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

Percentage Accurate: 69.7% → 99.8%

Time: 18.3s

Alternatives: 23

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.1%

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

   1. associate-+r+ 67.1%

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

   2. *-commutative 67.1%

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

   3. frac-times 88.2%

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

   4. associate-*l/ 81.8%

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

   5. times-frac 99.8%

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

   6. associate-+r+ 99.8%

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

   7. +-commutative 99.8%

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

   8. associate-+l+ 99.8%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 83.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{y + x}\\ \mathbf{if}\;y \leq 3.9 \cdot 10^{-200}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-95}:\\ \;\;\;\;\frac{t_0}{x + 1}\\ \mathbf{elif}\;y \leq 36000000000:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y + x} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y x))))
   (if (<= y 3.9e-200)
     (/ (/ y (+ x 1.0)) x)
     (if (<= y 1.15e-134)
       (/ (/ x (+ y (+ x 1.0))) (+ y x))
       (if (<= y 5e-95)
         (/ t_0 (+ x 1.0))
         (if (<= y 36000000000.0)
           (/ x (* (+ y x) (+ y 1.0)))
           (* (/ t_0 (+ y x)) (/ x y))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (y <= 3.9e-200) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 1.15e-134) {
		tmp = (x / (y + (x + 1.0))) / (y + x);
	} else if (y <= 5e-95) {
		tmp = t_0 / (x + 1.0);
	} else if (y <= 36000000000.0) {
		tmp = x / ((y + x) * (y + 1.0));
	} else {
		tmp = (t_0 / (y + x)) * (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) :: t_0
    real(8) :: tmp
    t_0 = y / (y + x)
    if (y <= 3.9d-200) then
        tmp = (y / (x + 1.0d0)) / x
    else if (y <= 1.15d-134) then
        tmp = (x / (y + (x + 1.0d0))) / (y + x)
    else if (y <= 5d-95) then
        tmp = t_0 / (x + 1.0d0)
    else if (y <= 36000000000.0d0) then
        tmp = x / ((y + x) * (y + 1.0d0))
    else
        tmp = (t_0 / (y + x)) * (x / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (y <= 3.9e-200) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 1.15e-134) {
		tmp = (x / (y + (x + 1.0))) / (y + x);
	} else if (y <= 5e-95) {
		tmp = t_0 / (x + 1.0);
	} else if (y <= 36000000000.0) {
		tmp = x / ((y + x) * (y + 1.0));
	} else {
		tmp = (t_0 / (y + x)) * (x / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (y + x)
	tmp = 0
	if y <= 3.9e-200:
		tmp = (y / (x + 1.0)) / x
	elif y <= 1.15e-134:
		tmp = (x / (y + (x + 1.0))) / (y + x)
	elif y <= 5e-95:
		tmp = t_0 / (x + 1.0)
	elif y <= 36000000000.0:
		tmp = x / ((y + x) * (y + 1.0))
	else:
		tmp = (t_0 / (y + x)) * (x / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(y + x))
	tmp = 0.0
	if (y <= 3.9e-200)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	elseif (y <= 1.15e-134)
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) / Float64(y + x));
	elseif (y <= 5e-95)
		tmp = Float64(t_0 / Float64(x + 1.0));
	elseif (y <= 36000000000.0)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(t_0 / Float64(y + x)) * Float64(x / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (y + x);
	tmp = 0.0;
	if (y <= 3.9e-200)
		tmp = (y / (x + 1.0)) / x;
	elseif (y <= 1.15e-134)
		tmp = (x / (y + (x + 1.0))) / (y + x);
	elseif (y <= 5e-95)
		tmp = t_0 / (x + 1.0);
	elseif (y <= 36000000000.0)
		tmp = x / ((y + x) * (y + 1.0));
	else
		tmp = (t_0 / (y + x)) * (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_] := Block[{t$95$0 = N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 3.9e-200], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 1.15e-134], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e-95], N[(t$95$0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 36000000000.0], N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < 3.89999999999999999e-200

   1. Initial program 65.9%

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

      1. associate-*l* 65.9%

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

      2. +-commutative 65.9%

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

      3. +-commutative 65.9%

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

      4. +-commutative 65.9%

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

      5. associate-*l* 65.9%

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

      6. *-commutative 65.9%

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

      7. times-frac 84.3%

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

      8. +-commutative 84.3%

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

      9. +-commutative 84.3%

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

      10. +-commutative 84.3%

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

      11. associate-+l+ 84.3%

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

   3. Simplified 84.3%

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

   4. Taylor expanded in y around 0 56.4%

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

      1. +-commutative 56.4%

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

   6. Simplified 56.4%

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

      1. *-un-lft-identity 56.4%

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

      2. times-frac 59.2%

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

   8. Applied egg-rr 59.2%

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

      1. associate-*l/ 59.3%

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

      2. *-lft-identity 59.3%

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

   10. Simplified 59.3%

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

   if 3.89999999999999999e-200 < y < 1.15e-134

   1. Initial program 70.0%

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

      1. associate-+r+ 70.0%

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

      2. *-commutative 70.0%

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

      3. frac-times 92.1%

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

      4. associate-*l/ 70.0%

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

      5. times-frac 100.0%

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

      6. associate-+r+ 100.0%

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

      7. +-commutative 100.0%

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

      8. associate-+l+ 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in y around inf 34.4%

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

   if 1.15e-134 < y < 4.9999999999999998e-95

   1. Initial program 86.1%

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

      1. associate-+r+ 86.1%

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

      2. *-commutative 86.1%

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

      3. frac-times 99.8%

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

      4. associate-*l/ 86.1%

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

      5. times-frac 100.0%

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

      6. associate-+r+ 100.0%

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

      7. +-commutative 100.0%

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

      8. associate-+l+ 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. clear-num 100.0%

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

      2. un-div-inv 100.0%

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

      3. +-commutative 100.0%

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

      4. div-inv 99.8%

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

      5. clear-num 99.8%

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

      6. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around 0 77.7%

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

      1. +-commutative 77.7%

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

   8. Simplified 77.7%

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

   if 4.9999999999999998e-95 < y < 3.6e10

   1. Initial program 87.3%

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

      1. associate-*l* 87.3%

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

      2. +-commutative 87.3%

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

      3. +-commutative 87.3%

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

      4. +-commutative 87.3%

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

      5. associate-*l* 87.3%

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

      6. associate-*l/ 94.4%

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

      7. *-commutative 94.4%

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

      8. *-commutative 94.4%

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

      9. distribute-rgt1-in 90.6%

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

      10. fma-def 94.4%

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

      11. +-commutative 94.4%

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

      12. +-commutative 94.4%

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

      13. cube-unmult 94.5%

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

      14. +-commutative 94.5%

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

   3. Simplified 94.5%

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

      1. fma-udef 90.7%

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

      2. cube-mult 90.6%

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

      3. distribute-rgt1-in 94.4%

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

      4. associate-+r+ 94.4%

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

      5. *-commutative 94.4%

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

      6. associate-*l* 94.4%

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

      7. associate-+r+ 94.4%

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

      8. +-commutative 94.4%

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

      9. associate-+l+ 94.4%

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

   5. Applied egg-rr 94.4%

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

   6. Taylor expanded in x around 0 38.2%

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

      1. expm1-log1p-u 38.2%

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

      2. expm1-udef 5.4%

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

      3. associate-/r* 5.4%

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

      4. distribute-lft-in 5.4%

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

      5. *-rgt-identity 5.4%

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

      6. pow2 5.4%

         \[\leadsto e^{\mathsf{log1p}\left(y \cdot \frac{\frac{x}{x + y}}{y + \color{blue}{{y}^{2}}}\right)} - 1 \]

   8. Applied egg-rr 5.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(y \cdot \frac{\frac{x}{x + y}}{y + {y}^{2}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 38.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \frac{\frac{x}{x + y}}{y + {y}^{2}}\right)\right)} \]

      2. expm1-log1p 38.5%

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

      3. associate-*r/ 38.5%

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

      4. associate-*r/ 31.2%

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

      5. associate-/l/ 31.1%

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

      6. +-commutative 31.1%

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

      7. unpow2 31.1%

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

      8. *-rgt-identity 31.1%

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

      9. distribute-lft-in 31.1%

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

      10. associate-*r* 31.1%

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

      11. times-frac 38.5%

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

      12. *-inverses 38.5%

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

   10. Simplified 38.5%

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

   if 3.6e10 < y

   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. associate-*l* 59.4%

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

      2. +-commutative 59.4%

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

      3. +-commutative 59.4%

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

      4. +-commutative 59.4%

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

      5. associate-*l* 59.4%

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

      6. *-commutative 59.4%

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

      7. times-frac 90.0%

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

      8. +-commutative 90.0%

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

      9. +-commutative 90.0%

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

      10. +-commutative 90.0%

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

      11. associate-+l+ 90.0%

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

   3. Simplified 90.0%

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

   4. Taylor expanded in y around inf 78.4%

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

      1. associate-/r* 82.0%

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

      2. div-inv 82.0%

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

   6. Applied egg-rr 82.0%

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

      1. associate-*r/ 82.0%

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

      2. *-rgt-identity 82.0%

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

   8. Simplified 82.0%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{-200}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{elif}\;y \leq 36000000000:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + x} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 3: 97.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{\frac{x}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{1}{\frac{y + \left(x + 1\right)}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2e-21)
   (* (/ y (+ y x)) (/ (/ x (+ x 1.0)) (+ y x)))
   (if (<= y 7.8e+154)
     (* (/ y (* (+ y x) (+ y x))) (/ x (+ x (+ y 1.0))))
     (* (/ 1.0 y) (/ 1.0 (/ (+ y (+ x 1.0)) x))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2e-21) {
		tmp = (y / (y + x)) * ((x / (x + 1.0)) / (y + x));
	} else if (y <= 7.8e+154) {
		tmp = (y / ((y + x) * (y + x))) * (x / (x + (y + 1.0)));
	} else {
		tmp = (1.0 / y) * (1.0 / ((y + (x + 1.0)) / x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2d-21) then
        tmp = (y / (y + x)) * ((x / (x + 1.0d0)) / (y + x))
    else if (y <= 7.8d+154) then
        tmp = (y / ((y + x) * (y + x))) * (x / (x + (y + 1.0d0)))
    else
        tmp = (1.0d0 / y) * (1.0d0 / ((y + (x + 1.0d0)) / x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2e-21) {
		tmp = (y / (y + x)) * ((x / (x + 1.0)) / (y + x));
	} else if (y <= 7.8e+154) {
		tmp = (y / ((y + x) * (y + x))) * (x / (x + (y + 1.0)));
	} else {
		tmp = (1.0 / y) * (1.0 / ((y + (x + 1.0)) / x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2e-21:
		tmp = (y / (y + x)) * ((x / (x + 1.0)) / (y + x))
	elif y <= 7.8e+154:
		tmp = (y / ((y + x) * (y + x))) * (x / (x + (y + 1.0)))
	else:
		tmp = (1.0 / y) * (1.0 / ((y + (x + 1.0)) / x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2e-21)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(Float64(x / Float64(x + 1.0)) / Float64(y + x)));
	elseif (y <= 7.8e+154)
		tmp = Float64(Float64(y / Float64(Float64(y + x) * Float64(y + x))) * Float64(x / Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(1.0 / y) * Float64(1.0 / Float64(Float64(y + Float64(x + 1.0)) / x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2e-21)
		tmp = (y / (y + x)) * ((x / (x + 1.0)) / (y + x));
	elseif (y <= 7.8e+154)
		tmp = (y / ((y + x) * (y + x))) * (x / (x + (y + 1.0)));
	else
		tmp = (1.0 / y) * (1.0 / ((y + (x + 1.0)) / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 2e-21], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.8e+154], N[(N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(1.0 / N[(N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.99999999999999982e-21

   1. Initial program 69.6%

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

      1. associate-+r+ 69.6%

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

      2. *-commutative 69.6%

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

      3. frac-times 87.2%

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

      4. associate-*l/ 78.2%

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

      5. times-frac 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in y around 0 85.3%

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

      1. +-commutative 85.3%

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

   6. Simplified 85.3%

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

   if 1.99999999999999982e-21 < y < 7.8000000000000006e154

   1. Initial program 62.0%

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

      1. associate-*l* 62.0%

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

      2. +-commutative 62.0%

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

      3. +-commutative 62.0%

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

      4. +-commutative 62.0%

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

      5. associate-*l* 62.0%

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

      6. *-commutative 62.0%

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

      7. times-frac 93.1%

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

      8. +-commutative 93.1%

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

      9. +-commutative 93.1%

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

      10. +-commutative 93.1%

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

      11. associate-+l+ 93.1%

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

   3. Simplified 93.1%

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

   if 7.8000000000000006e154 < y

   1. Initial program 59.8%

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

      1. associate-*l* 59.8%

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

      2. +-commutative 59.8%

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

      3. +-commutative 59.8%

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

      4. +-commutative 59.8%

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

      5. associate-*l* 59.8%

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

      6. *-commutative 59.8%

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

      7. times-frac 88.2%

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

      8. +-commutative 88.2%

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

      9. +-commutative 88.2%

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

      10. +-commutative 88.2%

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

      11. associate-+l+ 88.2%

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

   3. Simplified 88.2%

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

   4. Taylor expanded in y around inf 96.0%

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

      1. clear-num 96.1%

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

      2. inv-pow 96.1%

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

      3. associate-+r+ 96.1%

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

      4. +-commutative 96.1%

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

      5. associate-+l+ 96.1%

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

   6. Applied egg-rr 96.1%

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

      1. unpow-1 96.1%

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

   8. Simplified 96.1%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{\frac{x}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{1}{\frac{y + \left(x + 1\right)}{x}}\\ \end{array} \]

