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

Percentage Accurate: 68.8% → 99.8%

Time: 15.7s

Alternatives: 17

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.8% 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

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

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)) (+ y (+ x 1.0)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 68.8%

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

   1. *-commutative 68.8%

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

   2. associate-*l* 68.8%

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

   3. times-frac 93.4%

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

   4. +-commutative 93.4%

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

   5. +-commutative 93.4%

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

   6. associate-+r+ 93.4%

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

   7. +-commutative 93.4%

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

   8. associate-+l+ 93.4%

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

4. Applied egg-rr 93.4%

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

   1. div-inv 93.3%

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

   2. distribute-rgt-in 88.8%

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

   3. +-commutative 88.8%

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

   4. distribute-rgt-in 93.3%

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

   5. +-commutative 93.3%

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

6. Applied egg-rr 93.3%

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

   1. associate-*r/ 93.4%

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

   2. *-rgt-identity 93.4%

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

   3. associate-/r* 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 92.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{\frac{x}{y + x}}{y + \left(x + 1\right)}\\ \mathbf{if}\;x \leq -8.8 \cdot 10^{+73}:\\ \;\;\;\;t\_0 \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-163}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ x (+ y x)) (+ y (+ x 1.0)))))
   (if (<= x -8.8e+73)
     (* t_0 (/ y x))
     (if (<= x -4.4e-163)
       (* x (/ y (* (+ x (+ y 1.0)) (* (+ y x) (+ y x)))))
       t_0))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (x / (y + x)) / (y + (x + 1.0));
	double tmp;
	if (x <= -8.8e+73) {
		tmp = t_0 * (y / x);
	} else if (x <= -4.4e-163) {
		tmp = x * (y / ((x + (y + 1.0)) * ((y + x) * (y + x))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / (y + x)) / (y + (x + 1.0d0))
    if (x <= (-8.8d+73)) then
        tmp = t_0 * (y / x)
    else if (x <= (-4.4d-163)) then
        tmp = x * (y / ((x + (y + 1.0d0)) * ((y + x) * (y + x))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (x / (y + x)) / (y + (x + 1.0));
	double tmp;
	if (x <= -8.8e+73) {
		tmp = t_0 * (y / x);
	} else if (x <= -4.4e-163) {
		tmp = x * (y / ((x + (y + 1.0)) * ((y + x) * (y + x))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (x / (y + x)) / (y + (x + 1.0))
	tmp = 0
	if x <= -8.8e+73:
		tmp = t_0 * (y / x)
	elif x <= -4.4e-163:
		tmp = x * (y / ((x + (y + 1.0)) * ((y + x) * (y + x))))
	else:
		tmp = t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(x / Float64(y + x)) / Float64(y + Float64(x + 1.0)))
	tmp = 0.0
	if (x <= -8.8e+73)
		tmp = Float64(t_0 * Float64(y / x));
	elseif (x <= -4.4e-163)
		tmp = Float64(x * Float64(y / Float64(Float64(x + Float64(y + 1.0)) * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (x / (y + x)) / (y + (x + 1.0));
	tmp = 0.0;
	if (x <= -8.8e+73)
		tmp = t_0 * (y / x);
	elseif (x <= -4.4e-163)
		tmp = x * (y / ((x + (y + 1.0)) * ((y + x) * (y + x))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.8e+73], N[(t$95$0 * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.4e-163], N[(x * N[(y / N[(N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.8e73

   1. Initial program 40.6%

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

      1. *-commutative 40.6%

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

      2. associate-*l* 40.6%

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

      3. times-frac 81.2%

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

      4. +-commutative 81.2%

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

      5. +-commutative 81.2%

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

      6. associate-+r+ 81.2%

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

      7. +-commutative 81.2%

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

      8. associate-+l+ 81.2%

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

   4. Applied egg-rr 81.2%

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

      1. div-inv 81.2%

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

      2. distribute-rgt-in 69.6%

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

      3. +-commutative 69.6%

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

      4. distribute-rgt-in 81.2%

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

      5. +-commutative 81.2%

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

   6. Applied egg-rr 81.2%

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

      1. associate-*r/ 81.2%

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

      2. *-rgt-identity 81.2%

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

      3. associate-/r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in y around 0 89.0%

