. Simplified 45.6%

      \[\leadsto \color{blue}{\frac{x}{y} - x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+141}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-153}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 96000000:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 17: 82.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-84}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -2.8e+118)
   (/ (/ y x) x)
   (if (<= y 3.2e-84) (/ y (* x (+ x 1.0))) (/ (/ x (+ y 1.0)) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -2.8e+118) {
		tmp = (y / x) / x;
	} else if (y <= 3.2e-84) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.8d+118)) then
        tmp = (y / x) / x
    else if (y <= 3.2d-84) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= -2.8e+118) {
		tmp = (y / x) / x;
	} else if (y <= 3.2e-84) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -2.8e+118:
		tmp = (y / x) / x
	elif y <= 3.2e-84:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -2.8e+118)
		tmp = Float64(Float64(y / x) / x);
	elseif (y <= 3.2e-84)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.8e+118)
		tmp = (y / x) / x;
	elseif (y <= 3.2e-84)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -2.8e+118], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 3.2e-84], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.79999999999999986e118

   1. Initial program 42.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 53.7%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. +-commutative 53.7%

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 53.7%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      4. +-commutative 53.7%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

      5. associate-/r* 42.9%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. associate-*l/ 70.9%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

      7. *-commutative 70.9%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 70.9%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 15.9%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 70.9%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 70.9%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 70.9%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 70.9%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 70.9%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 70.9%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   4. Taylor expanded in x around inf 27.1%

      \[\leadsto y \cdot \color{blue}{\frac{1}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 27.1%

         \[\leadsto y \cdot \frac{1}{\color{blue}{x \cdot x}} \]

   6. Simplified 27.1%

      \[\leadsto y \cdot \color{blue}{\frac{1}{x \cdot x}} \]
   7. Step-by-step derivation

      1. un-div-inv 27.1%

         \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

      2. associate-/r* 21.2%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   8. Applied egg-rr 21.2%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   if -2.79999999999999986e118 < y < 3.1999999999999999e-84

   1. Initial program 74.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 87.8%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+l+ 87.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 87.8%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 70.1%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   if 3.1999999999999999e-84 < y

   1. Initial program 66.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 71.1%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. +-commutative 71.1%

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 71.1%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      4. +-commutative 71.1%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

      5. associate-/l/ 66.7%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. times-frac 85.5%

         \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) + 1} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}} \]

      7. *-commutative 85.5%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 85.5%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 85.5%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. +-commutative 85.5%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ 85.5%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 85.5%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around inf 71.7%

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{x + \left(y + 1\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 71.7%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + \left(y + 1\right)}}{y}} \]

      2. *-un-lft-identity 71.7%

         \[\leadsto \frac{\color{blue}{\frac{x}{x + \left(y + 1\right)}}}{y} \]

   6. Applied egg-rr 71.7%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + \left(y + 1\right)}}{y}} \]

   7. Taylor expanded in x around 0 71.4%

      \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{y} \]
   8. Step-by-step derivation

      1. +-commutative 71.4%

         \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{y} \]

   9. Simplified 71.4%

      \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-84}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]

Alternative 18: 82.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.1e-84) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ x y)) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-84) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (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 <= 3.1d-84) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (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 <= 3.1e-84) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.1e-84:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (x + y)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.1e-84)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / 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 <= 3.1e-84)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (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, 3.1e-84], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.10000000000000002e-84

   1. Initial program 67.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 86.3%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. 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 y around 0 59.4%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 58.0%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 58.0%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   6. Simplified 58.0%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if 3.10000000000000002e-84 < y

   1. Initial program 66.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 71.1%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. +-commutative 71.1%

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 71.1%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      4. +-commutative 71.1%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

      5. associate-/r* 66.7%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. associate-*l/ 78.4%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

      7. *-commutative 78.4%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 78.4%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 74.9%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 78.4%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 78.4%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 78.4%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 78.4%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 78.4%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 78.4%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 66.7%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 64.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 64.8%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 66.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. associate-+r+ 66.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 66.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. *-commutative 66.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      8. frac-times 85.4%

         \[\leadsto \color{blue}{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      9. associate-/r* 99.8%

         \[\leadsto \frac{y}{x + \left(y + 1\right)} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      10. clear-num 99.8%

          \[\leadsto \color{blue}{\frac{1}{\frac{x + \left(y + 1\right)}{y}}} \cdot \frac{\frac{x}{x + y}}{x + y} \]

      11. frac-times 99.3%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}} \]

      12. *-un-lft-identity 99.3%

          \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)} \]

   5. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}} \]

   6. Taylor expanded in x around 0 71.8%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{1 + y}} \]
   7. Step-by-step derivation

      1. +-commutative 71.8%

         \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{y + 1}} \]

   8. Simplified 71.8%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{y + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]

Alternative 19: 82.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + \left(y + 1\right)}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4.8e-86) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ x (+ y 1.0))) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.8e-86) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (x + (y + 1.0))) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.8d-86) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (x + (y + 1.0d0))) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 4.8e-86) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (x + (y + 1.0))) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 4.8e-86:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (x + (y + 1.0))) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4.8e-86)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(x + Float64(y + 1.0))) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.8e-86)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (x + (y + 1.0))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 4.8e-86], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.80000000000000026e-86