Alternative 4: 97.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ t_1 := \frac{y}{y + x}\\ \mathbf{if}\;y \leq -3.05 \cdot 10^{+26}:\\ \;\;\;\;\frac{t_1}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+154}:\\ \;\;\;\;t_1 \cdot \frac{x}{\left(y + x\right) \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{1}{\frac{t_0}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))) (t_1 (/ y (+ y x))))
   (if (<= y -3.05e+26)
     (/ t_1 (+ x (+ y (+ y 1.0))))
     (if (<= y 7.2e+154)
       (* t_1 (/ x (* (+ y x) t_0)))
       (* (/ 1.0 y) (/ 1.0 (/ t_0 x)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double t_1 = y / (y + x);
	double tmp;
	if (y <= -3.05e+26) {
		tmp = t_1 / (x + (y + (y + 1.0)));
	} else if (y <= 7.2e+154) {
		tmp = t_1 * (x / ((y + x) * t_0));
	} else {
		tmp = (1.0 / y) * (1.0 / (t_0 / x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    t_1 = y / (y + x)
    if (y <= (-3.05d+26)) then
        tmp = t_1 / (x + (y + (y + 1.0d0)))
    else if (y <= 7.2d+154) then
        tmp = t_1 * (x / ((y + x) * t_0))
    else
        tmp = (1.0d0 / y) * (1.0d0 / (t_0 / x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double t_1 = y / (y + x);
	double tmp;
	if (y <= -3.05e+26) {
		tmp = t_1 / (x + (y + (y + 1.0)));
	} else if (y <= 7.2e+154) {
		tmp = t_1 * (x / ((y + x) * t_0));
	} else {
		tmp = (1.0 / y) * (1.0 / (t_0 / x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	t_1 = y / (y + x)
	tmp = 0
	if y <= -3.05e+26:
		tmp = t_1 / (x + (y + (y + 1.0)))
	elif y <= 7.2e+154:
		tmp = t_1 * (x / ((y + x) * t_0))
	else:
		tmp = (1.0 / y) * (1.0 / (t_0 / x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	t_1 = Float64(y / Float64(y + x))
	tmp = 0.0
	if (y <= -3.05e+26)
		tmp = Float64(t_1 / Float64(x + Float64(y + Float64(y + 1.0))));
	elseif (y <= 7.2e+154)
		tmp = Float64(t_1 * Float64(x / Float64(Float64(y + x) * t_0)));
	else
		tmp = Float64(Float64(1.0 / y) * Float64(1.0 / Float64(t_0 / x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	t_1 = y / (y + x);
	tmp = 0.0;
	if (y <= -3.05e+26)
		tmp = t_1 / (x + (y + (y + 1.0)));
	elseif (y <= 7.2e+154)
		tmp = t_1 * (x / ((y + x) * t_0));
	else
		tmp = (1.0 / y) * (1.0 / (t_0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.05e+26], N[(t$95$1 / N[(x + N[(y + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.2e+154], N[(t$95$1 * N[(x / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(1.0 / N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.0500000000000001e26

   1. Initial program 59.6%

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

      1. associate-+r+ 59.6%

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

      2. *-commutative 59.6%

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

      3. frac-times 87.8%

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

      4. associate-*l/ 87.8%

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

      5. times-frac 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. clear-num 99.8%

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

      2. un-div-inv 99.8%

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

      3. +-commutative 99.8%

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

      4. div-inv 99.8%

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

      5. clear-num 99.8%

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

      6. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around -inf 41.2%

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

      1. neg-mul-1 41.2%

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

      2. distribute-lft-in 41.2%

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

      3. mul-1-neg 41.2%

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

      4. remove-double-neg 41.2%

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

      5. distribute-lft-in 41.2%

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

      6. metadata-eval 41.2%

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

      7. neg-mul-1 41.2%

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

      8. distribute-lft-in 41.2%

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

      9. metadata-eval 41.2%

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

      10. mul-1-neg 41.2%

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

      11. remove-double-neg 41.2%

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

      12. +-commutative 41.2%

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

   8. Simplified 41.2%

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

   if -3.0500000000000001e26 < y < 7.2000000000000001e154

   1. Initial program 71.0%

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

      1. associate-+r+ 71.0%

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

      2. *-commutative 71.0%

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

      3. associate-*l* 71.0%

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

      4. times-frac 98.4%

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

      5. associate-+r+ 98.4%

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

      6. +-commutative 98.4%

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

      7. associate-+l+ 98.4%

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

   3. Applied egg-rr 98.4%

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

   if 7.2000000000000001e154 < y

   1. Initial program 59.8%

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

      1. associate-*l* 59.8%

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

      2. +-commutative 59.8%

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

      3. +-commutative 59.8%

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

      4. +-commutative 59.8%

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

      5. associate-*l* 59.8%

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

      6. *-commutative 59.8%

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

      7. times-frac 88.2%

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

      8. +-commutative 88.2%

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

      9. +-commutative 88.2%

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

      10. +-commutative 88.2%

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

      11. associate-+l+ 88.2%

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

   3. Simplified 88.2%

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

   4. Taylor expanded in y around inf 96.0%

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

      1. clear-num 96.1%

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

      2. inv-pow 96.1%

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

      3. associate-+r+ 96.1%

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

      4. +-commutative 96.1%

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

      5. associate-+l+ 96.1%

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

   6. Applied egg-rr 96.1%

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

      1. unpow-1 96.1%

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

   8. Simplified 96.1%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.05 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{1}{\frac{y + \left(x + 1\right)}{x}}\\ \end{array} \]

Alternative 5: 92.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{y + x}\\ \mathbf{if}\;y \leq 5.9 \cdot 10^{-161}:\\ \;\;\;\;\frac{t_0}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;y \leq 180000000:\\ \;\;\;\;\frac{x}{x + 1} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y + x} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y x))))
   (if (<= y 5.9e-161)
     (/ t_0 (+ x (+ y (+ y 1.0))))
     (if (<= y 180000000.0)
       (* (/ x (+ x 1.0)) (/ y (* (+ y x) (+ y x))))
       (* (/ t_0 (+ y x)) (/ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (y <= 5.9e-161) {
		tmp = t_0 / (x + (y + (y + 1.0)));
	} else if (y <= 180000000.0) {
		tmp = (x / (x + 1.0)) * (y / ((y + x) * (y + x)));
	} else {
		tmp = (t_0 / (y + x)) * (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) :: t_0
    real(8) :: tmp
    t_0 = y / (y + x)
    if (y <= 5.9d-161) then
        tmp = t_0 / (x + (y + (y + 1.0d0)))
    else if (y <= 180000000.0d0) then
        tmp = (x / (x + 1.0d0)) * (y / ((y + x) * (y + x)))
    else
        tmp = (t_0 / (y + x)) * (x / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (y <= 5.9e-161) {
		tmp = t_0 / (x + (y + (y + 1.0)));
	} else if (y <= 180000000.0) {
		tmp = (x / (x + 1.0)) * (y / ((y + x) * (y + x)));
	} else {
		tmp = (t_0 / (y + x)) * (x / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (y + x)
	tmp = 0
	if y <= 5.9e-161:
		tmp = t_0 / (x + (y + (y + 1.0)))
	elif y <= 180000000.0:
		tmp = (x / (x + 1.0)) * (y / ((y + x) * (y + x)))
	else:
		tmp = (t_0 / (y + x)) * (x / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(y + x))
	tmp = 0.0
	if (y <= 5.9e-161)
		tmp = Float64(t_0 / Float64(x + Float64(y + Float64(y + 1.0))));
	elseif (y <= 180000000.0)
		tmp = Float64(Float64(x / Float64(x + 1.0)) * Float64(y / Float64(Float64(y + x) * Float64(y + x))));
	else
		tmp = Float64(Float64(t_0 / Float64(y + x)) * Float64(x / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (y + x);
	tmp = 0.0;
	if (y <= 5.9e-161)
		tmp = t_0 / (x + (y + (y + 1.0)));
	elseif (y <= 180000000.0)
		tmp = (x / (x + 1.0)) * (y / ((y + x) * (y + x)));
	else
		tmp = (t_0 / (y + x)) * (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_] := Block[{t$95$0 = N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 5.9e-161], N[(t$95$0 / N[(x + N[(y + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 180000000.0], N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.9000000000000002e-161

   1. Initial program 66.8%

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

      1. associate-+r+ 66.8%

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

      2. *-commutative 66.8%

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

      3. frac-times 84.3%

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

      4. associate-*l/ 77.2%

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

      5. times-frac 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. clear-num 99.7%

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

      2. un-div-inv 99.8%

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

      3. +-commutative 99.8%

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

      4. div-inv 99.7%

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

      5. clear-num 99.7%

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

      6. +-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x around -inf 61.5%

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

      1. neg-mul-1 61.5%

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

      2. distribute-lft-in 61.5%

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

      3. mul-1-neg 61.5%

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

      4. remove-double-neg 61.5%

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

      5. distribute-lft-in 61.5%

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

      6. metadata-eval 61.5%

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

      7. neg-mul-1 61.5%

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

      8. distribute-lft-in 61.5%

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

      9. metadata-eval 61.5%

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

      10. mul-1-neg 61.5%

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

      11. remove-double-neg 61.5%

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

      12. +-commutative 61.5%

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

   8. Simplified 61.5%

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

   if 5.9000000000000002e-161 < y < 1.8e8

   1. Initial program 83.6%

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

      1. associate-*l* 83.6%

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

      2. +-commutative 83.6%

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

      3. +-commutative 83.6%

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

      4. +-commutative 83.6%

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

      5. associate-*l* 83.6%

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

      6. *-commutative 83.6%

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

      7. times-frac 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 98.5%

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

      1. +-commutative 98.5%

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

   6. Simplified 98.5%

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

   if 1.8e8 < y

   1. Initial program 58.7%

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

      1. associate-*l* 58.7%

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

      2. +-commutative 58.7%

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

      3. +-commutative 58.7%

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

      4. +-commutative 58.7%

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

      5. associate-*l* 58.7%

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

      6. *-commutative 58.7%

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

      7. times-frac 90.1%

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

      8. +-commutative 90.1%

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

      9. +-commutative 90.1%

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

      10. +-commutative 90.1%

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

      11. associate-+l+ 90.1%

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

   3. Simplified 90.1%

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

   4. Taylor expanded in y around inf 77.3%

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

      1. associate-/r* 80.9%

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

      2. div-inv 80.9%

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

   6. Applied egg-rr 80.9%

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

      1. associate-*r/ 80.9%

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

      2. *-rgt-identity 80.9%

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

   8. Simplified 80.9%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.9 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;y \leq 180000000:\\ \;\;\;\;\frac{x}{x + 1} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + x} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 6: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-200}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-134}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-95}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* x (+ x 1.0)))))
   (if (<= y -3.2e+26)
     (* (/ y x) (/ 1.0 x))
     (if (<= y 3.9e-200)
       t_0
       (if (<= y 1.05e-134)
         (/ x y)
         (if (<= y 5e-95)
           t_0
           (if (<= y 7.2e+154)
             (/ x (* y (+ y 1.0)))
             (* (/ 1.0 y) (/ x y)))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x * (x + 1.0));
	double tmp;
	if (y <= -3.2e+26) {
		tmp = (y / x) * (1.0 / x);
	} else if (y <= 3.9e-200) {
		tmp = t_0;
	} else if (y <= 1.05e-134) {
		tmp = x / y;
	} else if (y <= 5e-95) {
		tmp = t_0;
	} else if (y <= 7.2e+154) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (1.0 / y) * (x / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (x * (x + 1.0d0))
    if (y <= (-3.2d+26)) then
        tmp = (y / x) * (1.0d0 / x)
    else if (y <= 3.9d-200) then
        tmp = t_0
    else if (y <= 1.05d-134) then
        tmp = x / y
    else if (y <= 5d-95) then
        tmp = t_0
    else if (y <= 7.2d+154) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (1.0d0 / y) * (x / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (x * (x + 1.0));
	double tmp;
	if (y <= -3.2e+26) {
		tmp = (y / x) * (1.0 / x);
	} else if (y <= 3.9e-200) {
		tmp = t_0;
	} else if (y <= 1.05e-134) {
		tmp = x / y;
	} else if (y <= 5e-95) {
		tmp = t_0;
	} else if (y <= 7.2e+154) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (1.0 / y) * (x / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (x * (x + 1.0))
	tmp = 0
	if y <= -3.2e+26:
		tmp = (y / x) * (1.0 / x)
	elif y <= 3.9e-200:
		tmp = t_0
	elif y <= 1.05e-134:
		tmp = x / y
	elif y <= 5e-95:
		tmp = t_0
	elif y <= 7.2e+154:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (1.0 / y) * (x / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(x * Float64(x + 1.0)))
	tmp = 0.0
	if (y <= -3.2e+26)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	elseif (y <= 3.9e-200)
		tmp = t_0;
	elseif (y <= 1.05e-134)
		tmp = Float64(x / y);
	elseif (y <= 5e-95)
		tmp = t_0;
	elseif (y <= 7.2e+154)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(1.0 / y) * Float64(x / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (x * (x + 1.0));
	tmp = 0.0;
	if (y <= -3.2e+26)
		tmp = (y / x) * (1.0 / x);
	elseif (y <= 3.9e-200)
		tmp = t_0;
	elseif (y <= 1.05e-134)
		tmp = x / y;
	elseif (y <= 5e-95)
		tmp = t_0;
	elseif (y <= 7.2e+154)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (1.0 / y) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.2e+26], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.9e-200], t$95$0, If[LessEqual[y, 1.05e-134], N[(x / y), $MachinePrecision], If[LessEqual[y, 5e-95], t$95$0, If[LessEqual[y, 7.2e+154], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -3.20000000000000029e26