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

   if -8.8e73 < x < -4.40000000000000022e-163

   1. Initial program 89.3%

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

      1. associate-/l* 96.7%

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

      2. associate-+l+ 96.7%

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

   3. Simplified 96.7%

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

   if -4.40000000000000022e-163 < x

   1. Initial program 69.1%

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

      1. *-commutative 69.1%

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

      2. associate-*l* 69.1%

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

      3. times-frac 95.0%

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

      4. +-commutative 95.0%

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

      5. +-commutative 95.0%

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

      6. associate-+r+ 95.0%

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

      7. +-commutative 95.0%

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

      8. associate-+l+ 95.0%

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

   4. Applied egg-rr 95.0%

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

      1. div-inv 94.8%

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

      2. distribute-rgt-in 92.1%

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

      3. +-commutative 92.1%

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

      4. distribute-rgt-in 94.8%

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

      5. +-commutative 94.8%

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

   6. Applied egg-rr 94.8%

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

      1. associate-*r/ 95.0%

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

      2. *-rgt-identity 95.0%

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

      3. associate-/r* 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in y around inf 54.9%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{+73}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + 1\right)} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-163}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{+164}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot t\_0} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -5e+164)
     (* (/ y (+ y x)) (/ 1.0 x))
     (if (<= x -5.05e-138)
       (* (/ x (* (+ y x) t_0)) (/ y x))
       (/ (/ x (+ y x)) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -5e+164) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -5.05e-138) {
		tmp = (x / ((y + x) * t_0)) * (y / x);
	} else {
		tmp = (x / (y + x)) / t_0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (x <= (-5d+164)) then
        tmp = (y / (y + x)) * (1.0d0 / x)
    else if (x <= (-5.05d-138)) then
        tmp = (x / ((y + x) * t_0)) * (y / x)
    else
        tmp = (x / (y + x)) / t_0
    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 (x <= -5e+164) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -5.05e-138) {
		tmp = (x / ((y + x) * t_0)) * (y / x);
	} else {
		tmp = (x / (y + x)) / t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -5e+164:
		tmp = (y / (y + x)) * (1.0 / x)
	elif x <= -5.05e-138:
		tmp = (x / ((y + x) * t_0)) * (y / x)
	else:
		tmp = (x / (y + x)) / t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -5e+164)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / x));
	elseif (x <= -5.05e-138)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * t_0)) * Float64(y / x));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / t_0);
	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 (x <= -5e+164)
		tmp = (y / (y + x)) * (1.0 / x);
	elseif (x <= -5.05e-138)
		tmp = (x / ((y + x) * t_0)) * (y / x);
	else
		tmp = (x / (y + x)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5e+164], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.05e-138], N[(N[(x / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999995e164

   1. Initial program 42.1%

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

      1. *-commutative 42.1%

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

      2. associate-*l* 42.1%

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

      3. times-frac 72.2%

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

      4. +-commutative 72.2%

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

      5. +-commutative 72.2%

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

      6. associate-+r+ 72.2%

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

      7. +-commutative 72.2%

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

      8. associate-+l+ 72.2%

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

   4. Applied egg-rr 72.2%

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

   5. Taylor expanded in x around inf 87.0%

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

   if -4.9999999999999995e164 < x < -5.0499999999999997e-138

   1. Initial program 77.2%

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

      1. *-commutative 77.2%

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

      2. associate-*l* 77.2%

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

      3. times-frac 98.4%

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

      4. +-commutative 98.4%

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

      5. +-commutative 98.4%

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

      6. associate-+r+ 98.4%

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

      7. +-commutative 98.4%

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

      8. associate-+l+ 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in y around 0 72.2%

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

   if -5.0499999999999997e-138 < x

   1. Initial program 70.2%

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

      1. *-commutative 70.2%

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

      2. associate-*l* 70.2%

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

      3. times-frac 95.2%

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

      4. +-commutative 95.2%

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

      5. +-commutative 95.2%

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

      6. associate-+r+ 95.2%

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

      7. +-commutative 95.2%

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

      8. associate-+l+ 95.2%

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

   4. Applied egg-rr 95.2%

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

      1. div-inv 95.0%

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

      2. distribute-rgt-in 92.4%

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

      3. +-commutative 92.4%

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

      4. distribute-rgt-in 95.0%

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

      5. +-commutative 95.0%

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

   6. Applied egg-rr 95.0%

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

      1. associate-*r/ 95.2%

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

      2. *-rgt-identity 95.2%

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

      3. associate-/r* 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in y around inf 56.5%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+164}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+164}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -2500000000:\\ \;\;\;\;\frac{y}{x \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5e+164)
   (* (/ y (+ y x)) (/ 1.0 x))
   (if (<= x -2500000000.0)
     (/ y (* x (+ y x)))
     (if (<= x -1.1e-138) (/ y (* x (+ x 1.0))) (/ (/ x (+ y x)) (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5e+164) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -2500000000.0) {
		tmp = y / (x * (y + x));
	} else if (x <= -1.1e-138) {
		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 (x <= (-5d+164)) then
        tmp = (y / (y + x)) * (1.0d0 / x)
    else if (x <= (-2500000000.0d0)) then
        tmp = y / (x * (y + x))
    else if (x <= (-1.1d-138)) 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 (x <= -5e+164) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -2500000000.0) {
		tmp = y / (x * (y + x));
	} else if (x <= -1.1e-138) {
		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 x <= -5e+164:
		tmp = (y / (y + x)) * (1.0 / x)
	elif x <= -2500000000.0:
		tmp = y / (x * (y + x))
	elif x <= -1.1e-138:
		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 (x <= -5e+164)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / x));
	elseif (x <= -2500000000.0)
		tmp = Float64(y / Float64(x * Float64(y + x)));
	elseif (x <= -1.1e-138)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5e+164)
		tmp = (y / (y + x)) * (1.0 / x);
	elseif (x <= -2500000000.0)
		tmp = y / (x * (y + x));
	elseif (x <= -1.1e-138)
		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[x, -5e+164], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2500000000.0], N[(y / N[(x * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.1e-138], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.9999999999999995e164

   1. Initial program 42.1%

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

      1. *-commutative 42.1%

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

      2. associate-*l* 42.1%

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

      3. times-frac 72.2%

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

      4. +-commutative 72.2%

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

      5. +-commutative 72.2%

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

      6. associate-+r+ 72.2%

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

      7. +-commutative 72.2%

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

      8. associate-+l+ 72.2%

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

   4. Applied egg-rr 72.2%

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

   5. Taylor expanded in x around inf 87.0%

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

   if -4.9999999999999995e164 < x < -2.5e9

   1. Initial program 57.6%

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

      1. *-commutative 57.6%

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

      2. associate-*l* 57.6%

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

      3. times-frac 96.7%

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

      4. +-commutative 96.7%

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

      5. +-commutative 96.7%

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

      6. associate-+r+ 96.7%

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

      7. +-commutative 96.7%

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

      8. associate-+l+ 96.7%

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

   4. Applied egg-rr 96.7%

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

   5. Taylor expanded in x around inf 42.9%

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

      1. frac-times 68.6%

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

      2. *-rgt-identity 68.6%

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

   7. Applied egg-rr 68.6%

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

   if -2.5e9 < x < -1.0999999999999999e-138

   1. Initial program 93.9%

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

      1. associate-/l* 97.5%

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

      2. associate-+l+ 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in y around 0 53.1%