   1. Initial program 67.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 86.3%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. 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 y around 0 59.4%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 58.0%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 58.0%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   6. Simplified 58.0%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if 4.80000000000000026e-86 < y

   1. Initial program 66.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 71.1%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. +-commutative 71.1%

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 71.1%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      4. +-commutative 71.1%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

      5. associate-/l/ 66.7%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. times-frac 85.5%

         \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) + 1} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}} \]

      7. *-commutative 85.5%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 85.5%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 85.5%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. +-commutative 85.5%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ 85.5%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 85.5%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around inf 71.7%

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{x + \left(y + 1\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 71.7%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + \left(y + 1\right)}}{y}} \]

      2. *-un-lft-identity 71.7%

         \[\leadsto \frac{\color{blue}{\frac{x}{x + \left(y + 1\right)}}}{y} \]

   6. Applied egg-rr 71.7%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + \left(y + 1\right)}}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + \left(y + 1\right)}}{y}\\ \end{array} \]

Alternative 20: 82.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.2e-84) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ y 1.0)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-84) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.2d-84) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-84) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.2e-84:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.2e-84)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.2e-84)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.2e-84], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.1999999999999999e-84

   1. Initial program 67.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 86.3%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. 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 y around 0 59.4%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 58.0%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 58.0%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   6. Simplified 58.0%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if 3.1999999999999999e-84 < y

   1. Initial program 66.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 71.1%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. +-commutative 71.1%

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 71.1%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      4. +-commutative 71.1%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

      5. associate-/l/ 66.7%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. times-frac 85.5%

         \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) + 1} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}} \]

      7. *-commutative 85.5%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 85.5%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 85.5%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. +-commutative 85.5%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ 85.5%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 85.5%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around inf 71.7%

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{x + \left(y + 1\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 71.7%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + \left(y + 1\right)}}{y}} \]

      2. *-un-lft-identity 71.7%

         \[\leadsto \frac{\color{blue}{\frac{x}{x + \left(y + 1\right)}}}{y} \]

   6. Applied egg-rr 71.7%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + \left(y + 1\right)}}{y}} \]

   7. Taylor expanded in x around 0 71.4%

      \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{y} \]
   8. Step-by-step derivation

      1. +-commutative 71.4%

         \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{y} \]

   9. Simplified 71.4%

      \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]

Alternative 21: 44.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{-153}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 1.75e-153) (/ y x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.75e-153) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.75d-153) then
        tmp = y / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.75e-153) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.75e-153:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.75e-153)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.75e-153)
		tmp = y / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.75e-153], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.7499999999999999e-153

   1. Initial program 65.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 69.9%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. +-commutative 69.9%

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 69.9%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      4. +-commutative 69.9%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

      5. associate-/r* 65.4%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. associate-*l/ 79.3%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

      7. *-commutative 79.3%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 79.3%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 53.7%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 79.3%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 79.3%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 79.3%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 79.3%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 79.3%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 79.3%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 65.4%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 45.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 45.1%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 65.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. associate-+r+ 65.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 65.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. *-commutative 65.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      8. frac-times 85.5%

         \[\leadsto \color{blue}{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      9. associate-/r* 99.8%

         \[\leadsto \frac{y}{x + \left(y + 1\right)} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      10. clear-num 99.7%

          \[\leadsto \color{blue}{\frac{1}{\frac{x + \left(y + 1\right)}{y}}} \cdot \frac{\frac{x}{x + y}}{x + y} \]

      11. frac-times 99.4%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}} \]

      12. *-un-lft-identity 99.4%

          \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)} \]

   5. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}} \]

   6. Taylor expanded in y around 0 57.8%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 57.8%

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   8. Simplified 57.8%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]

   9. Taylor expanded in x around 0 33.5%

      \[\leadsto \color{blue}{\frac{y}{x}} \]

   if 1.7499999999999999e-153 < 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. times-frac 87.1%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+l+ 87.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 87.1%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around 0 61.2%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. distribute-rgt-in 61.2%

         \[\leadsto \frac{x}{\color{blue}{1 \cdot y + y \cdot y}} \]

      2. *-lft-identity 61.2%

         \[\leadsto \frac{x}{\color{blue}{y} + y \cdot y} \]

   6. Simplified 61.2%

      \[\leadsto \color{blue}{\frac{x}{y + y \cdot y}} \]

   7. Taylor expanded in y around 0 35.7%

      \[\leadsto \frac{x}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{-153}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 22: 4.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{0.5}{x} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ 0.5 x))

C

assert(x < y);
double code(double x, double y) {
	return 0.5 / 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 = 0.5d0 / x
end function

Java

assert x < y;
public static double code(double x, double y) {
	return 0.5 / x;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return 0.5 / x

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(0.5 / x)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 0.5 / x;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(0.5 / x), $MachinePrecision]

Derivation

1. Initial program 66.9%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 71.2%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