   1. Initial program 60.6%

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

      1. associate-+r+ 60.6%

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

      2. *-commutative 60.6%

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

      3. frac-times 87.6%

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

      4. associate-*l/ 87.6%

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

      5. times-frac 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around inf 37.8%

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

   5. Taylor expanded in y around 0 37.1%

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

   if -3.20000000000000029e26 < y < 3.89999999999999999e-200 or 1.05e-134 < y < 4.9999999999999998e-95

   1. Initial program 70.0%

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

      1. associate-*l* 70.1%

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

      2. +-commutative 70.1%

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

      3. +-commutative 70.1%

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

      4. +-commutative 70.1%

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

      5. associate-*l* 70.0%

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

      6. *-commutative 70.0%

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

      7. times-frac 83.7%

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

      8. +-commutative 83.7%

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

      9. +-commutative 83.7%

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

      10. +-commutative 83.7%

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

      11. associate-+l+ 83.7%

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

   3. Simplified 83.7%

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

   4. Taylor expanded in y around 0 71.9%

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

      1. +-commutative 71.9%

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

   6. Simplified 71.9%

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

   if 3.89999999999999999e-200 < y < 1.05e-134

   1. Initial program 70.0%

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

      1. associate-*l* 70.0%

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

      2. +-commutative 70.0%

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

      3. +-commutative 70.0%

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

      4. +-commutative 70.0%

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

      5. associate-*l* 70.0%

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

      6. *-commutative 70.0%

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

      7. times-frac 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%

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

   4. Taylor expanded in x around 0 32.4%

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

      1. +-commutative 32.4%

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

   6. Simplified 32.4%

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

   7. Taylor expanded in y around 0 32.4%

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

   if 4.9999999999999998e-95 < y < 7.2000000000000001e154

   1. Initial program 71.7%

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

      1. associate-*l* 71.7%

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

      2. +-commutative 71.7%

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

      3. +-commutative 71.7%

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

      4. +-commutative 71.7%

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

      5. associate-*l* 71.7%

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

      6. *-commutative 71.7%

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

      7. times-frac 95.4%

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

      8. +-commutative 95.4%

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

      9. +-commutative 95.4%

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

      10. +-commutative 95.4%

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

      11. associate-+l+ 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in x around 0 47.6%

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

      1. +-commutative 47.6%

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

   6. Simplified 47.6%

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

   if 7.2000000000000001e154 < y

   1. Initial program 59.8%

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

      1. associate-*l* 59.8%

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

      2. +-commutative 59.8%

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

      3. +-commutative 59.8%

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

      4. +-commutative 59.8%

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

      5. associate-*l* 59.8%

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

      6. *-commutative 59.8%

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

      7. times-frac 88.2%

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

      8. +-commutative 88.2%

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

      9. +-commutative 88.2%

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

      10. +-commutative 88.2%

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

      11. associate-+l+ 88.2%

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

   3. Simplified 88.2%

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

   4. Taylor expanded in y around inf 96.0%

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

   5. Taylor expanded in y around inf 96.0%

      \[\leadsto \frac{1}{y} \cdot \color{blue}{\frac{x}{y}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-200}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-134}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 7: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{-200}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-134}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.9e-200)
   (/ (/ y (+ x 1.0)) x)
   (if (<= y 1.05e-134)
     (/ x y)
     (if (<= y 2.6e-95)
       (/ y (* x (+ x 1.0)))
       (if (<= y 7.2e+154)
         (/ x (* y (+ y 1.0)))
         (* (/ x y) (/ 1.0 (+ y x))))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.9e-200) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 1.05e-134) {
		tmp = x / y;
	} else if (y <= 2.6e-95) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 7.2e+154) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / (y + x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.9d-200) then
        tmp = (y / (x + 1.0d0)) / x
    else if (y <= 1.05d-134) then
        tmp = x / y
    else if (y <= 2.6d-95) then
        tmp = y / (x * (x + 1.0d0))
    else if (y <= 7.2d+154) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) * (1.0d0 / (y + x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.9e-200) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 1.05e-134) {
		tmp = x / y;
	} else if (y <= 2.6e-95) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 7.2e+154) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.9e-200:
		tmp = (y / (x + 1.0)) / x
	elif y <= 1.05e-134:
		tmp = x / y
	elif y <= 2.6e-95:
		tmp = y / (x * (x + 1.0))
	elif y <= 7.2e+154:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) * (1.0 / (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.9e-200)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	elseif (y <= 1.05e-134)
		tmp = Float64(x / y);
	elseif (y <= 2.6e-95)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (y <= 7.2e+154)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.9e-200)
		tmp = (y / (x + 1.0)) / x;
	elseif (y <= 1.05e-134)
		tmp = x / y;
	elseif (y <= 2.6e-95)
		tmp = y / (x * (x + 1.0));
	elseif (y <= 7.2e+154)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) * (1.0 / (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.9e-200], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 1.05e-134], N[(x / y), $MachinePrecision], If[LessEqual[y, 2.6e-95], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.2e+154], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < 3.89999999999999999e-200

   1. Initial program 65.9%

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

      1. associate-*l* 65.9%

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

      2. +-commutative 65.9%

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

      3. +-commutative 65.9%

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

      4. +-commutative 65.9%

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

      5. associate-*l* 65.9%

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

      6. *-commutative 65.9%

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

      7. times-frac 84.3%

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

      8. +-commutative 84.3%

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

      9. +-commutative 84.3%

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

      10. +-commutative 84.3%

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

      11. associate-+l+ 84.3%

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

   3. Simplified 84.3%

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

   4. Taylor expanded in y around 0 56.4%

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

      1. +-commutative 56.4%

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

   6. Simplified 56.4%

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

      1. *-un-lft-identity 56.4%

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

      2. times-frac 59.2%

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

   8. Applied egg-rr 59.2%

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

      1. associate-*l/ 59.3%

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

      2. *-lft-identity 59.3%

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

   10. Simplified 59.3%

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

   if 3.89999999999999999e-200 < y < 1.05e-134

   1. Initial program 70.0%

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

      1. associate-*l* 70.0%

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

      2. +-commutative 70.0%

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

      3. +-commutative 70.0%

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

      4. +-commutative 70.0%

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

      5. associate-*l* 70.0%

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

      6. *-commutative 70.0%

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

      7. times-frac 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%

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

   4. Taylor expanded in x around 0 32.4%

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

      1. +-commutative 32.4%

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

   6. Simplified 32.4%

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

   7. Taylor expanded in y around 0 32.4%

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

   if 1.05e-134 < y < 2.60000000000000001e-95

   1. Initial program 86.1%

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

      1. associate-*l* 86.1%

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

      2. +-commutative 86.1%

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

      3. +-commutative 86.1%

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

      4. +-commutative 86.1%

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

      5. associate-*l* 86.1%

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

      6. *-commutative 86.1%

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

      7. times-frac 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 77.2%

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

      1. +-commutative 77.2%

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

   6. Simplified 77.2%

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

   if 2.60000000000000001e-95 < y < 7.2000000000000001e154

   1. Initial program 71.7%

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

      1. associate-*l* 71.7%

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

      2. +-commutative 71.7%

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

      3. +-commutative 71.7%

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

      4. +-commutative 71.7%

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

      5. associate-*l* 71.7%

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

      6. *-commutative 71.7%

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

      7. times-frac 95.4%

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

      8. +-commutative 95.4%

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

      9. +-commutative 95.4%

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

      10. +-commutative 95.4%

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

      11. associate-+l+ 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in x around 0 47.6%

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

      1. +-commutative 47.6%

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

   6. Simplified 47.6%

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

   if 7.2000000000000001e154 < y

   1. Initial program 59.8%

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

      1. associate-*l* 59.8%

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

      2. +-commutative 59.8%

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

      3. +-commutative 59.8%

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

      4. +-commutative 59.8%

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

      5. associate-*l* 59.8%

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

      6. *-commutative 59.8%

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

      7. times-frac 88.2%

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

      8. +-commutative 88.2%

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

      9. +-commutative 88.2%

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

      10. +-commutative 88.2%

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

      11. associate-+l+ 88.2%

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

   3. Simplified 88.2%

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

   4. Taylor expanded in y around inf 88.2%

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

      1. associate-/r* 96.1%

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

      2. div-inv 96.1%

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

   6. Applied egg-rr 96.1%

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

      1. associate-*r/ 96.1%

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

      2. *-rgt-identity 96.1%

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

   8. Simplified 96.1%

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

   9. Taylor expanded in y around inf 96.0%

      \[\leadsto \frac{\color{blue}{1}}{x + y} \cdot \frac{x}{y} \]
3. Recombined 5 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{-200}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-134}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y + x}\\ \end{array} \]

Alternative 8: 82.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{if}\;y \leq 3.9 \cdot 10^{-200}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-134}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y (+ y (+ x 1.0))))))
   (if (<= y 3.9e-200)
     (/ (/ y (+ x 1.0)) x)
     (if (<= y 1.05e-134)
       t_0
       (if (<= y 5e-95)
         (/ y (* x (+ x 1.0)))
         (if (<= y 8.8e+154) t_0 (* (/ x y) (/ 1.0 (+ y x)))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y * (y + (x + 1.0)));
	double tmp;
	if (y <= 3.9e-200) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 1.05e-134) {
		tmp = t_0;
	} else if (y <= 5e-95) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 8.8e+154) {
		tmp = t_0;
	} else {
		tmp = (x / y) * (1.0 / (y + x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y * (y + (x + 1.0d0)))
    if (y <= 3.9d-200) then
        tmp = (y / (x + 1.0d0)) / x
    else if (y <= 1.05d-134) then
        tmp = t_0
    else if (y <= 5d-95) then
        tmp = y / (x * (x + 1.0d0))
    else if (y <= 8.8d+154) then
        tmp = t_0
    else
        tmp = (x / y) * (1.0d0 / (y + x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y * (y + (x + 1.0)));
	double tmp;
	if (y <= 3.9e-200) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 1.05e-134) {
		tmp = t_0;
	} else if (y <= 5e-95) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 8.8e+154) {
		tmp = t_0;
	} else {
		tmp = (x / y) * (1.0 / (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y * (y + (x + 1.0)))
	tmp = 0
	if y <= 3.9e-200:
		tmp = (y / (x + 1.0)) / x
	elif y <= 1.05e-134:
		tmp = t_0
	elif y <= 5e-95:
		tmp = y / (x * (x + 1.0))
	elif y <= 8.8e+154:
		tmp = t_0
	else:
		tmp = (x / y) * (1.0 / (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y * Float64(y + Float64(x + 1.0))))
	tmp = 0.0
	if (y <= 3.9e-200)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	elseif (y <= 1.05e-134)
		tmp = t_0;
	elseif (y <= 5e-95)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (y <= 8.8e+154)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y * (y + (x + 1.0)));
	tmp = 0.0;
	if (y <= 3.9e-200)
		tmp = (y / (x + 1.0)) / x;
	elseif (y <= 1.05e-134)
		tmp = t_0;
	elseif (y <= 5e-95)
		tmp = y / (x * (x + 1.0));
	elseif (y <= 8.8e+154)
		tmp = t_0;
	else
		tmp = (x / y) * (1.0 / (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 3.9e-200], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 1.05e-134], t$95$0, If[LessEqual[y, 5e-95], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.8e+154], t$95$0, N[(N[(x / y), $MachinePrecision] * N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 3.89999999999999999e-200