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

   if -1.0999999999999999e-138 < x

   1. Initial program 70.2%

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

      1. *-commutative 70.2%

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

      2. associate-*l* 70.2%

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

      3. times-frac 95.2%

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

      4. +-commutative 95.2%

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

      5. +-commutative 95.2%

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

      6. associate-+r+ 95.2%

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

      7. +-commutative 95.2%

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

      8. associate-+l+ 95.2%

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

   4. Applied egg-rr 95.2%

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

      1. div-inv 95.0%

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

      2. distribute-rgt-in 92.4%

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

      3. +-commutative 92.4%

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

      4. distribute-rgt-in 95.0%

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

      5. +-commutative 95.0%

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

   6. Applied egg-rr 95.0%

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

      1. associate-*r/ 95.2%

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

      2. *-rgt-identity 95.2%

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

      3. associate-/r* 99.9%

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

   8. Simplified 99.9%

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

      1. clear-num 99.8%

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

      2. +-commutative 99.8%

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

      3. frac-times 99.5%

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

      4. *-un-lft-identity 99.5%

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

      5. +-commutative 99.5%

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

      6. +-commutative 99.5%

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

   10. Applied egg-rr 99.5%

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

   11. Taylor expanded in x around 0 55.9%

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

       1. +-commutative 55.9%

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

   13. Simplified 55.9%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+164}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -2500000000:\\ \;\;\;\;\frac{y}{x \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+18}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-9} \lor \neg \left(x \leq -5.05 \cdot 10^{-138}\right):\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.8e+18)
   (* (/ y x) (/ 1.0 x))
   (if (or (<= x -1.6e-9) (not (<= x -5.05e-138)))
     (/ x (* y (+ y 1.0)))
     (/ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.8e+18) {
		tmp = (y / x) * (1.0 / x);
	} else if ((x <= -1.6e-9) || !(x <= -5.05e-138)) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = y / x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.8d+18)) then
        tmp = (y / x) * (1.0d0 / x)
    else if ((x <= (-1.6d-9)) .or. (.not. (x <= (-5.05d-138)))) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = y / x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.8e+18) {
		tmp = (y / x) * (1.0 / x);
	} else if ((x <= -1.6e-9) || !(x <= -5.05e-138)) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = y / x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.8e+18:
		tmp = (y / x) * (1.0 / x)
	elif (x <= -1.6e-9) or not (x <= -5.05e-138):
		tmp = x / (y * (y + 1.0))
	else:
		tmp = y / x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.8e+18)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	elseif ((x <= -1.6e-9) || !(x <= -5.05e-138))
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(y / x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.8e+18)
		tmp = (y / x) * (1.0 / x);
	elseif ((x <= -1.6e-9) || ~((x <= -5.05e-138)))
		tmp = x / (y * (y + 1.0));
	else
		tmp = y / x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.8e+18], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -1.6e-9], N[Not[LessEqual[x, -5.05e-138]], $MachinePrecision]], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.8e18

   1. Initial program 48.3%

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

      1. *-commutative 48.3%

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

      2. associate-*l* 48.3%

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

      3. times-frac 84.1%

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

      4. +-commutative 84.1%

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

      5. +-commutative 84.1%

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

      6. associate-+r+ 84.1%

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

      7. +-commutative 84.1%

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

      8. associate-+l+ 84.1%

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

   4. Applied egg-rr 84.1%

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

   5. Taylor expanded in x around inf 66.7%

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

   6. Taylor expanded in y around 0 65.9%

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

   if -1.8e18 < x < -1.60000000000000006e-9 or -5.0499999999999997e-138 < x

   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* 83.8%

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

      2. associate-+l+ 83.8%

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

   3. Simplified 83.8%

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

   5. Taylor expanded in x around 0 53.7%

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

      1. +-commutative 53.7%

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

   7. Simplified 53.7%

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

   if -1.60000000000000006e-9 < x < -5.0499999999999997e-138

   1. Initial program 94.9%

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

      1. associate-/l* 99.4%

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

      2. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around 0 53.6%

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

      1. associate-/r* 53.6%

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

      2. +-commutative 53.6%

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

   7. Simplified 53.6%

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

   8. Taylor expanded in x around 0 53.2%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+18}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-9} \lor \neg \left(x \leq -5.05 \cdot 10^{-138}\right):\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+164}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -1550000000:\\ \;\;\;\;\frac{y}{x \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5e+164)
   (* (/ y x) (/ 1.0 x))
   (if (<= x -1550000000.0)
     (/ y (* x (+ y x)))
     (if (<= x -5.05e-138) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5e+164) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -1550000000.0) {
		tmp = y / (x * (y + x));
	} else if (x <= -5.05e-138) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5d+164)) then
        tmp = (y / x) * (1.0d0 / x)
    else if (x <= (-1550000000.0d0)) then
        tmp = y / (x * (y + x))
    else if (x <= (-5.05d-138)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -5e+164) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -1550000000.0) {
		tmp = y / (x * (y + x));
	} else if (x <= -5.05e-138) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -5e+164:
		tmp = (y / x) * (1.0 / x)
	elif x <= -1550000000.0:
		tmp = y / (x * (y + x))
	elif x <= -5.05e-138:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5e+164)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	elseif (x <= -1550000000.0)
		tmp = Float64(y / Float64(x * Float64(y + x)));
	elseif (x <= -5.05e-138)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5e+164)
		tmp = (y / x) * (1.0 / x);
	elseif (x <= -1550000000.0)
		tmp = y / (x * (y + x));
	elseif (x <= -5.05e-138)
		tmp = y / (x * (x + 1.0));
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -5e+164], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1550000000.0], N[(y / N[(x * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.05e-138], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.9999999999999995e164