   2. +-commutative 71.2%

      \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

   3. +-commutative 71.2%

      \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

   4. +-commutative 71.2%

      \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

   5. associate-/r* 66.9%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

   6. associate-*l/ 79.9%

      \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

   7. *-commutative 79.9%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

   8. *-commutative 79.9%

      \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   9. distribute-rgt1-in 62.1%

      \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   10. fma-def 79.9%

       \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

   11. +-commutative 79.9%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   12. +-commutative 79.9%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   13. cube-unmult 79.9%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

   14. +-commutative 79.9%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

3. Simplified 79.9%

   \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ 66.9%

      \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   2. fma-udef 53.0%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   3. cube-mult 53.0%

      \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   4. distribute-rgt1-in 66.9%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   5. associate-+r+ 66.9%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

   6. *-commutative 66.9%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

   7. *-commutative 66.9%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   8. frac-times 86.0%

      \[\leadsto \color{blue}{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   9. associate-/r* 99.8%

      \[\leadsto \frac{y}{x + \left(y + 1\right)} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

   10. clear-num 99.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{x + \left(y + 1\right)}{y}}} \cdot \frac{\frac{x}{x + y}}{x + y} \]

   11. frac-times 99.4%

       \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}} \]

   12. *-un-lft-identity 99.4%

       \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)} \]

5. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}} \]

6. Taylor expanded in y around -inf 54.2%

   \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{y + -1 \cdot \left(-1 \cdot x + -1 \cdot \left(1 + x\right)\right)}} \]
7. Step-by-step derivation

   1. mul-1-neg 54.2%

      \[\leadsto \frac{\frac{x}{x + y}}{y + \color{blue}{\left(-\left(-1 \cdot x + -1 \cdot \left(1 + x\right)\right)\right)}} \]

   2. unsub-neg 54.2%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{y - \left(-1 \cdot x + -1 \cdot \left(1 + x\right)\right)}} \]

   3. neg-mul-1 54.2%

      \[\leadsto \frac{\frac{x}{x + y}}{y - \left(\color{blue}{\left(-x\right)} + -1 \cdot \left(1 + x\right)\right)} \]

   4. +-commutative 54.2%

      \[\leadsto \frac{\frac{x}{x + y}}{y - \color{blue}{\left(-1 \cdot \left(1 + x\right) + \left(-x\right)\right)}} \]

   5. unsub-neg 54.2%

      \[\leadsto \frac{\frac{x}{x + y}}{y - \color{blue}{\left(-1 \cdot \left(1 + x\right) - x\right)}} \]

   6. distribute-lft-in 54.2%

      \[\leadsto \frac{\frac{x}{x + y}}{y - \left(\color{blue}{\left(-1 \cdot 1 + -1 \cdot x\right)} - x\right)} \]

   7. metadata-eval 54.2%

      \[\leadsto \frac{\frac{x}{x + y}}{y - \left(\left(\color{blue}{-1} + -1 \cdot x\right) - x\right)} \]

   8. neg-mul-1 54.2%

      \[\leadsto \frac{\frac{x}{x + y}}{y - \left(\left(-1 + \color{blue}{\left(-x\right)}\right) - x\right)} \]

   9. unsub-neg 54.2%

      \[\leadsto \frac{\frac{x}{x + y}}{y - \left(\color{blue}{\left(-1 - x\right)} - x\right)} \]

8. Simplified 54.2%

   \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{y - \left(\left(-1 - x\right) - x\right)}} \]

9. Taylor expanded in x around inf 4.2%

   \[\leadsto \color{blue}{\frac{0.5}{x}} \]

10. Final simplification 4.2%

    \[\leadsto \frac{0.5}{x} \]

Alternative 23: 26.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{x}{y} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ x y))

C

assert(x < y);
double code(double x, double y) {
	return x / y;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / y
end function

Java

assert x < y;
public static double code(double x, double y) {
	return x / y;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return x / y

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x / y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x / y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 66.9%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. times-frac 86.0%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. associate-+l+ 86.0%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

3. Simplified 86.0%

   \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

4. Taylor expanded in x around 0 50.7%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
5. Step-by-step derivation

   1. distribute-rgt-in 50.7%

      \[\leadsto \frac{x}{\color{blue}{1 \cdot y + y \cdot y}} \]

   2. *-lft-identity 50.7%

      \[\leadsto \frac{x}{\color{blue}{y} + y \cdot y} \]

6. Simplified 50.7%

   \[\leadsto \color{blue}{\frac{x}{y + y \cdot y}} \]

7. Taylor expanded in y around 0 29.8%

   \[\leadsto \frac{x}{\color{blue}{y}} \]

8. Final simplification 29.8%

   \[\leadsto \frac{x}{y} \]

Developer target: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x)))))

C

double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x / ((y + 1.0d0) + x)) / (y + x)) / (1.0d0 / (y / (y + x)))
end function

Java

public static double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Python

def code(x, y):
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)))

Julia

function code(x, y)
	return Float64(Float64(Float64(x / Float64(Float64(y + 1.0) + x)) / Float64(y + x)) / Float64(1.0 / Float64(y / Float64(y + x))))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
end

Wolfram

code[x_, y_] := N[(N[(N[(x / N[(N[(y + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023290 
(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))))