   1. Initial program 65.9%

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

      1. associate-*l* 65.9%

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

      2. +-commutative 65.9%

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

      3. +-commutative 65.9%

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

      4. +-commutative 65.9%

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

      5. associate-*l* 65.9%

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

      6. *-commutative 65.9%

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

      7. times-frac 84.3%

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

      8. +-commutative 84.3%

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

      9. +-commutative 84.3%

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

      10. +-commutative 84.3%

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

      11. associate-+l+ 84.3%

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

   3. Simplified 84.3%

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

   4. Taylor expanded in y around 0 56.4%

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

      1. +-commutative 56.4%

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

   6. Simplified 56.4%

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

      1. *-un-lft-identity 56.4%

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

      2. times-frac 59.2%

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

   8. Applied egg-rr 59.2%

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

      1. associate-*l/ 59.3%

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

      2. *-lft-identity 59.3%

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

   10. Simplified 59.3%

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

   if 3.89999999999999999e-200 < y < 1.05e-134 or 4.9999999999999998e-95 < y < 8.8000000000000004e154

   1. Initial program 71.4%

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

      1. associate-*l* 71.4%

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

      2. +-commutative 71.4%

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

      3. +-commutative 71.4%

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

      4. +-commutative 71.4%

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

      5. associate-*l* 71.4%

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

      6. *-commutative 71.4%

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

      7. times-frac 94.8%

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

      8. +-commutative 94.8%

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

      9. +-commutative 94.8%

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

      10. +-commutative 94.8%

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

      11. associate-+l+ 94.8%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in y around inf 44.0%

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

      1. *-commutative 44.0%

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

      2. associate-+r+ 44.0%

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

      3. +-commutative 44.0%

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

      4. associate-+r+ 44.0%

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

      5. frac-times 45.5%

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

      6. *-rgt-identity 45.5%

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

   6. Applied egg-rr 45.5%

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

   if 1.05e-134 < y < 4.9999999999999998e-95

   1. Initial program 86.1%

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

      1. associate-*l* 86.1%

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

      2. +-commutative 86.1%

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

      3. +-commutative 86.1%

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

      4. +-commutative 86.1%

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

      5. associate-*l* 86.1%

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

      6. *-commutative 86.1%

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

      7. times-frac 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 77.2%

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

      1. +-commutative 77.2%

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

   6. Simplified 77.2%

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

   if 8.8000000000000004e154 < y

   1. Initial program 59.8%

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

      1. associate-*l* 59.8%

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

      2. +-commutative 59.8%

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

      3. +-commutative 59.8%

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

      4. +-commutative 59.8%

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

      5. associate-*l* 59.8%

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

      6. *-commutative 59.8%

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

      7. times-frac 88.2%

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

      8. +-commutative 88.2%

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

      9. +-commutative 88.2%

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

      10. +-commutative 88.2%

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

      11. associate-+l+ 88.2%

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

   3. Simplified 88.2%

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

   4. Taylor expanded in y around inf 88.2%

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

      1. associate-/r* 96.1%

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

      2. div-inv 96.1%

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

   6. Applied egg-rr 96.1%

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

      1. associate-*r/ 96.1%

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

      2. *-rgt-identity 96.1%

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

   8. Simplified 96.1%

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

   9. Taylor expanded in y around inf 96.0%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{-200}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-134}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y + x}\\ \end{array} \]

Alternative 9: 86.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y + \left(x + 1\right)}\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-156}:\\ \;\;\;\;\frac{t_0}{y + x} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y (+ x 1.0)))))
   (if (<= x -3.4e-156) (* (/ t_0 (+ y x)) (/ y x)) (/ t_0 y))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y + (x + 1.0));
	double tmp;
	if (x <= -3.4e-156) {
		tmp = (t_0 / (y + x)) * (y / x);
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = x / (y + (x + 1.0d0))
    if (x <= (-3.4d-156)) then
        tmp = (t_0 / (y + x)) * (y / x)
    else
        tmp = t_0 / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y + (x + 1.0));
	double tmp;
	if (x <= -3.4e-156) {
		tmp = (t_0 / (y + x)) * (y / x);
	} else {
		tmp = t_0 / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y + (x + 1.0))
	tmp = 0
	if x <= -3.4e-156:
		tmp = (t_0 / (y + x)) * (y / x)
	else:
		tmp = t_0 / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y + Float64(x + 1.0)))
	tmp = 0.0
	if (x <= -3.4e-156)
		tmp = Float64(Float64(t_0 / Float64(y + x)) * Float64(y / x));
	else
		tmp = Float64(t_0 / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y + (x + 1.0));
	tmp = 0.0;
	if (x <= -3.4e-156)
		tmp = (t_0 / (y + x)) * (y / x);
	else
		tmp = t_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_] := Block[{t$95$0 = N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.4e-156], N[(N[(t$95$0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.3999999999999999e-156

   1. Initial program 62.7%

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

      1. associate-+r+ 62.7%

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

      2. *-commutative 62.7%

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

      3. frac-times 93.4%

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

      4. associate-*l/ 88.7%

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

      5. times-frac 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in y around 0 78.3%

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

   if -3.3999999999999999e-156 < x

   1. Initial program 69.2%

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

      1. associate-*l* 69.2%

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

      2. +-commutative 69.2%

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

      3. +-commutative 69.2%

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

      4. +-commutative 69.2%

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

      5. associate-*l* 69.2%

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

      6. *-commutative 69.2%

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

      7. times-frac 85.6%

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

      8. +-commutative 85.6%

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

      9. +-commutative 85.6%

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

      10. +-commutative 85.6%

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

      11. associate-+l+ 85.6%

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

   3. Simplified 85.6%

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

   4. Taylor expanded in y around inf 55.5%

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

      1. associate-*l/ 55.6%

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

      2. *-un-lft-identity 55.6%

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

      3. associate-+r+ 55.6%

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

      4. +-commutative 55.6%

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

      5. associate-+r+ 55.6%

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

   6. Applied egg-rr 55.6%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-156}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \]

Alternative 10: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{y + x}\\ \mathbf{if}\;y \leq 54000000:\\ \;\;\;\;t_0 \cdot \frac{\frac{x}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y + x} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y x))))
   (if (<= y 54000000.0)
     (* t_0 (/ (/ x (+ x 1.0)) (+ y x)))
     (* (/ t_0 (+ y x)) (/ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (y <= 54000000.0) {
		tmp = t_0 * ((x / (x + 1.0)) / (y + x));
	} else {
		tmp = (t_0 / (y + x)) * (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) :: t_0
    real(8) :: tmp
    t_0 = y / (y + x)
    if (y <= 54000000.0d0) then
        tmp = t_0 * ((x / (x + 1.0d0)) / (y + x))
    else
        tmp = (t_0 / (y + x)) * (x / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (y <= 54000000.0) {
		tmp = t_0 * ((x / (x + 1.0)) / (y + x));
	} else {
		tmp = (t_0 / (y + x)) * (x / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (y + x)
	tmp = 0
	if y <= 54000000.0:
		tmp = t_0 * ((x / (x + 1.0)) / (y + x))
	else:
		tmp = (t_0 / (y + x)) * (x / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(y + x))
	tmp = 0.0
	if (y <= 54000000.0)
		tmp = Float64(t_0 * Float64(Float64(x / Float64(x + 1.0)) / Float64(y + x)));
	else
		tmp = Float64(Float64(t_0 / Float64(y + x)) * Float64(x / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (y + x);
	tmp = 0.0;
	if (y <= 54000000.0)
		tmp = t_0 * ((x / (x + 1.0)) / (y + x));
	else
		tmp = (t_0 / (y + x)) * (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_] := Block[{t$95$0 = N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 54000000.0], N[(t$95$0 * N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 5.4e7

   1. Initial program 70.2%

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

      1. associate-+r+ 70.2%

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

      2. *-commutative 70.2%

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

      3. frac-times 87.4%

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

      4. associate-*l/ 78.7%

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

      5. times-frac 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in y around 0 85.4%

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

      1. +-commutative 85.4%

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

   6. Simplified 85.4%

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

   if 5.4e7 < y

   1. Initial program 58.7%

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

      1. associate-*l* 58.7%

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

      2. +-commutative 58.7%

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

      3. +-commutative 58.7%

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

      4. +-commutative 58.7%

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

      5. associate-*l* 58.7%

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

      6. *-commutative 58.7%

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

      7. times-frac 90.1%

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

      8. +-commutative 90.1%

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

      9. +-commutative 90.1%

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

      10. +-commutative 90.1%

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

      11. associate-+l+ 90.1%

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

   3. Simplified 90.1%

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

   4. Taylor expanded in y around inf 77.3%

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

      1. associate-/r* 80.9%

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

      2. div-inv 80.9%

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

   6. Applied egg-rr 80.9%

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

      1. associate-*r/ 80.9%

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

      2. *-rgt-identity 80.9%

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

   8. Simplified 80.9%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 54000000:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{\frac{x}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + x} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 11: 74.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{if}\;y \leq -9.4 \cdot 10^{-135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-200}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ y x) (/ 1.0 x))))
   (if (<= y -9.4e-135)
     t_0
     (if (<= y 3.9e-200)
       (/ y x)
       (if (<= y 3.3e-20)
         (/ x y)
         (if (<= y 3.4e+27) t_0 (* (/ 1.0 y) (/ x y))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y / x) * (1.0 / x);
	double tmp;
	if (y <= -9.4e-135) {
		tmp = t_0;
	} else if (y <= 3.9e-200) {
		tmp = y / x;
	} else if (y <= 3.3e-20) {
		tmp = x / y;
	} else if (y <= 3.4e+27) {
		tmp = t_0;
	} else {
		tmp = (1.0 / y) * (x / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y / x) * (1.0d0 / x)
    if (y <= (-9.4d-135)) then
        tmp = t_0
    else if (y <= 3.9d-200) then
        tmp = y / x
    else if (y <= 3.3d-20) then
        tmp = x / y
    else if (y <= 3.4d+27) then
        tmp = t_0
    else
        tmp = (1.0d0 / y) * (x / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y / x) * (1.0 / x);
	double tmp;
	if (y <= -9.4e-135) {
		tmp = t_0;
	} else if (y <= 3.9e-200) {
		tmp = y / x;
	} else if (y <= 3.3e-20) {
		tmp = x / y;
	} else if (y <= 3.4e+27) {
		tmp = t_0;
	} else {
		tmp = (1.0 / y) * (x / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y / x) * (1.0 / x)
	tmp = 0
	if y <= -9.4e-135:
		tmp = t_0
	elif y <= 3.9e-200:
		tmp = y / x
	elif y <= 3.3e-20:
		tmp = x / y
	elif y <= 3.4e+27:
		tmp = t_0
	else:
		tmp = (1.0 / y) * (x / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y / x) * Float64(1.0 / x))
	tmp = 0.0
	if (y <= -9.4e-135)
		tmp = t_0;
	elseif (y <= 3.9e-200)
		tmp = Float64(y / x);
	elseif (y <= 3.3e-20)
		tmp = Float64(x / y);
	elseif (y <= 3.4e+27)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / y) * Float64(x / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (y / x) * (1.0 / x);
	tmp = 0.0;
	if (y <= -9.4e-135)
		tmp = t_0;
	elseif (y <= 3.9e-200)
		tmp = y / x;
	elseif (y <= 3.3e-20)
		tmp = x / y;
	elseif (y <= 3.4e+27)
		tmp = t_0;
	else
		tmp = (1.0 / y) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9.4e-135], t$95$0, If[LessEqual[y, 3.9e-200], N[(y / x), $MachinePrecision], If[LessEqual[y, 3.3e-20], N[(x / y), $MachinePrecision], If[LessEqual[y, 3.4e+27], t$95$0, N[(N[(1.0 / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -9.39999999999999952e-135 or 3.3e-20 < y < 3.4e27

   1. Initial program 70.7%

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

      1. associate-+r+ 70.7%

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

      2. *-commutative 70.7%

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

      3. frac-times 93.2%

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

      4. associate-*l/ 89.8%

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

      5. times-frac 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around inf 44.7%

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

   5. Taylor expanded in y around 0 44.1%

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

   if -9.39999999999999952e-135 < y < 3.89999999999999999e-200

   1. Initial program 58.8%

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

      1. associate-*l* 58.8%

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

      2. +-commutative 58.8%

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

      3. +-commutative 58.8%

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

      4. +-commutative 58.8%

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

      5. associate-*l* 58.8%

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

      6. *-commutative 58.8%

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

      7. times-frac 71.8%

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

      8. +-commutative 71.8%

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

      9. +-commutative 71.8%

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

      10. +-commutative 71.8%

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

      11. associate-+l+ 71.8%

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

   3. Simplified 71.8%

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

   4. Taylor expanded in y around 0 86.3%

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

      1. +-commutative 86.3%

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

   6. Simplified 86.3%

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

   7. Taylor expanded in x around 0 80.1%

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

   if 3.89999999999999999e-200 < y < 3.3e-20

   1. Initial program 82.4%

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

      1. associate-*l* 82.4%

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

      2. +-commutative 82.4%

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

      3. +-commutative 82.4%

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

      4. +-commutative 82.4%

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

      5. associate-*l* 82.4%

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

      6. *-commutative 82.4%

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

      7. times-frac 97.2%

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

      8. +-commutative 97.2%

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

      9. +-commutative 97.2%

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

      10. +-commutative 97.2%

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

      11. associate-+l+ 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in x around 0 34.2%