   1. Initial program 42.1%

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

      1. *-commutative 42.1%

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

      2. associate-*l* 42.1%

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

      3. times-frac 72.2%

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

      4. +-commutative 72.2%

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

      5. +-commutative 72.2%

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

      6. associate-+r+ 72.2%

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

      7. +-commutative 72.2%

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

      8. associate-+l+ 72.2%

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

   4. Applied egg-rr 72.2%

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

   5. Taylor expanded in x around inf 87.0%

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

   6. Taylor expanded in y around 0 86.7%

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

   if -4.9999999999999995e164 < x < -1.55e9

   1. Initial program 57.6%

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

      1. *-commutative 57.6%

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

      2. associate-*l* 57.6%

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

      3. times-frac 96.7%

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

      4. +-commutative 96.7%

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

      5. +-commutative 96.7%

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

      6. associate-+r+ 96.7%

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

      7. +-commutative 96.7%

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

      8. associate-+l+ 96.7%

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

   4. Applied egg-rr 96.7%

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

   5. Taylor expanded in x around inf 42.9%

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

      1. frac-times 68.6%

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

      2. *-rgt-identity 68.6%

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

   7. Applied egg-rr 68.6%

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

   if -1.55e9 < x < -5.0499999999999997e-138

   1. Initial program 93.9%

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

      1. associate-/l* 97.5%

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

      2. associate-+l+ 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in y around 0 53.1%

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

   if -5.0499999999999997e-138 < x

   1. Initial program 70.2%

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

      1. associate-/l* 83.4%

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

      2. associate-+l+ 83.4%

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

   3. Simplified 83.4%

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

   5. Taylor expanded in x around 0 53.5%

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

      1. +-commutative 53.5%

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

   7. Simplified 53.5%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+164}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -1550000000:\\ \;\;\;\;\frac{y}{x \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+164}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -2500000000:\\ \;\;\;\;\frac{y}{x \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -8e+164)
   (* (/ y x) (/ 1.0 x))
   (if (<= x -2500000000.0)
     (/ y (* x (+ y x)))
     (if (<= x -5.05e-138) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -8e+164) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -2500000000.0) {
		tmp = y / (x * (y + x));
	} else if (x <= -5.05e-138) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-8d+164)) then
        tmp = (y / x) * (1.0d0 / x)
    else if (x <= (-2500000000.0d0)) then
        tmp = y / (x * (y + x))
    else if (x <= (-5.05d-138)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -8e+164) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -2500000000.0) {
		tmp = y / (x * (y + x));
	} else if (x <= -5.05e-138) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -8e+164:
		tmp = (y / x) * (1.0 / x)
	elif x <= -2500000000.0:
		tmp = y / (x * (y + x))
	elif x <= -5.05e-138:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -8e+164)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	elseif (x <= -2500000000.0)
		tmp = Float64(y / Float64(x * Float64(y + x)));
	elseif (x <= -5.05e-138)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8e+164)
		tmp = (y / x) * (1.0 / x);
	elseif (x <= -2500000000.0)
		tmp = y / (x * (y + x));
	elseif (x <= -5.05e-138)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -8e+164], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2500000000.0], N[(y / N[(x * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.05e-138], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8e164

   1. Initial program 42.1%

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

      1. *-commutative 42.1%

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

      2. associate-*l* 42.1%

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

      3. times-frac 72.2%

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

      4. +-commutative 72.2%

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

      5. +-commutative 72.2%

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

      6. associate-+r+ 72.2%

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

      7. +-commutative 72.2%

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

      8. associate-+l+ 72.2%

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

   4. Applied egg-rr 72.2%

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

   5. Taylor expanded in x around inf 87.0%

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

   6. Taylor expanded in y around 0 86.7%

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

   if -8e164 < x < -2.5e9

   1. Initial program 57.6%

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

      1. *-commutative 57.6%

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

      2. associate-*l* 57.6%

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

      3. times-frac 96.7%

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

      4. +-commutative 96.7%

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

      5. +-commutative 96.7%

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

      6. associate-+r+ 96.7%

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

      7. +-commutative 96.7%

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

      8. associate-+l+ 96.7%

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

   4. Applied egg-rr 96.7%

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

   5. Taylor expanded in x around inf 42.9%

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

      1. frac-times 68.6%

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

      2. *-rgt-identity 68.6%

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

   7. Applied egg-rr 68.6%

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

   if -2.5e9 < x < -5.0499999999999997e-138

   1. Initial program 93.9%

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

      1. associate-/l* 97.5%

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

      2. associate-+l+ 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in y around 0 53.1%

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

   if -5.0499999999999997e-138 < x

   1. Initial program 70.2%

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

      1. *-commutative 70.2%

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

      2. associate-*l* 70.2%

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

      3. times-frac 95.2%

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

      4. +-commutative 95.2%

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

      5. +-commutative 95.2%

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

      6. associate-+r+ 95.2%

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

      7. +-commutative 95.2%

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

      8. associate-+l+ 95.2%

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

   4. Applied egg-rr 95.2%

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

      1. div-inv 95.0%

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

      2. distribute-rgt-in 92.4%

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

      3. +-commutative 92.4%

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

      4. distribute-rgt-in 95.0%

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

      5. +-commutative 95.0%

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

   6. Applied egg-rr 95.0%

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

      1. associate-*r/ 95.2%

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

      2. *-rgt-identity 95.2%

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

      3. associate-/r* 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around 0 53.5%

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

       1. associate-/r* 55.7%

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

       2. +-commutative 55.7%

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

   11. Simplified 55.7%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+164}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -2500000000:\\ \;\;\;\;\frac{y}{x \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+164}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -950000000:\\ \;\;\;\;\frac{y}{x \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5e+164)
   (* (/ y (+ y x)) (/ 1.0 x))
   (if (<= x -950000000.0)
     (/ y (* x (+ y x)))
     (if (<= x -5.05e-138) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5e+164) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -950000000.0) {
		tmp = y / (x * (y + x));
	} else if (x <= -5.05e-138) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5d+164)) then
        tmp = (y / (y + x)) * (1.0d0 / x)
    else if (x <= (-950000000.0d0)) then
        tmp = y / (x * (y + x))
    else if (x <= (-5.05d-138)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -5e+164) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -950000000.0) {
		tmp = y / (x * (y + x));
	} else if (x <= -5.05e-138) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -5e+164:
		tmp = (y / (y + x)) * (1.0 / x)
	elif x <= -950000000.0:
		tmp = y / (x * (y + x))
	elif x <= -5.05e-138:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5e+164)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / x));
	elseif (x <= -950000000.0)
		tmp = Float64(y / Float64(x * Float64(y + x)));
	elseif (x <= -5.05e-138)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5e+164)
		tmp = (y / (y + x)) * (1.0 / x);
	elseif (x <= -950000000.0)
		tmp = y / (x * (y + x));
	elseif (x <= -5.05e-138)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -5e+164], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -950000000.0], N[(y / N[(x * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.05e-138], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.9999999999999995e164

