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

Percentage Accurate: 69.8% → 99.8%

Time: 22.9s

Alternatives: 21

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: 69.8% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.4%

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

   1. *-commutative 65.4%

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

   2. associate-*l* 65.4%

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

   3. times-frac 93.8%

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

   4. +-commutative 93.8%

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

   5. +-commutative 93.8%

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

   6. associate-+r+ 93.8%

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

   7. +-commutative 93.8%

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

   8. associate-+l+ 93.8%

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

4. Applied egg-rr 93.8%

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

   1. associate-*l/ 93.8%

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

   2. associate-/r* 99.8%

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

   3. +-commutative 99.8%

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

   4. +-commutative 99.8%

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

   5. associate-+r+ 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+l+ 99.8%

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

   8. +-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 78.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+57}:\\ \;\;\;\;\frac{y \cdot \frac{1}{x}}{y + x}\\ \mathbf{elif}\;x \leq -38000000:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-38}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x + 1}{y}}\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-153}:\\ \;\;\;\;\frac{1}{x + 1} \cdot \frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2e+57)
   (/ (* y (/ 1.0 x)) (+ y x))
   (if (<= x -38000000.0)
     (/ (/ x (+ y 1.0)) (+ y x))
     (if (<= x -1.3e-38)
       (/ 1.0 (* x (/ (+ x 1.0) y)))
       (if (<= x -1.9e-85)
         (/ x (* y (+ y 1.0)))
         (if (<= x -4.1e-153)
           (* (/ 1.0 (+ x 1.0)) (/ y (+ y x)))
           (/ (/ x y) (+ y 1.0))))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2e+57) {
		tmp = (y * (1.0 / x)) / (y + x);
	} else if (x <= -38000000.0) {
		tmp = (x / (y + 1.0)) / (y + x);
	} else if (x <= -1.3e-38) {
		tmp = 1.0 / (x * ((x + 1.0) / y));
	} else if (x <= -1.9e-85) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -4.1e-153) {
		tmp = (1.0 / (x + 1.0)) * (y / (y + x));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2d+57)) then
        tmp = (y * (1.0d0 / x)) / (y + x)
    else if (x <= (-38000000.0d0)) then
        tmp = (x / (y + 1.0d0)) / (y + x)
    else if (x <= (-1.3d-38)) then
        tmp = 1.0d0 / (x * ((x + 1.0d0) / y))
    else if (x <= (-1.9d-85)) then
        tmp = x / (y * (y + 1.0d0))
    else if (x <= (-4.1d-153)) then
        tmp = (1.0d0 / (x + 1.0d0)) * (y / (y + x))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2e+57) {
		tmp = (y * (1.0 / x)) / (y + x);
	} else if (x <= -38000000.0) {
		tmp = (x / (y + 1.0)) / (y + x);
	} else if (x <= -1.3e-38) {
		tmp = 1.0 / (x * ((x + 1.0) / y));
	} else if (x <= -1.9e-85) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -4.1e-153) {
		tmp = (1.0 / (x + 1.0)) * (y / (y + x));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2e+57:
		tmp = (y * (1.0 / x)) / (y + x)
	elif x <= -38000000.0:
		tmp = (x / (y + 1.0)) / (y + x)
	elif x <= -1.3e-38:
		tmp = 1.0 / (x * ((x + 1.0) / y))
	elif x <= -1.9e-85:
		tmp = x / (y * (y + 1.0))
	elif x <= -4.1e-153:
		tmp = (1.0 / (x + 1.0)) * (y / (y + x))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2e+57)
		tmp = Float64(Float64(y * Float64(1.0 / x)) / Float64(y + x));
	elseif (x <= -38000000.0)
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	elseif (x <= -1.3e-38)
		tmp = Float64(1.0 / Float64(x * Float64(Float64(x + 1.0) / y)));
	elseif (x <= -1.9e-85)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	elseif (x <= -4.1e-153)
		tmp = Float64(Float64(1.0 / Float64(x + 1.0)) * Float64(y / Float64(y + x)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2e+57)
		tmp = (y * (1.0 / x)) / (y + x);
	elseif (x <= -38000000.0)
		tmp = (x / (y + 1.0)) / (y + x);
	elseif (x <= -1.3e-38)
		tmp = 1.0 / (x * ((x + 1.0) / y));
	elseif (x <= -1.9e-85)
		tmp = x / (y * (y + 1.0));
	elseif (x <= -4.1e-153)
		tmp = (1.0 / (x + 1.0)) * (y / (y + x));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2e+57], N[(N[(y * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -38000000.0], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.3e-38], N[(1.0 / N[(x * N[(N[(x + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.9e-85], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.1e-153], N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -2.0000000000000001e57

   1. Initial program 52.5%

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

      1. *-commutative 52.5%

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

      2. associate-*l* 52.5%

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

      3. times-frac 87.6%

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

      4. +-commutative 87.6%

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

      5. +-commutative 87.6%

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

      6. associate-+r+ 87.6%

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

      7. +-commutative 87.6%

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

      8. associate-+l+ 87.6%

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

   4. Applied egg-rr 87.6%

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

      1. associate-*l/ 87.6%

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

      2. associate-/r* 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+l+ 99.8%

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

      8. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf 87.8%

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

   if -2.0000000000000001e57 < x < -3.8e7

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-*l* 80.7%

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

      3. times-frac 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.7%

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

      2. associate-/r* 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-+r+ 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+l+ 99.7%