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

      1. +-commutative 34.2%

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

   6. Simplified 34.2%

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

   7. Taylor expanded in y around 0 34.2%

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

   if 3.4e27 < y

   1. Initial program 59.2%

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

      1. associate-*l* 59.2%

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

      2. +-commutative 59.2%

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

      3. +-commutative 59.2%

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

      4. +-commutative 59.2%

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

      5. associate-*l* 59.2%

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

      6. *-commutative 59.2%

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

      7. times-frac 89.2%

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

      8. +-commutative 89.2%

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

      9. +-commutative 89.2%

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

      10. +-commutative 89.2%

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

      11. associate-+l+ 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in y around inf 79.5%

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

   5. Taylor expanded in y around inf 79.2%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.4 \cdot 10^{-135}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-200}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+27}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 12: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{-200}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-134}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.9e-200)
   (/ (/ y (+ x 1.0)) x)
   (if (<= y 1.05e-134)
     (/ x y)
     (if (<= y 5e-95)
       (/ y (* x (+ x 1.0)))
       (if (<= y 8.2e+154) (/ x (* y (+ y 1.0))) (* (/ 1.0 y) (/ x y)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.9e-200) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 1.05e-134) {
		tmp = x / y;
	} else if (y <= 5e-95) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 8.2e+154) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (1.0 / y) * (x / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.9d-200) then
        tmp = (y / (x + 1.0d0)) / x
    else if (y <= 1.05d-134) then
        tmp = x / y
    else if (y <= 5d-95) then
        tmp = y / (x * (x + 1.0d0))
    else if (y <= 8.2d+154) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (1.0d0 / y) * (x / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.9e-200) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 1.05e-134) {
		tmp = x / y;
	} else if (y <= 5e-95) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 8.2e+154) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (1.0 / y) * (x / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.9e-200:
		tmp = (y / (x + 1.0)) / x
	elif y <= 1.05e-134:
		tmp = x / y
	elif y <= 5e-95:
		tmp = y / (x * (x + 1.0))
	elif y <= 8.2e+154:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (1.0 / y) * (x / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.9e-200)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	elseif (y <= 1.05e-134)
		tmp = Float64(x / y);
	elseif (y <= 5e-95)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (y <= 8.2e+154)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(1.0 / y) * Float64(x / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.9e-200)
		tmp = (y / (x + 1.0)) / x;
	elseif (y <= 1.05e-134)
		tmp = x / y;
	elseif (y <= 5e-95)
		tmp = y / (x * (x + 1.0));
	elseif (y <= 8.2e+154)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (1.0 / y) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.9e-200], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 1.05e-134], N[(x / y), $MachinePrecision], If[LessEqual[y, 5e-95], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.2e+154], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < 3.89999999999999999e-200

   1. Initial program 65.9%

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

      1. associate-*l* 65.9%

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

      2. +-commutative 65.9%

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

      3. +-commutative 65.9%

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

      4. +-commutative 65.9%

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

      5. associate-*l* 65.9%

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

      6. *-commutative 65.9%

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

      7. times-frac 84.3%

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

      8. +-commutative 84.3%

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

      9. +-commutative 84.3%

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

      10. +-commutative 84.3%

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

      11. associate-+l+ 84.3%

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

   3. Simplified 84.3%

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

   4. Taylor expanded in y around 0 56.4%

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

      1. +-commutative 56.4%

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

   6. Simplified 56.4%

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

      1. *-un-lft-identity 56.4%

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

      2. times-frac 59.2%

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

   8. Applied egg-rr 59.2%

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

      1. associate-*l/ 59.3%

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

      2. *-lft-identity 59.3%

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

   10. Simplified 59.3%

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

   if 3.89999999999999999e-200 < y < 1.05e-134

   1. Initial program 70.0%

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

      1. associate-*l* 70.0%

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

      2. +-commutative 70.0%

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

      3. +-commutative 70.0%

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

      4. +-commutative 70.0%

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

      5. associate-*l* 70.0%

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

      6. *-commutative 70.0%

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

      7. times-frac 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%

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

   4. Taylor expanded in x around 0 32.4%

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

      1. +-commutative 32.4%

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

   6. Simplified 32.4%

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

   7. Taylor expanded in y around 0 32.4%

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

   if 1.05e-134 < y < 4.9999999999999998e-95

   1. Initial program 86.1%

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

      1. associate-*l* 86.1%

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

      2. +-commutative 86.1%

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

      3. +-commutative 86.1%

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

      4. +-commutative 86.1%

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

      5. associate-*l* 86.1%

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

      6. *-commutative 86.1%

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

      7. times-frac 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. +-commutative 99.8%

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

      11. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 77.2%

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

      1. +-commutative 77.2%

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

   6. Simplified 77.2%

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

   if 4.9999999999999998e-95 < y < 8.2e154

   1. Initial program 71.7%

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

      1. associate-*l* 71.7%

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

      2. +-commutative 71.7%

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

      3. +-commutative 71.7%

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

      4. +-commutative 71.7%

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

      5. associate-*l* 71.7%

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

      6. *-commutative 71.7%

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

      7. times-frac 95.4%

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

      8. +-commutative 95.4%

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

      9. +-commutative 95.4%

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

      10. +-commutative 95.4%

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

      11. associate-+l+ 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in x around 0 47.6%

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

      1. +-commutative 47.6%

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

   6. Simplified 47.6%

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

   if 8.2e154 < y

   1. Initial program 59.8%

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

      1. associate-*l* 59.8%

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

      2. +-commutative 59.8%

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

      3. +-commutative 59.8%

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

      4. +-commutative 59.8%

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

      5. associate-*l* 59.8%

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

      6. *-commutative 59.8%

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

      7. times-frac 88.2%

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

      8. +-commutative 88.2%

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

      9. +-commutative 88.2%

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

      10. +-commutative 88.2%

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

      11. associate-+l+ 88.2%

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

   3. Simplified 88.2%

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

   4. Taylor expanded in y around inf 96.0%

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

   5. Taylor expanded in y around inf 96.0%

      \[\leadsto \frac{1}{y} \cdot \color{blue}{\frac{x}{y}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{-200}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-134}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 13: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{-200}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-134}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.9e-200)
   (/ (/ y (+ x 1.0)) x)
   (if (<= y 1.05e-134)
     (/ x (* y (+ y (+ x 1.0))))
     (if (<= y 5e-95) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ x (+ y 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.9e-200) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 1.05e-134) {
		tmp = x / (y * (y + (x + 1.0)));
	} else if (y <= 5e-95) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (x + (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.9d-200) then
        tmp = (y / (x + 1.0d0)) / x
    else if (y <= 1.05d-134) then
        tmp = x / (y * (y + (x + 1.0d0)))
    else if (y <= 5d-95) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (x + (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.9e-200) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 1.05e-134) {
		tmp = x / (y * (y + (x + 1.0)));
	} else if (y <= 5e-95) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (x + (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.9e-200:
		tmp = (y / (x + 1.0)) / x
	elif y <= 1.05e-134:
		tmp = x / (y * (y + (x + 1.0)))
	elif y <= 5e-95:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (x + (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.9e-200)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	elseif (y <= 1.05e-134)
		tmp = Float64(x / Float64(y * Float64(y + Float64(x + 1.0))));
	elseif (y <= 5e-95)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(x + Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.9e-200)
		tmp = (y / (x + 1.0)) / x;
	elseif (y <= 1.05e-134)
		tmp = x / (y * (y + (x + 1.0)));
	elseif (y <= 5e-95)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (x + (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.9e-200], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 1.05e-134], N[(x / N[(y * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e-95], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 3.89999999999999999e-200

   1. Initial program 65.9%

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

      1. associate-*l* 65.9%

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

      2. +-commutative 65.9%

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

      3. +-commutative 65.9%

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

      4. +-commutative 65.9%

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

      5. associate-*l* 65.9%

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

      6. *-commutative 65.9%

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

      7. times-frac 84.3%

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

      8. +-commutative 84.3%

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

      9. +-commutative 84.3%

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

      10. +-commutative 84.3%

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

      11. associate-+l+ 84.3%

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

   3. Simplified 84.3%

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

   4. Taylor expanded in y around 0 56.4%

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

      1. +-commutative 56.4%

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

   6. Simplified 56.4%

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

      1. *-un-lft-identity 56.4%

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

      2. times-frac 59.2%

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

   8. Applied egg-rr 59.2%

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

      1. associate-*l/ 59.3%

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

      2. *-lft-identity 59.3%

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

   10. Simplified 59.3%

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

   if 3.89999999999999999e-200 < y < 1.05e-134

   1. Initial program 70.0%

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

      1. associate-*l* 70.0%

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

      2. +-commutative 70.0%

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

      3. +-commutative 70.0%

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

      4. +-commutative 70.0%

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

      5. associate-*l* 70.0%

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

      6. *-commutative 70.0%

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

      7. times-frac 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%

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

   4. Taylor expanded in y around inf 32.5%

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

      1. *-commutative 32.5%

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

      2. associate-+r+ 32.5%

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

      3. +-commutative 32.5%

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} + 1} \cdot \frac{1}{y} \]

      4. associate-+r+ 32.5%

         \[\leadsto \frac{x}{\color{blue}{y + \left(x + 1\right)}} \cdot \frac{1}{y} \]

      5. frac-times 32.8%

         \[\leadsto \color{blue}{\frac{x \cdot 1}{\left(y + \left(x + 1\right)\right) \cdot y}} \]

      6. *-rgt-identity 32.8%

         \[\leadsto \frac{\color{blue}{x}}{\left(y + \left(x + 1\right)\right) \cdot y} \]

   6. Applied egg-rr 32.8%

      \[\leadsto \color{blue}{\frac{x}{\left(y + \left(x + 1\right)\right) \cdot y}} \]

   if 1.05e-134 < y < 4.9999999999999998e-95

   1. Initial program 86.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 86.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 86.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 86.1%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 86.1%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 86.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 86.1%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. times-frac 99.8%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 99.8%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. +-commutative 99.8%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ 99.8%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 77.2%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 77.2%

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   6. Simplified 77.2%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]

   if 4.9999999999999998e-95 < y

   1. Initial program 67.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 67.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 67.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 67.1%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 67.1%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 67.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 67.1%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. times-frac 92.6%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 92.6%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 92.6%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. +-commutative 92.6%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ 92.6%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 92.6%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around inf 65.8%

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{x + \left(y + 1\right)} \]
   5. Step-by-step derivation

      1. *-commutative 65.8%

         \[\leadsto \color{blue}{\frac{x}{x + \left(y + 1\right)} \cdot \frac{1}{y}} \]

      2. associate-+r+ 65.8%

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{1}{y} \]

      3. +-commutative 65.8%

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} + 1} \cdot \frac{1}{y} \]

      4. associate-+r+ 65.8%

         \[\leadsto \frac{x}{\color{blue}{y + \left(x + 1\right)}} \cdot \frac{1}{y} \]

      5. frac-times 63.9%

         \[\leadsto \color{blue}{\frac{x \cdot 1}{\left(y + \left(x + 1\right)\right) \cdot y}} \]

      6. *-rgt-identity 63.9%

         \[\leadsto \frac{\color{blue}{x}}{\left(y + \left(x + 1\right)\right) \cdot y} \]

   6. Applied egg-rr 63.9%

      \[\leadsto \color{blue}{\frac{x}{\left(y + \left(x + 1\right)\right) \cdot y}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 63.9%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{\left(y + \left(x + 1\right)\right) \cdot y} \]

      2. times-frac 65.9%

         \[\leadsto \color{blue}{\frac{1}{y + \left(x + 1\right)} \cdot \frac{x}{y}} \]

      3. associate-+r+ 65.9%

         \[\leadsto \frac{1}{\color{blue}{\left(y + x\right) + 1}} \cdot \frac{x}{y} \]

      4. +-commutative 65.9%

         \[\leadsto \frac{1}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{y} \]

      5. associate-+l+ 65.9%

         \[\leadsto \frac{1}{\color{blue}{x + \left(y + 1\right)}} \cdot \frac{x}{y} \]

   8. Applied egg-rr 65.9%

      \[\leadsto \color{blue}{\frac{1}{x + \left(y + 1\right)} \cdot \frac{x}{y}} \]
   9. Step-by-step derivation