   1. Initial program 42.1%

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

      1. *-commutative 42.1%

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

      2. associate-*l* 42.1%

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

      3. times-frac 72.2%

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

      4. +-commutative 72.2%

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

      5. +-commutative 72.2%

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

      6. associate-+r+ 72.2%

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

      7. +-commutative 72.2%

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

      8. associate-+l+ 72.2%

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

   4. Applied egg-rr 72.2%

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

   5. Taylor expanded in x around inf 87.0%

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

   if -4.9999999999999995e164 < x < -9.5e8

   1. Initial program 57.6%

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

      1. *-commutative 57.6%

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

      2. associate-*l* 57.6%

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

      3. times-frac 96.7%

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

      4. +-commutative 96.7%

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

      5. +-commutative 96.7%

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

      6. associate-+r+ 96.7%

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

      7. +-commutative 96.7%

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

      8. associate-+l+ 96.7%

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

   4. Applied egg-rr 96.7%

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

   5. Taylor expanded in x around inf 42.9%

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

      1. frac-times 68.6%

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

      2. *-rgt-identity 68.6%

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

   7. Applied egg-rr 68.6%

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

   if -9.5e8 < x < -5.0499999999999997e-138

   1. Initial program 93.9%

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

      1. associate-/l* 97.5%

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

      2. associate-+l+ 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in y around 0 53.1%

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

   if -5.0499999999999997e-138 < x

   1. Initial program 70.2%

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

      1. *-commutative 70.2%

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

      2. associate-*l* 70.2%

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

      3. times-frac 95.2%

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

      4. +-commutative 95.2%

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

      5. +-commutative 95.2%

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

      6. associate-+r+ 95.2%

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

      7. +-commutative 95.2%

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

      8. associate-+l+ 95.2%

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

   4. Applied egg-rr 95.2%

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

      1. div-inv 95.0%

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

      2. distribute-rgt-in 92.4%

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

      3. +-commutative 92.4%

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

      4. distribute-rgt-in 95.0%

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

      5. +-commutative 95.0%

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

   6. Applied egg-rr 95.0%

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

      1. associate-*r/ 95.2%

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

      2. *-rgt-identity 95.2%

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

      3. associate-/r* 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around 0 53.5%

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

       1. associate-/r* 55.7%

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

       2. +-commutative 55.7%

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

   11. Simplified 55.7%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+164}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -950000000:\\ \;\;\;\;\frac{y}{x \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 94.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+123}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + 1\right)} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.4e+123)
   (* (/ (/ x (+ y x)) (+ y (+ x 1.0))) (/ y x))
   (* x (/ (/ y (* (+ y x) (+ x (+ y 1.0)))) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.4e+123) {
		tmp = ((x / (y + x)) / (y + (x + 1.0))) * (y / x);
	} else {
		tmp = x * ((y / ((y + x) * (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 (x <= (-2.4d+123)) then
        tmp = ((x / (y + x)) / (y + (x + 1.0d0))) * (y / x)
    else
        tmp = x * ((y / ((y + x) * (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 (x <= -2.4e+123) {
		tmp = ((x / (y + x)) / (y + (x + 1.0))) * (y / x);
	} else {
		tmp = x * ((y / ((y + x) * (x + (y + 1.0)))) / (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.4e+123:
		tmp = ((x / (y + x)) / (y + (x + 1.0))) * (y / x)
	else:
		tmp = x * ((y / ((y + x) * (x + (y + 1.0)))) / (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.4e+123)
		tmp = Float64(Float64(Float64(x / Float64(y + x)) / Float64(y + Float64(x + 1.0))) * Float64(y / x));
	else
		tmp = Float64(x * Float64(Float64(y / Float64(Float64(y + x) * Float64(x + Float64(y + 1.0)))) / Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.4e+123)
		tmp = ((x / (y + x)) / (y + (x + 1.0))) * (y / x);
	else
		tmp = x * ((y / ((y + x) * (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[x, -2.4e+123], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y / N[(N[(y + x), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.39999999999999989e123

   1. Initial program 38.7%

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

      1. *-commutative 38.7%

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

      2. associate-*l* 38.7%

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

      3. times-frac 74.8%

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

      4. +-commutative 74.8%

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

      5. +-commutative 74.8%

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

      6. associate-+r+ 74.8%

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

      7. +-commutative 74.8%

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

      8. associate-+l+ 74.8%

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

   4. Applied egg-rr 74.8%

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

      1. div-inv 74.8%

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

      2. distribute-rgt-in 62.3%

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

      3. +-commutative 62.3%

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

      4. distribute-rgt-in 74.8%

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

      5. +-commutative 74.8%

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

   6. Applied egg-rr 74.8%

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

      1. associate-*r/ 74.8%

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

      2. *-rgt-identity 74.8%

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

      3. associate-/r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in y around 0 88.2%