      1. associate-*l/ 65.9%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y}}{x + \left(y + 1\right)}} \]

      2. *-lft-identity 65.9%

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{x + \left(y + 1\right)} \]

   10. Simplified 65.9%

       \[\leadsto \color{blue}{\frac{\frac{x}{y}}{x + \left(y + 1\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{-200}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-134}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \]

Alternative 14: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;y \leq 3.9 \cdot 10^{-200}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-134}:\\ \;\;\;\;\frac{x}{y \cdot t_0}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t_0}}{y}\\ \end{array} \end{array} \]

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 1.0))))
   (if (<= y 3.9e-200)
     (/ (/ y (+ x 1.0)) x)
     (if (<= y 1.05e-134)
       (/ x (* y t_0))
       (if (<= y 4.3e-95) (/ y (* x (+ x 1.0))) (/ (/ x t_0) y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= 3.9e-200) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 1.05e-134) {
		tmp = x / (y * t_0);
	} else if (y <= 4.3e-95) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / t_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) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (y <= 3.9d-200) then
        tmp = (y / (x + 1.0d0)) / x
    else if (y <= 1.05d-134) then
        tmp = x / (y * t_0)
    else if (y <= 4.3d-95) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / t_0) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= 3.9e-200) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 1.05e-134) {
		tmp = x / (y * t_0);
	} else if (y <= 4.3e-95) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / t_0) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if y <= 3.9e-200:
		tmp = (y / (x + 1.0)) / x
	elif y <= 1.05e-134:
		tmp = x / (y * t_0)
	elif y <= 4.3e-95:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / t_0) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (y <= 3.9e-200)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	elseif (y <= 1.05e-134)
		tmp = Float64(x / Float64(y * t_0));
	elseif (y <= 4.3e-95)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / t_0) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (y <= 3.9e-200)
		tmp = (y / (x + 1.0)) / x;
	elseif (y <= 1.05e-134)
		tmp = x / (y * t_0);
	elseif (y <= 4.3e-95)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / t_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_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 3.9e-200], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 1.05e-134], N[(x / N[(y * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.3e-95], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / t$95$0), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 3.89999999999999999e-200

   1. Initial program 65.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 65.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 65.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 65.9%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 65.9%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 65.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 65.9%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. times-frac 84.3%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 84.3%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 84.3%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. +-commutative 84.3%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ 84.3%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 84.3%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 56.4%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 56.4%

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   6. Simplified 56.4%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 56.4%

         \[\leadsto \frac{\color{blue}{1 \cdot y}}{x \cdot \left(x + 1\right)} \]

      2. times-frac 59.2%

         \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{y}{x + 1}} \]

   8. Applied egg-rr 59.2%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{y}{x + 1}} \]
   9. Step-by-step derivation

      1. associate-*l/ 59.3%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{y}{x + 1}}{x}} \]

      2. *-lft-identity 59.3%

         \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{x} \]

   10. Simplified 59.3%

       \[\leadsto \color{blue}{\frac{\frac{y}{x + 1}}{x}} \]

   if 3.89999999999999999e-200 < y < 1.05e-134

   1. Initial program 70.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 70.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 70.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 70.0%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 70.0%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 70.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 70.0%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. times-frac 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%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around inf 32.5%

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{x + \left(y + 1\right)} \]
   5. Step-by-step derivation

      1. *-commutative 32.5%

         \[\leadsto \color{blue}{\frac{x}{x + \left(y + 1\right)} \cdot \frac{1}{y}} \]

      2. associate-+r+ 32.5%

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{1}{y} \]

      3. +-commutative 32.5%

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} + 1} \cdot \frac{1}{y} \]

      4. associate-+r+ 32.5%

         \[\leadsto \frac{x}{\color{blue}{y + \left(x + 1\right)}} \cdot \frac{1}{y} \]

      5. frac-times 32.8%

         \[\leadsto \color{blue}{\frac{x \cdot 1}{\left(y + \left(x + 1\right)\right) \cdot y}} \]

      6. *-rgt-identity 32.8%

         \[\leadsto \frac{\color{blue}{x}}{\left(y + \left(x + 1\right)\right) \cdot y} \]

   6. Applied egg-rr 32.8%

      \[\leadsto \color{blue}{\frac{x}{\left(y + \left(x + 1\right)\right) \cdot y}} \]

   if 1.05e-134 < y < 4.29999999999999997e-95

   1. Initial program 86.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 86.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 86.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 86.1%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 86.1%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 86.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 86.1%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. times-frac 99.8%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 99.8%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. +-commutative 99.8%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ 99.8%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 77.2%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 77.2%

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   6. Simplified 77.2%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]

   if 4.29999999999999997e-95 < y

   1. Initial program 67.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 67.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 67.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 67.1%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 67.1%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 67.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 67.1%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. times-frac 92.6%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 92.6%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 92.6%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. +-commutative 92.6%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ 92.6%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 92.6%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around inf 65.8%

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{x + \left(y + 1\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 65.9%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + \left(y + 1\right)}}{y}} \]

      2. *-un-lft-identity 65.9%

         \[\leadsto \frac{\color{blue}{\frac{x}{x + \left(y + 1\right)}}}{y} \]

      3. associate-+r+ 65.9%

         \[\leadsto \frac{\frac{x}{\color{blue}{\left(x + y\right) + 1}}}{y} \]

      4. +-commutative 65.9%

         \[\leadsto \frac{\frac{x}{\color{blue}{\left(y + x\right)} + 1}}{y} \]

      5. associate-+r+ 65.9%

         \[\leadsto \frac{\frac{x}{\color{blue}{y + \left(x + 1\right)}}}{y} \]

   6. Applied egg-rr 65.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + \left(x + 1\right)}}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{-200}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-134}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \]

Alternative 15: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;y \leq 3.9 \cdot 10^{-200}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-134}:\\ \;\;\;\;\frac{x}{y \cdot t_0}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t_0}}{y}\\ \end{array} \end{array} \]

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 1.0))))
   (if (<= y 3.9e-200)
     (/ (/ y (+ x 1.0)) x)
     (if (<= y 1.3e-134)
       (/ x (* y t_0))
       (if (<= y 5e-95) (/ (/ y (+ y x)) (+ x 1.0)) (/ (/ x t_0) y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= 3.9e-200) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 1.3e-134) {
		tmp = x / (y * t_0);
	} else if (y <= 5e-95) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = (x / t_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) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (y <= 3.9d-200) then
        tmp = (y / (x + 1.0d0)) / x
    else if (y <= 1.3d-134) then
        tmp = x / (y * t_0)
    else if (y <= 5d-95) then
        tmp = (y / (y + x)) / (x + 1.0d0)
    else
        tmp = (x / t_0) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= 3.9e-200) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 1.3e-134) {
		tmp = x / (y * t_0);
	} else if (y <= 5e-95) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = (x / t_0) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if y <= 3.9e-200:
		tmp = (y / (x + 1.0)) / x
	elif y <= 1.3e-134:
		tmp = x / (y * t_0)
	elif y <= 5e-95:
		tmp = (y / (y + x)) / (x + 1.0)
	else:
		tmp = (x / t_0) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (y <= 3.9e-200)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	elseif (y <= 1.3e-134)
		tmp = Float64(x / Float64(y * t_0));
	elseif (y <= 5e-95)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / t_0) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (y <= 3.9e-200)
		tmp = (y / (x + 1.0)) / x;
	elseif (y <= 1.3e-134)
		tmp = x / (y * t_0);
	elseif (y <= 5e-95)
		tmp = (y / (y + x)) / (x + 1.0);
	else
		tmp = (x / t_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_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 3.9e-200], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 1.3e-134], N[(x / N[(y * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e-95], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / t$95$0), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 3.89999999999999999e-200

   1. Initial program 65.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 65.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 65.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 65.9%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 65.9%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 65.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 65.9%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. times-frac 84.3%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 84.3%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 84.3%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. +-commutative 84.3%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ 84.3%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 84.3%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 56.4%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 56.4%

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   6. Simplified 56.4%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 56.4%

         \[\leadsto \frac{\color{blue}{1 \cdot y}}{x \cdot \left(x + 1\right)} \]

      2. times-frac 59.2%

         \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{y}{x + 1}} \]

   8. Applied egg-rr 59.2%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{y}{x + 1}} \]
   9. Step-by-step derivation

      1. associate-*l/ 59.3%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{y}{x + 1}}{x}} \]

      2. *-lft-identity 59.3%

         \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{x} \]

   10. Simplified 59.3%

       \[\leadsto \color{blue}{\frac{\frac{y}{x + 1}}{x}} \]

   if 3.89999999999999999e-200 < y < 1.30000000000000011e-134

   1. Initial program 70.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 70.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 70.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 70.0%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 70.0%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 70.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 70.0%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. times-frac 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%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around inf 32.5%

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{x + \left(y + 1\right)} \]
   5. Step-by-step derivation

      1. *-commutative 32.5%

         \[\leadsto \color{blue}{\frac{x}{x + \left(y + 1\right)} \cdot \frac{1}{y}} \]

      2. associate-+r+ 32.5%

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{1}{y} \]

      3. +-commutative 32.5%

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} + 1} \cdot \frac{1}{y} \]

      4. associate-+r+ 32.5%

         \[\leadsto \frac{x}{\color{blue}{y + \left(x + 1\right)}} \cdot \frac{1}{y} \]

      5. frac-times 32.8%

         \[\leadsto \color{blue}{\frac{x \cdot 1}{\left(y + \left(x + 1\right)\right) \cdot y}} \]

      6. *-rgt-identity 32.8%

         \[\leadsto \frac{\color{blue}{x}}{\left(y + \left(x + 1\right)\right) \cdot y} \]

   6. Applied egg-rr 32.8%

      \[\leadsto \color{blue}{\frac{x}{\left(y + \left(x + 1\right)\right) \cdot y}} \]

   if 1.30000000000000011e-134 < y < 4.9999999999999998e-95

   1. Initial program 86.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-+r+ 86.1%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      2. *-commutative 86.1%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)} \]

      3. frac-times 99.8%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

      4. associate-*l/ 86.1%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      5. times-frac 100.0%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{\frac{x}{x + \left(y + 1\right)}}{x + y}} \]

      6. associate-+r+ 100.0%

         \[\leadsto \frac{y}{x + y} \cdot \frac{\frac{x}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      7. +-commutative 100.0%

         \[\leadsto \frac{y}{x + y} \cdot \frac{\frac{x}{\color{blue}{\left(y + x\right)} + 1}}{x + y} \]

      8. associate-+l+ 100.0%

         \[\leadsto \frac{y}{x + y} \cdot \frac{\frac{x}{\color{blue}{y + \left(x + 1\right)}}}{x + y} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{\frac{x}{y + \left(x + 1\right)}}{x + y}} \]
   4. Step-by-step derivation

      1. clear-num 100.0%

         \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{\frac{x + y}{\frac{x}{y + \left(x + 1\right)}}}} \]

      2. un-div-inv 100.0%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{\frac{x + y}{\frac{x}{y + \left(x + 1\right)}}}} \]

      3. +-commutative 100.0%

         \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{\frac{x + y}{\frac{x}{y + \left(x + 1\right)}}} \]

      4. div-inv 99.8%

         \[\leadsto \frac{\frac{y}{y + x}}{\color{blue}{\left(x + y\right) \cdot \frac{1}{\frac{x}{y + \left(x + 1\right)}}}} \]

      5. clear-num 99.8%

         \[\leadsto \frac{\frac{y}{y + x}}{\left(x + y\right) \cdot \color{blue}{\frac{y + \left(x + 1\right)}{x}}} \]

      6. +-commutative 99.8%

         \[\leadsto \frac{\frac{y}{y + x}}{\color{blue}{\left(y + x\right)} \cdot \frac{y + \left(x + 1\right)}{x}} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\left(y + x\right) \cdot \frac{y + \left(x + 1\right)}{x}}} \]

   6. Taylor expanded in y around 0 77.7%

      \[\leadsto \frac{\frac{y}{y + x}}{\color{blue}{1 + x}} \]
   7. Step-by-step derivation

      1. +-commutative 77.7%

         \[\leadsto \frac{\frac{y}{y + x}}{\color{blue}{x + 1}} \]

   8. Simplified 77.7%

      \[\leadsto \frac{\frac{y}{y + x}}{\color{blue}{x + 1}} \]

   if 4.9999999999999998e-95 < y

   1. Initial program 67.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 67.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 67.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 67.1%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 67.1%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 67.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 67.1%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. times-frac 92.6%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 92.6%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 92.6%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. +-commutative 92.6%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ 92.6%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 92.6%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around inf 65.8%

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{x + \left(y + 1\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 65.9%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + \left(y + 1\right)}}{y}} \]

      2. *-un-lft-identity 65.9%

         \[\leadsto \frac{\color{blue}{\frac{x}{x + \left(y + 1\right)}}}{y} \]