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

   if -2.39999999999999989e123 < x

   1. Initial program 73.1%

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

      1. *-commutative 73.1%

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

      2. associate-*l* 73.1%

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

      3. times-frac 96.0%

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

      4. +-commutative 96.0%

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

      5. +-commutative 96.0%

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

      6. associate-+r+ 96.0%

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

      7. +-commutative 96.0%

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

      8. associate-+l+ 96.0%

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

   4. Applied egg-rr 96.0%

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

      1. div-inv 95.9%

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

      2. distribute-rgt-in 92.6%

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

      3. +-commutative 92.6%

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

      4. distribute-rgt-in 95.9%

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

      5. +-commutative 95.9%

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

   6. Applied egg-rr 95.9%

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

      1. associate-*r/ 96.0%

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

      2. *-rgt-identity 96.0%

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

      3. associate-/r* 99.8%

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

   8. Simplified 99.8%

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

      1. clear-num 99.7%

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

      2. +-commutative 99.7%

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

      3. frac-times 99.5%

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

      4. *-un-lft-identity 99.5%

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

      5. +-commutative 99.5%

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

      6. +-commutative 99.5%

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

   10. Applied egg-rr 99.5%

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

       1. div-inv 99.5%

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

       2. div-inv 99.4%

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

       3. associate-*l* 93.3%

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

       4. +-commutative 93.3%

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

       5. associate-/r* 93.3%

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

       6. clear-num 93.4%

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

       7. +-commutative 93.4%

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

       8. associate-+r+ 93.4%

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

       9. +-commutative 93.4%

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

       10. associate-+l+ 93.4%

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

   12. Applied egg-rr 93.4%

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

       1. associate-*l/ 93.5%

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

       2. *-lft-identity 93.5%

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

       3. associate-/l/ 93.5%

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

   14. Simplified 93.5%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+123}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + 1\right)} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 95.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;y \leq 1.8 \cdot 10^{+155}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= y 1.8e+155)
     (* (/ y (+ y x)) (/ x (* (+ y x) t_0)))
     (/ (/ x (+ y x)) t_0))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= 1.8e+155) {
		tmp = (y / (y + x)) * (x / ((y + x) * t_0));
	} else {
		tmp = (x / (y + x)) / t_0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (y <= 1.8d+155) then
        tmp = (y / (y + x)) * (x / ((y + x) * t_0))
    else
        tmp = (x / (y + x)) / t_0
    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 <= 1.8e+155) {
		tmp = (y / (y + x)) * (x / ((y + x) * t_0));
	} else {
		tmp = (x / (y + x)) / t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if y <= 1.8e+155:
		tmp = (y / (y + x)) * (x / ((y + x) * t_0))
	else:
		tmp = (x / (y + x)) / t_0
	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 <= 1.8e+155)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(Float64(y + x) * t_0)));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / t_0);
	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 <= 1.8e+155)
		tmp = (y / (y + x)) * (x / ((y + x) * t_0));
	else
		tmp = (x / (y + x)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.8e+155], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.80000000000000004e155

   1. Initial program 69.1%

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

      1. *-commutative 69.1%

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

      2. associate-*l* 69.1%

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

      3. times-frac 95.2%

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

      4. +-commutative 95.2%

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

      5. +-commutative 95.2%

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

      6. associate-+r+ 95.2%

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

      7. +-commutative 95.2%

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

      8. associate-+l+ 95.2%

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

   4. Applied egg-rr 95.2%

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

   if 1.80000000000000004e155 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 67.1%

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

      2. associate-*l* 67.1%

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

      3. times-frac 82.3%

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

      4. +-commutative 82.3%

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

      5. +-commutative 82.3%

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

      6. associate-+r+ 82.3%

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

      7. +-commutative 82.3%

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

      8. associate-+l+ 82.3%

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

   4. Applied egg-rr 82.3%

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

      1. div-inv 82.3%

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

      2. distribute-rgt-in 73.1%

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

      3. +-commutative 73.1%

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

      4. distribute-rgt-in 82.3%

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

      5. +-commutative 82.3%

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

   6. Applied egg-rr 82.3%

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

      1. associate-*r/ 82.3%

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

      2. *-rgt-identity 82.3%

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

      3. associate-/r* 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in y around inf 96.9%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{+155}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 87.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{-175}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1e-175)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 2.5e+154) (/ x (* (+ y x) (+ y (+ x 1.0)))) (/ (/ x (+ y x)) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1e-175) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 2.5e+154) {
		tmp = x / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = (x / (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 <= 1d-175) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 2.5d+154) then
        tmp = x / ((y + x) * (y + (x + 1.0d0)))
    else
        tmp = (x / (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 <= 1e-175) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 2.5e+154) {
		tmp = x / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = (x / (y + x)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1e-175:
		tmp = (y / x) / (x + 1.0)
	elif y <= 2.5e+154:
		tmp = x / ((y + x) * (y + (x + 1.0)))
	else:
		tmp = (x / (y + x)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1e-175)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 2.5e+154)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1e-175)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 2.5e+154)
		tmp = x / ((y + x) * (y + (x + 1.0)));
	else
		tmp = (x / (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, 1e-175], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.5e+154], N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1e-175

   1. Initial program 69.3%

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

      1. associate-/l* 86.5%

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

      2. associate-+l+ 86.5%

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

   3. Simplified 86.5%

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

   5. Taylor expanded in y around 0 58.7%

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

      1. associate-/r* 60.2%

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

      2. +-commutative 60.2%

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

   7. Simplified 60.2%

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

   if 1e-175 < y < 2.50000000000000002e154

   1. Initial program 70.5%

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

      1. *-commutative 70.5%

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

      2. associate-*l* 70.5%

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

      3. times-frac 94.7%

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

      4. +-commutative 94.7%

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

      5. +-commutative 94.7%

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

      6. associate-+r+ 94.7%

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

      7. +-commutative 94.7%

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

      8. associate-+l+ 94.7%

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

   4. Applied egg-rr 94.7%

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

   5. Taylor expanded in y around inf 66.7%

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

   if 2.50000000000000002e154 < y

   1. Initial program 63.6%

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

      1. *-commutative 63.6%

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

      2. associate-*l* 63.6%

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

      3. times-frac 78.3%

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

      4. +-commutative 78.3%

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

      5. +-commutative 78.3%

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

      6. associate-+r+ 78.3%

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

      7. +-commutative 78.3%

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

      8. associate-+l+ 78.3%

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

   4. Applied egg-rr 78.3%

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

      1. div-inv 78.3%

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

      2. distribute-rgt-in 69.3%

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

      3. +-commutative 69.3%

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

      4. distribute-rgt-in 78.3%

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

      5. +-commutative 78.3%

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

   6. Applied egg-rr 78.3%

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

      1. associate-*r/ 78.3%

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

      2. *-rgt-identity 78.3%

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

      3. associate-/r* 99.9%

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

   8. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. +-commutative 99.9%

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

      3. frac-times 99.8%

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

      4. *-un-lft-identity 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in y around inf 92.1%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-175}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 87.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;y \leq 9.5 \cdot 10^{-176}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+155}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= y 9.5e-176)
     (/ (/ y x) (+ x 1.0))
     (if (<= y 7.2e+155) (/ x (* (+ y x) t_0)) (/ (/ x (+ y x)) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= 9.5e-176) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 7.2e+155) {
		tmp = x / ((y + x) * t_0);
	} else {
		tmp = (x / (y + x)) / t_0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (y <= 9.5d-176) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 7.2d+155) then
        tmp = x / ((y + x) * t_0)
    else
        tmp = (x / (y + x)) / t_0
    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 <= 9.5e-176) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 7.2e+155) {
		tmp = x / ((y + x) * t_0);
	} else {
		tmp = (x / (y + x)) / t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if y <= 9.5e-176:
		tmp = (y / x) / (x + 1.0)
	elif y <= 7.2e+155:
		tmp = x / ((y + x) * t_0)
	else:
		tmp = (x / (y + x)) / t_0
	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 <= 9.5e-176)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 7.2e+155)
		tmp = Float64(x / Float64(Float64(y + x) * t_0));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / t_0);
	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 <= 9.5e-176)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 7.2e+155)
		tmp = x / ((y + x) * t_0);
	else
		tmp = (x / (y + x)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 9.5e-176], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.2e+155], N[(x / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 9.5e-176

   1. Initial program 69.3%

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

      1. associate-/l* 86.5%

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

      2. associate-+l+ 86.5%

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

   3. Simplified 86.5%

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

   5. Taylor expanded in y around 0 58.7%

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

      1. associate-/r* 60.2%

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

      2. +-commutative 60.2%

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

   7. Simplified 60.2%

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

   if 9.5e-176 < y < 7.20000000000000015e155

   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. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 68.5%

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

      2. associate-*l* 68.6%

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

      3. times-frac 92.2%

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

      4. +-commutative 92.2%

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

      5. +-commutative 92.2%

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

      6. associate-+r+ 92.2%

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

      7. +-commutative 92.2%

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

      8. associate-+l+ 92.2%

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

   4. Applied egg-rr 92.2%

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

   5. Taylor expanded in y around inf 65.0%

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

   if 7.20000000000000015e155 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 67.1%

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

      2. associate-*l* 67.1%

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

      3. times-frac 82.3%

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

      4. +-commutative 82.3%

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

      5. +-commutative 82.3%

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

      6. associate-+r+ 82.3%

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

      7. +-commutative 82.3%

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

      8. associate-+l+ 82.3%

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

   4. Applied egg-rr 82.3%

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

      1. div-inv 82.3%

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

      2. distribute-rgt-in 73.1%

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

      3. +-commutative 73.1%

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

      4. distribute-rgt-in 82.3%

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

      5. +-commutative 82.3%

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

   6. Applied egg-rr 82.3%

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

      1. associate-*r/ 82.3%

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

      2. *-rgt-identity 82.3%

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

      3. associate-/r* 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in y around inf 96.9%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{-176}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+155}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 87.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{\frac{x}{y + x}}{y + \left(x + 1\right)}\\ \mathbf{if}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;t\_0 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ x (+ y x)) (+ y (+ x 1.0)))))
   (if (<= x -5.05e-138) (* t_0 (/ y x)) t_0)))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (x / (y + x)) / (y + (x + 1.0));
	double tmp;
	if (x <= -5.05e-138) {
		tmp = t_0 * (y / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / (y + x)) / (y + (x + 1.0d0))
    if (x <= (-5.05d-138)) then
        tmp = t_0 * (y / x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (x / (y + x)) / (y + (x + 1.0));
	double tmp;
	if (x <= -5.05e-138) {
		tmp = t_0 * (y / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (x / (y + x)) / (y + (x + 1.0))
	tmp = 0
	if x <= -5.05e-138:
		tmp = t_0 * (y / x)
	else:
		tmp = t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(x / Float64(y + x)) / Float64(y + Float64(x + 1.0)))
	tmp = 0.0
	if (x <= -5.05e-138)
		tmp = Float64(t_0 * Float64(y / x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (x / (y + x)) / (y + (x + 1.0));
	tmp = 0.0;
	if (x <= -5.05e-138)
		tmp = t_0 * (y / x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.05e-138], N[(t$95$0 * N[(y / x), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if x < -5.0499999999999997e-138

   1. Initial program 66.4%

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

      1. *-commutative 66.4%

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

      2. associate-*l* 66.4%

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

      3. times-frac 90.3%

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

      4. +-commutative 90.3%

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

      5. +-commutative 90.3%

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

      6. associate-+r+ 90.3%

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

      7. +-commutative 90.3%

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

      8. associate-+l+ 90.3%

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

   4. Applied egg-rr 90.3%

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

      1. div-inv 90.2%

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

      2. distribute-rgt-in 82.8%

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

      3. +-commutative 82.8%

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

      4. distribute-rgt-in 90.2%

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

      5. +-commutative 90.2%

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

   6. Applied egg-rr 90.2%

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

      1. associate-*r/ 90.3%

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

      2. *-rgt-identity 90.3%

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

      3. associate-/r* 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in y around 0 76.7%