      3. associate-+r+ 65.9%

         \[\leadsto \frac{\frac{x}{\color{blue}{\left(x + y\right) + 1}}}{y} \]

      4. +-commutative 65.9%

         \[\leadsto \frac{\frac{x}{\color{blue}{\left(y + x\right)} + 1}}{y} \]

      5. associate-+r+ 65.9%

         \[\leadsto \frac{\frac{x}{\color{blue}{y + \left(x + 1\right)}}}{y} \]

   6. Applied egg-rr 65.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + \left(x + 1\right)}}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{-200}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-134}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \]

Alternative 16: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y + \left(x + 1\right)}\\ \mathbf{if}\;y \leq 3.9 \cdot 10^{-200}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-134}:\\ \;\;\;\;\frac{t_0}{y + x}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y (+ x 1.0)))))
   (if (<= y 3.9e-200)
     (/ (/ y (+ x 1.0)) x)
     (if (<= y 1.05e-134)
       (/ t_0 (+ y x))
       (if (<= y 5e-95) (/ (/ y (+ y x)) (+ x 1.0)) (/ t_0 y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y + (x + 1.0));
	double tmp;
	if (y <= 3.9e-200) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 1.05e-134) {
		tmp = t_0 / (y + x);
	} else if (y <= 5e-95) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = x / (y + (x + 1.0d0))
    if (y <= 3.9d-200) then
        tmp = (y / (x + 1.0d0)) / x
    else if (y <= 1.05d-134) then
        tmp = t_0 / (y + x)
    else if (y <= 5d-95) then
        tmp = (y / (y + x)) / (x + 1.0d0)
    else
        tmp = t_0 / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y + (x + 1.0));
	double tmp;
	if (y <= 3.9e-200) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 1.05e-134) {
		tmp = t_0 / (y + x);
	} else if (y <= 5e-95) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = t_0 / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y + (x + 1.0))
	tmp = 0
	if y <= 3.9e-200:
		tmp = (y / (x + 1.0)) / x
	elif y <= 1.05e-134:
		tmp = t_0 / (y + x)
	elif y <= 5e-95:
		tmp = (y / (y + x)) / (x + 1.0)
	else:
		tmp = t_0 / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y + Float64(x + 1.0)))
	tmp = 0.0
	if (y <= 3.9e-200)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	elseif (y <= 1.05e-134)
		tmp = Float64(t_0 / Float64(y + x));
	elseif (y <= 5e-95)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + 1.0));
	else
		tmp = Float64(t_0 / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y + (x + 1.0));
	tmp = 0.0;
	if (y <= 3.9e-200)
		tmp = (y / (x + 1.0)) / x;
	elseif (y <= 1.05e-134)
		tmp = t_0 / (y + x);
	elseif (y <= 5e-95)
		tmp = (y / (y + x)) / (x + 1.0);
	else
		tmp = t_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_] := Block[{t$95$0 = N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 3.9e-200], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 1.05e-134], N[(t$95$0 / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e-95], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / y), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 3.89999999999999999e-200

   1. Initial program 65.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 65.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 65.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 65.9%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 65.9%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 65.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 65.9%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. times-frac 84.3%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 84.3%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 84.3%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. +-commutative 84.3%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ 84.3%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 84.3%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 56.4%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 56.4%

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   6. Simplified 56.4%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 56.4%

         \[\leadsto \frac{\color{blue}{1 \cdot y}}{x \cdot \left(x + 1\right)} \]

      2. times-frac 59.2%

         \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{y}{x + 1}} \]

   8. Applied egg-rr 59.2%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{y}{x + 1}} \]
   9. Step-by-step derivation

      1. associate-*l/ 59.3%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{y}{x + 1}}{x}} \]

      2. *-lft-identity 59.3%

         \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{x} \]

   10. Simplified 59.3%

       \[\leadsto \color{blue}{\frac{\frac{y}{x + 1}}{x}} \]

   if 3.89999999999999999e-200 < y < 1.05e-134

   1. Initial program 70.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-+r+ 70.0%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      2. *-commutative 70.0%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)} \]

      3. frac-times 92.1%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

      4. associate-*l/ 70.0%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      5. times-frac 100.0%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{\frac{x}{x + \left(y + 1\right)}}{x + y}} \]

      6. associate-+r+ 100.0%

         \[\leadsto \frac{y}{x + y} \cdot \frac{\frac{x}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      7. +-commutative 100.0%

         \[\leadsto \frac{y}{x + y} \cdot \frac{\frac{x}{\color{blue}{\left(y + x\right)} + 1}}{x + y} \]

      8. associate-+l+ 100.0%

         \[\leadsto \frac{y}{x + y} \cdot \frac{\frac{x}{\color{blue}{y + \left(x + 1\right)}}}{x + y} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{\frac{x}{y + \left(x + 1\right)}}{x + y}} \]

   4. Taylor expanded in y around inf 34.4%

      \[\leadsto \color{blue}{1} \cdot \frac{\frac{x}{y + \left(x + 1\right)}}{x + y} \]

   if 1.05e-134 < y < 4.9999999999999998e-95

   1. Initial program 86.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-+r+ 86.1%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      2. *-commutative 86.1%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)} \]

      3. frac-times 99.8%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

      4. associate-*l/ 86.1%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      5. times-frac 100.0%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{\frac{x}{x + \left(y + 1\right)}}{x + y}} \]

      6. associate-+r+ 100.0%

         \[\leadsto \frac{y}{x + y} \cdot \frac{\frac{x}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      7. +-commutative 100.0%

         \[\leadsto \frac{y}{x + y} \cdot \frac{\frac{x}{\color{blue}{\left(y + x\right)} + 1}}{x + y} \]

      8. associate-+l+ 100.0%

         \[\leadsto \frac{y}{x + y} \cdot \frac{\frac{x}{\color{blue}{y + \left(x + 1\right)}}}{x + y} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{\frac{x}{y + \left(x + 1\right)}}{x + y}} \]
   4. Step-by-step derivation

      1. clear-num 100.0%

         \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{\frac{x + y}{\frac{x}{y + \left(x + 1\right)}}}} \]

      2. un-div-inv 100.0%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{\frac{x + y}{\frac{x}{y + \left(x + 1\right)}}}} \]

      3. +-commutative 100.0%

         \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{\frac{x + y}{\frac{x}{y + \left(x + 1\right)}}} \]

      4. div-inv 99.8%

         \[\leadsto \frac{\frac{y}{y + x}}{\color{blue}{\left(x + y\right) \cdot \frac{1}{\frac{x}{y + \left(x + 1\right)}}}} \]

      5. clear-num 99.8%

         \[\leadsto \frac{\frac{y}{y + x}}{\left(x + y\right) \cdot \color{blue}{\frac{y + \left(x + 1\right)}{x}}} \]

      6. +-commutative 99.8%

         \[\leadsto \frac{\frac{y}{y + x}}{\color{blue}{\left(y + x\right)} \cdot \frac{y + \left(x + 1\right)}{x}} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\left(y + x\right) \cdot \frac{y + \left(x + 1\right)}{x}}} \]

   6. Taylor expanded in y around 0 77.7%

      \[\leadsto \frac{\frac{y}{y + x}}{\color{blue}{1 + x}} \]
   7. Step-by-step derivation

      1. +-commutative 77.7%

         \[\leadsto \frac{\frac{y}{y + x}}{\color{blue}{x + 1}} \]

   8. Simplified 77.7%

      \[\leadsto \frac{\frac{y}{y + x}}{\color{blue}{x + 1}} \]

   if 4.9999999999999998e-95 < y

   1. Initial program 67.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 67.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 67.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 67.1%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 67.1%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 67.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 67.1%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. times-frac 92.6%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 92.6%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 92.6%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. +-commutative 92.6%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ 92.6%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 92.6%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around inf 65.8%

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{x + \left(y + 1\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 65.9%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + \left(y + 1\right)}}{y}} \]

      2. *-un-lft-identity 65.9%

         \[\leadsto \frac{\color{blue}{\frac{x}{x + \left(y + 1\right)}}}{y} \]

      3. associate-+r+ 65.9%

         \[\leadsto \frac{\frac{x}{\color{blue}{\left(x + y\right) + 1}}}{y} \]

      4. +-commutative 65.9%

         \[\leadsto \frac{\frac{x}{\color{blue}{\left(y + x\right)} + 1}}{y} \]

      5. associate-+r+ 65.9%

         \[\leadsto \frac{\frac{x}{\color{blue}{y + \left(x + 1\right)}}}{y} \]

   6. Applied egg-rr 65.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + \left(x + 1\right)}}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{-200}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \]

Alternative 17: 68.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{y \cdot x}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-200}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 4.6:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -1.2e+41)
   (/ x (* y x))
   (if (<= y 3.4e-200)
     (/ y x)
     (if (<= y 4.6) (- (/ x y) x) (* (/ 1.0 y) (/ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -1.2e+41) {
		tmp = x / (y * x);
	} else if (y <= 3.4e-200) {
		tmp = y / x;
	} else if (y <= 4.6) {
		tmp = (x / y) - x;
	} else {
		tmp = (1.0 / y) * (x / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.2d+41)) then
        tmp = x / (y * x)
    else if (y <= 3.4d-200) then
        tmp = y / x
    else if (y <= 4.6d0) then
        tmp = (x / y) - x
    else
        tmp = (1.0d0 / y) * (x / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= -1.2e+41) {
		tmp = x / (y * x);
	} else if (y <= 3.4e-200) {
		tmp = y / x;
	} else if (y <= 4.6) {
		tmp = (x / y) - x;
	} else {
		tmp = (1.0 / y) * (x / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -1.2e+41:
		tmp = x / (y * x)
	elif y <= 3.4e-200:
		tmp = y / x
	elif y <= 4.6:
		tmp = (x / y) - x
	else:
		tmp = (1.0 / y) * (x / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -1.2e+41)
		tmp = Float64(x / Float64(y * x));
	elseif (y <= 3.4e-200)
		tmp = Float64(y / x);
	elseif (y <= 4.6)
		tmp = Float64(Float64(x / y) - x);
	else
		tmp = Float64(Float64(1.0 / y) * Float64(x / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.2e+41)
		tmp = x / (y * x);
	elseif (y <= 3.4e-200)
		tmp = y / x;
	elseif (y <= 4.6)
		tmp = (x / y) - x;
	else
		tmp = (1.0 / y) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -1.2e+41], N[(x / N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.4e-200], N[(y / x), $MachinePrecision], If[LessEqual[y, 4.6], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.2000000000000001e41

   1. Initial program 56.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 56.3%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 56.3%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 56.3%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 56.3%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 56.3%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 56.3%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. times-frac 86.3%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 86.3%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 86.3%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. +-commutative 86.3%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ 86.3%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 86.3%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around inf 68.4%

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{x + \left(y + 1\right)} \]
   5. Step-by-step derivation

      1. *-commutative 68.4%

         \[\leadsto \color{blue}{\frac{x}{x + \left(y + 1\right)} \cdot \frac{1}{y}} \]

      2. associate-+r+ 68.4%

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{1}{y} \]

      3. +-commutative 68.4%

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} + 1} \cdot \frac{1}{y} \]

      4. associate-+r+ 68.4%

         \[\leadsto \frac{x}{\color{blue}{y + \left(x + 1\right)}} \cdot \frac{1}{y} \]

      5. frac-times 81.9%

         \[\leadsto \color{blue}{\frac{x \cdot 1}{\left(y + \left(x + 1\right)\right) \cdot y}} \]

      6. *-rgt-identity 81.9%

         \[\leadsto \frac{\color{blue}{x}}{\left(y + \left(x + 1\right)\right) \cdot y} \]

   6. Applied egg-rr 81.9%

      \[\leadsto \color{blue}{\frac{x}{\left(y + \left(x + 1\right)\right) \cdot y}} \]

   7. Taylor expanded in x around inf 27.6%

      \[\leadsto \frac{x}{\color{blue}{x \cdot y}} \]

   if -1.2000000000000001e41 < y < 3.4000000000000003e-200

   1. Initial program 70.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 70.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 70.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 70.4%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 70.4%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 70.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 70.4%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. times-frac 83.3%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 83.3%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 83.3%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. +-commutative 83.3%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ 83.3%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 83.3%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 70.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 70.9%

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   6. Simplified 70.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]

   7. Taylor expanded in x around 0 50.8%

      \[\leadsto \color{blue}{\frac{y}{x}} \]

   if 3.4000000000000003e-200 < y < 4.5999999999999996

   1. Initial program 83.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 83.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 83.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 83.5%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 83.5%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 83.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 83.5%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. times-frac 97.4%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 97.4%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 97.4%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. +-commutative 97.4%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ 97.4%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 97.4%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around 0 34.3%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 34.3%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   6. Simplified 34.3%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   7. Taylor expanded in y around 0 34.3%

      \[\leadsto \color{blue}{-1 \cdot x + \frac{x}{y}} \]
   8. Step-by-step derivation

      1. neg-mul-1 34.3%

         \[\leadsto \color{blue}{\left(-x\right)} + \frac{x}{y} \]