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

   if -5.0499999999999997e-138 < x

   1. Initial program 70.2%

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

      1. *-commutative 70.2%

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

      2. associate-*l* 70.2%

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

      3. times-frac 95.2%

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

      4. +-commutative 95.2%

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

      5. +-commutative 95.2%

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

      6. associate-+r+ 95.2%

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

      7. +-commutative 95.2%

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

      8. associate-+l+ 95.2%

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

   4. Applied egg-rr 95.2%

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

      1. div-inv 95.0%

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

      2. distribute-rgt-in 92.4%

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

      3. +-commutative 92.4%

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

      4. distribute-rgt-in 95.0%

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

      5. +-commutative 95.0%

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

   6. Applied egg-rr 95.0%

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

      1. associate-*r/ 95.2%

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

      2. *-rgt-identity 95.2%

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

      3. associate-/r* 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in y around inf 56.5%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + 1\right)} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+164}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5e+164)
   (* (/ y x) (/ 1.0 x))
   (if (<= x -5.05e-138) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5e+164) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -5.05e-138) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5d+164)) then
        tmp = (y / x) * (1.0d0 / x)
    else if (x <= (-5.05d-138)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -5e+164) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -5.05e-138) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -5e+164:
		tmp = (y / x) * (1.0 / x)
	elif x <= -5.05e-138:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5e+164)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	elseif (x <= -5.05e-138)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5e+164)
		tmp = (y / x) * (1.0 / x);
	elseif (x <= -5.05e-138)
		tmp = y / (x * (x + 1.0));
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -5e+164], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.05e-138], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999995e164

   1. Initial program 42.1%

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

      1. *-commutative 42.1%

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

      2. associate-*l* 42.1%

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

      3. times-frac 72.2%

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

      4. +-commutative 72.2%

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

      5. +-commutative 72.2%

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

      6. associate-+r+ 72.2%

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

      7. +-commutative 72.2%

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

      8. associate-+l+ 72.2%

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

   4. Applied egg-rr 72.2%

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

   5. Taylor expanded in x around inf 87.0%

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

   6. Taylor expanded in y around 0 86.7%

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

   if -4.9999999999999995e164 < x < -5.0499999999999997e-138

   1. Initial program 77.2%

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

      1. associate-/l* 90.0%

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

      2. associate-+l+ 90.0%

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

   3. Simplified 90.0%

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

   5. Taylor expanded in y around 0 47.8%

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

   if -5.0499999999999997e-138 < x

   1. Initial program 70.2%

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

      1. associate-/l* 83.4%

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

      2. associate-+l+ 83.4%

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

   3. Simplified 83.4%

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

   5. Taylor expanded in x around 0 53.5%

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

      1. +-commutative 53.5%

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

   7. Simplified 53.5%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+164}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 67.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) (* (/ y x) (/ 1.0 x)) (if (<= x -5.05e-138) (/ y x) (/ x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -5.05e-138) {
		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 (x <= (-1.0d0)) then
        tmp = (y / x) * (1.0d0 / x)
    else if (x <= (-5.05d-138)) 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 (x <= -1.0) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -5.05e-138) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (y / x) * (1.0 / x)
	elif x <= -5.05e-138:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	elseif (x <= -5.05e-138)
		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 (x <= -1.0)
		tmp = (y / x) * (1.0 / x);
	elseif (x <= -5.05e-138)
		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[x, -1.0], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.05e-138], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 54.5%

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

      1. *-commutative 54.5%

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

      2. associate-*l* 54.6%

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

      3. times-frac 86.0%

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

      4. +-commutative 86.0%

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

      5. +-commutative 86.0%

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

      6. associate-+r+ 86.0%

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

      7. +-commutative 86.0%

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

      8. associate-+l+ 86.0%

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

   4. Applied egg-rr 86.0%

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

   5. Taylor expanded in x around inf 61.3%

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

   6. Taylor expanded in y around 0 60.6%

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

   if -1 < x < -5.0499999999999997e-138

   1. Initial program 92.9%

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

      1. associate-/l* 97.2%

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

      2. associate-+l+ 97.2%

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

   3. Simplified 97.2%

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

   5. Taylor expanded in y around 0 51.8%

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

      1. associate-/r* 51.8%

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

      2. +-commutative 51.8%

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

   7. Simplified 51.8%

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

   8. Taylor expanded in x around 0 51.5%

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

   if -5.0499999999999997e-138 < x

   1. Initial program 70.2%

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

      1. associate-/l* 83.4%

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

      2. associate-+l+ 83.4%

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

   3. Simplified 83.4%

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

   5. Taylor expanded in x around 0 53.5%

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

      1. +-commutative 53.5%

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

   7. Simplified 53.5%

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

   8. Taylor expanded in y around 0 35.8%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -4.5e-138) (/ y x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4.5e-138) {
		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 (x <= (-4.5d-138)) 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 (x <= -4.5e-138) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -4.5e-138:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -4.5e-138)
		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 (x <= -4.5e-138)
		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[x, -4.5e-138], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.50000000000000008e-138

   1. Initial program 66.4%

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

      1. associate-/l* 84.5%

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

      2. associate-+l+ 84.5%

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

   3. Simplified 84.5%

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

   5. Taylor expanded in y around 0 55.4%

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

      1. associate-/r* 59.8%

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

      2. +-commutative 59.8%

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

   7. Simplified 59.8%

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

   8. Taylor expanded in x around 0 28.8%

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

   if -4.50000000000000008e-138 < x

   1. Initial program 70.2%

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

      1. associate-/l* 83.4%

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

      2. associate-+l+ 83.4%

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

   3. Simplified 83.4%

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

   5. Taylor expanded in x around 0 53.5%

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

      1. +-commutative 53.5%

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

   7. Simplified 53.5%

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

   8. Taylor expanded in y around 0 35.8%

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

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 26.5% 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 68.8%

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

   1. associate-/l* 83.8%

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

   2. associate-+l+ 83.8%

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

3. Simplified 83.8%

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

5. Taylor expanded in x around 0 47.7%

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

   1. +-commutative 47.7%

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

7. Simplified 47.7%

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

8. Taylor expanded in y around 0 25.6%

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

9. Final simplification 25.6%

   \[\leadsto \frac{x}{y} \]
10. Add Preprocessing

Developer target: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024096 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))