      2. +-commutative 34.3%

         \[\leadsto \color{blue}{\frac{x}{y} + \left(-x\right)} \]

      3. unsub-neg 34.3%

         \[\leadsto \color{blue}{\frac{x}{y} - x} \]

   9. Simplified 34.3%

      \[\leadsto \color{blue}{\frac{x}{y} - x} \]

   if 4.5999999999999996 < 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-*l* 59.3%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 59.3%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 59.3%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 59.3%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 59.3%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 59.3%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. times-frac 90.3%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 90.3%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 90.3%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. +-commutative 90.3%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ 90.3%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 90.3%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around inf 74.4%

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{x + \left(y + 1\right)} \]

   5. Taylor expanded in y around inf 73.9%

      \[\leadsto \frac{1}{y} \cdot \color{blue}{\frac{x}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{y \cdot x}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-200}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 4.6:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 18: 76.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{-135}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-200}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+155}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -2.05e-135)
   (* (/ y x) (/ 1.0 x))
   (if (<= y 3.9e-200)
     (/ y x)
     (if (<= y 1.2e+155) (/ x (* y (+ y 1.0))) (* (/ 1.0 y) (/ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -2.05e-135) {
		tmp = (y / x) * (1.0 / x);
	} else if (y <= 3.9e-200) {
		tmp = y / x;
	} else if (y <= 1.2e+155) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (1.0 / y) * (x / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.05d-135)) then
        tmp = (y / x) * (1.0d0 / x)
    else if (y <= 3.9d-200) then
        tmp = y / x
    else if (y <= 1.2d+155) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (1.0d0 / y) * (x / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= -2.05e-135) {
		tmp = (y / x) * (1.0 / x);
	} else if (y <= 3.9e-200) {
		tmp = y / x;
	} else if (y <= 1.2e+155) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (1.0 / y) * (x / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -2.05e-135:
		tmp = (y / x) * (1.0 / x)
	elif y <= 3.9e-200:
		tmp = y / x
	elif y <= 1.2e+155:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (1.0 / y) * (x / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -2.05e-135)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	elseif (y <= 3.9e-200)
		tmp = Float64(y / x);
	elseif (y <= 1.2e+155)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(1.0 / y) * Float64(x / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.05e-135)
		tmp = (y / x) * (1.0 / x);
	elseif (y <= 3.9e-200)
		tmp = y / x;
	elseif (y <= 1.2e+155)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (1.0 / y) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -2.05e-135], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.9e-200], N[(y / x), $MachinePrecision], If[LessEqual[y, 1.2e+155], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.05000000000000005e-135

   1. Initial program 70.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-+r+ 70.6%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      2. *-commutative 70.6%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)} \]

      3. frac-times 92.5%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

      4. associate-*l/ 88.6%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      5. times-frac 99.8%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{\frac{x}{x + \left(y + 1\right)}}{x + y}} \]

      6. associate-+r+ 99.8%

         \[\leadsto \frac{y}{x + y} \cdot \frac{\frac{x}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

      7. +-commutative 99.8%

         \[\leadsto \frac{y}{x + y} \cdot \frac{\frac{x}{\color{blue}{\left(y + x\right)} + 1}}{x + y} \]

      8. associate-+l+ 99.8%

         \[\leadsto \frac{y}{x + y} \cdot \frac{\frac{x}{\color{blue}{y + \left(x + 1\right)}}}{x + y} \]

   3. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{\frac{x}{y + \left(x + 1\right)}}{x + y}} \]

   4. Taylor expanded in x around inf 41.6%

      \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{x}} \]

   5. Taylor expanded in y around 0 41.0%

      \[\leadsto \color{blue}{\frac{y}{x}} \cdot \frac{1}{x} \]

   if -2.05000000000000005e-135 < y < 3.89999999999999999e-200

   1. Initial program 58.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 58.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 58.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 58.8%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 58.8%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 58.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 58.8%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. times-frac 71.8%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 71.8%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 71.8%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. +-commutative 71.8%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ 71.8%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 71.8%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 86.3%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 86.3%

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   6. Simplified 86.3%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]

   7. Taylor expanded in x around 0 80.1%

      \[\leadsto \color{blue}{\frac{y}{x}} \]

   if 3.89999999999999999e-200 < y < 1.2000000000000001e155

   1. Initial program 72.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 72.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 72.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 72.7%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 72.7%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 72.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 72.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. times-frac 95.2%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 95.2%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 95.2%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. +-commutative 95.2%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ 95.2%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 95.2%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around 0 42.4%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 42.4%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   6. Simplified 42.4%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   if 1.2000000000000001e155 < y

   1. Initial program 59.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 59.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 59.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 59.8%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 59.8%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 59.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 59.8%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. times-frac 88.2%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 88.2%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 88.2%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. +-commutative 88.2%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ 88.2%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 88.2%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around inf 96.0%

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{x + \left(y + 1\right)} \]

   5. Taylor expanded in y around inf 96.0%

      \[\leadsto \frac{1}{y} \cdot \color{blue}{\frac{x}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{-135}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-200}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+155}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 19: 46.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{y \cdot x}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-200}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -1.2e+41) (/ x (* y x)) (if (<= y 3.9e-200) (/ y x) (/ x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -1.2e+41) {
		tmp = x / (y * x);
	} else if (y <= 3.9e-200) {
		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 <= (-1.2d+41)) then
        tmp = x / (y * x)
    else if (y <= 3.9d-200) 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 <= -1.2e+41) {
		tmp = x / (y * x);
	} else if (y <= 3.9e-200) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -1.2e+41:
		tmp = x / (y * x)
	elif y <= 3.9e-200:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -1.2e+41)
		tmp = Float64(x / Float64(y * x));
	elseif (y <= 3.9e-200)
		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 <= -1.2e+41)
		tmp = x / (y * x);
	elseif (y <= 3.9e-200)
		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, -1.2e+41], N[(x / N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.9e-200], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2000000000000001e41

   1. Initial program 56.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 56.3%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 56.3%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 56.3%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 56.3%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 56.3%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 56.3%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. times-frac 86.3%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 86.3%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 86.3%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. +-commutative 86.3%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ 86.3%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 86.3%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around inf 68.4%

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{x + \left(y + 1\right)} \]
   5. Step-by-step derivation

      1. *-commutative 68.4%

         \[\leadsto \color{blue}{\frac{x}{x + \left(y + 1\right)} \cdot \frac{1}{y}} \]

      2. associate-+r+ 68.4%

         \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{1}{y} \]

      3. +-commutative 68.4%

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} + 1} \cdot \frac{1}{y} \]

      4. associate-+r+ 68.4%

         \[\leadsto \frac{x}{\color{blue}{y + \left(x + 1\right)}} \cdot \frac{1}{y} \]

      5. frac-times 81.9%

         \[\leadsto \color{blue}{\frac{x \cdot 1}{\left(y + \left(x + 1\right)\right) \cdot y}} \]

      6. *-rgt-identity 81.9%

         \[\leadsto \frac{\color{blue}{x}}{\left(y + \left(x + 1\right)\right) \cdot y} \]

   6. Applied egg-rr 81.9%

      \[\leadsto \color{blue}{\frac{x}{\left(y + \left(x + 1\right)\right) \cdot y}} \]

   7. Taylor expanded in x around inf 27.6%

      \[\leadsto \frac{x}{\color{blue}{x \cdot y}} \]

   if -1.2000000000000001e41 < y < 3.89999999999999999e-200

   1. Initial program 70.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 70.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 70.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 70.4%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 70.4%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 70.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 70.4%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. times-frac 83.3%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 83.3%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 83.3%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. +-commutative 83.3%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ 83.3%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 83.3%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 70.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 70.9%

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   6. Simplified 70.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]

   7. Taylor expanded in x around 0 50.8%

      \[\leadsto \color{blue}{\frac{y}{x}} \]

   if 3.89999999999999999e-200 < y

   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-*l* 68.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 68.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 68.5%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 68.5%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 68.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 68.5%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. times-frac 93.0%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 93.0%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 93.0%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. +-commutative 93.0%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ 93.0%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 93.0%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around 0 57.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 57.1%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   6. Simplified 57.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   7. Taylor expanded in y around 0 29.4%

      \[\leadsto \frac{x}{\color{blue}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{y \cdot x}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-200}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 20: 44.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{-200}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 3.9e-200) (/ y x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.9e-200) {
		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 <= 3.9d-200) 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 <= 3.9e-200) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.9e-200:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.9e-200)
		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 <= 3.9e-200)
		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, 3.9e-200], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.89999999999999999e-200

   1. Initial program 65.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 65.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 65.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 65.9%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 65.9%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 65.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 65.9%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. times-frac 84.3%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 84.3%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 84.3%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. +-commutative 84.3%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ 84.3%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 84.3%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 56.4%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 56.4%

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   6. Simplified 56.4%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]

   7. Taylor expanded in x around 0 35.4%

      \[\leadsto \color{blue}{\frac{y}{x}} \]

   if 3.89999999999999999e-200 < y

   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-*l* 68.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 68.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 68.5%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 68.5%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 68.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 68.5%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. times-frac 93.0%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 93.0%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 93.0%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. +-commutative 93.0%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ 93.0%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 93.0%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around 0 57.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 57.1%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   6. Simplified 57.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   7. Taylor expanded in y around 0 29.4%

      \[\leadsto \frac{x}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{-200}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 21: 4.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{1}{x} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ 1.0 x))

C

assert(x < y);
double code(double x, double y) {
	return 1.0 / x;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / x
end function

Java

assert x < y;
public static double code(double x, double y) {
	return 1.0 / x;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return 1.0 / x

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(1.0 / x)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 1.0 / x;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(1.0 / x), $MachinePrecision]

Derivation

1. Initial program 67.1%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-+r+ 67.1%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   2. *-commutative 67.1%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)} \]

   3. frac-times 88.2%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   4. associate-*l/ 81.8%

      \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   5. times-frac 99.8%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{\frac{x}{x + \left(y + 1\right)}}{x + y}} \]

   6. associate-+r+ 99.8%

      \[\leadsto \frac{y}{x + y} \cdot \frac{\frac{x}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

   7. +-commutative 99.8%

      \[\leadsto \frac{y}{x + y} \cdot \frac{\frac{x}{\color{blue}{\left(y + x\right)} + 1}}{x + y} \]

   8. associate-+l+ 99.8%

      \[\leadsto \frac{y}{x + y} \cdot \frac{\frac{x}{\color{blue}{y + \left(x + 1\right)}}}{x + y} \]

3. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{\frac{x}{y + \left(x + 1\right)}}{x + y}} \]

4. Taylor expanded in x around inf 40.6%

   \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{x}} \]

5. Taylor expanded in y around inf 4.4%

   \[\leadsto \color{blue}{\frac{1}{x}} \]

6. Final simplification 4.4%

   \[\leadsto \frac{1}{x} \]

Alternative 22: 4.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{1}{y} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ 1.0 y))

C

assert(x < y);
double code(double x, double y) {
	return 1.0 / y;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / y
end function

Java

assert x < y;
public static double code(double x, double y) {
	return 1.0 / y;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return 1.0 / y

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(1.0 / y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 1.0 / y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(1.0 / y), $MachinePrecision]

Derivation

1. Initial program 67.1%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 67.1%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

   2. +-commutative 67.1%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

   3. +-commutative 67.1%

      \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

   4. +-commutative 67.1%

      \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

   5. associate-*l* 67.1%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

   6. *-commutative 67.1%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

   7. times-frac 88.2%

      \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

   8. +-commutative 88.2%

      \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

   9. +-commutative 88.2%

      \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

   10. +-commutative 88.2%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

   11. associate-+l+ 88.2%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

3. Simplified 88.2%

   \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

4. Taylor expanded in y around inf 50.0%

   \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{x + \left(y + 1\right)} \]

5. Taylor expanded in x around inf 4.3%

   \[\leadsto \color{blue}{\frac{1}{y}} \]

6. Final simplification 4.3%

   \[\leadsto \frac{1}{y} \]

Alternative 23: 26.6% 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 67.1%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 67.1%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

   2. +-commutative 67.1%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

   3. +-commutative 67.1%

      \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

   4. +-commutative 67.1%

      \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

   5. associate-*l* 67.1%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

   6. *-commutative 67.1%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

   7. times-frac 88.2%

      \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

   8. +-commutative 88.2%

      \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

   9. +-commutative 88.2%

      \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

   10. +-commutative 88.2%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

   11. associate-+l+ 88.2%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

3. Simplified 88.2%

   \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

4. Taylor expanded in x around 0 50.9%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
5. Step-by-step derivation

   1. +-commutative 50.9%

      \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

6. Simplified 50.9%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

7. Taylor expanded in y around 0 25.9%

   \[\leadsto \frac{x}{\color{blue}{y}} \]

8. Final simplification 25.9%

   \[\leadsto \frac{x}{y} \]

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 2023322 
(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))))