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

      8. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0 61.4%

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

      1. +-commutative 61.4%

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

   9. Simplified 61.4%

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

   if -3.8e7 < x < -1.30000000000000005e-38

   1. Initial program 99.8%

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

      1. associate-/l* 100.0%

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

      2. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

      1. associate-*r/ 99.8%

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

      2. associate-+r+ 99.8%

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

      3. clear-num 99.3%

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

      4. associate-/l* 99.6%

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

      5. pow2 99.6%

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

      6. +-commutative 99.6%

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

      7. associate-+r+ 99.6%

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

      8. +-commutative 99.6%

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

      9. associate-+l+ 99.6%

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

      10. *-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. /-rgt-identity 99.6%

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

      2. /-rgt-identity 99.6%

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

      3. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in y around 0 58.3%

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

       1. associate-/l* 58.3%

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

       2. +-commutative 58.3%

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

   11. Simplified 58.3%

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

   if -1.30000000000000005e-38 < x < -1.8999999999999999e-85

   1. Initial program 93.1%

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

      1. associate-/l* 93.0%

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

      2. associate-+l+ 93.0%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in x around 0 77.2%

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

   if -1.8999999999999999e-85 < x < -4.1e-153

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

      2. associate-*l* 79.9%

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

      3. times-frac 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+r+ 99.9%

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

      7. +-commutative 99.9%

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

      8. associate-+l+ 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0 29.0%

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

      1. +-commutative 29.0%

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

   7. Simplified 29.0%

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

   if -4.1e-153 < x

   1. Initial program 63.7%

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

      1. associate-/l* 76.3%

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

      2. associate-+l+ 76.3%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in x around 0 59.5%

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

      1. associate-/r* 62.2%

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

      2. +-commutative 62.2%

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

   7. Simplified 62.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+57}:\\ \;\;\;\;\frac{y \cdot \frac{1}{x}}{y + x}\\ \mathbf{elif}\;x \leq -38000000:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-38}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x + 1}{y}}\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-153}:\\ \;\;\;\;\frac{1}{x + 1} \cdot \frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{if}\;x \leq -2.65 \cdot 10^{+57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -35000000:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{-38}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-85} \lor \neg \left(x \leq -4.1 \cdot 10^{-153}\right):\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* x (+ x 1.0)))))
   (if (<= x -2.65e+57)
     t_0
     (if (<= x -35000000.0)
       (/ (/ x y) (+ y x))
       (if (<= x -2.15e-38)
         t_0
         (if (or (<= x -1.2e-85) (not (<= x -4.1e-153)))
           (/ x (* y (+ y 1.0)))
           (/ y (+ y x))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x * (x + 1.0));
	double tmp;
	if (x <= -2.65e+57) {
		tmp = t_0;
	} else if (x <= -35000000.0) {
		tmp = (x / y) / (y + x);
	} else if (x <= -2.15e-38) {
		tmp = t_0;
	} else if ((x <= -1.2e-85) || !(x <= -4.1e-153)) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = y / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (x * (x + 1.0d0))
    if (x <= (-2.65d+57)) then
        tmp = t_0
    else if (x <= (-35000000.0d0)) then
        tmp = (x / y) / (y + x)
    else if (x <= (-2.15d-38)) then
        tmp = t_0
    else if ((x <= (-1.2d-85)) .or. (.not. (x <= (-4.1d-153)))) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = y / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (x * (x + 1.0));
	double tmp;
	if (x <= -2.65e+57) {
		tmp = t_0;
	} else if (x <= -35000000.0) {
		tmp = (x / y) / (y + x);
	} else if (x <= -2.15e-38) {
		tmp = t_0;
	} else if ((x <= -1.2e-85) || !(x <= -4.1e-153)) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = y / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (x * (x + 1.0))
	tmp = 0
	if x <= -2.65e+57:
		tmp = t_0
	elif x <= -35000000.0:
		tmp = (x / y) / (y + x)
	elif x <= -2.15e-38:
		tmp = t_0
	elif (x <= -1.2e-85) or not (x <= -4.1e-153):
		tmp = x / (y * (y + 1.0))
	else:
		tmp = y / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(x * Float64(x + 1.0)))
	tmp = 0.0
	if (x <= -2.65e+57)
		tmp = t_0;
	elseif (x <= -35000000.0)
		tmp = Float64(Float64(x / y) / Float64(y + x));
	elseif (x <= -2.15e-38)
		tmp = t_0;
	elseif ((x <= -1.2e-85) || !(x <= -4.1e-153))
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(y / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (x * (x + 1.0));
	tmp = 0.0;
	if (x <= -2.65e+57)
		tmp = t_0;
	elseif (x <= -35000000.0)
		tmp = (x / y) / (y + x);
	elseif (x <= -2.15e-38)
		tmp = t_0;
	elseif ((x <= -1.2e-85) || ~((x <= -4.1e-153)))
		tmp = x / (y * (y + 1.0));
	else
		tmp = y / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.65e+57], t$95$0, If[LessEqual[x, -35000000.0], N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.15e-38], t$95$0, If[Or[LessEqual[x, -1.2e-85], N[Not[LessEqual[x, -4.1e-153]], $MachinePrecision]], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.64999999999999993e57 or -3.5e7 < x < -2.1500000000000001e-38

   1. Initial program 58.4%

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

      1. associate-/l* 74.1%

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

      2. associate-+l+ 74.1%

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

   3. Simplified 74.1%

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

   5. Taylor expanded in y around 0 79.7%

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

      1. +-commutative 79.7%

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

   7. Simplified 79.7%

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

   if -2.64999999999999993e57 < x < -3.5e7

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-*l* 80.7%

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

      3. times-frac 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.7%

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

      2. associate-/r* 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-+r+ 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+l+ 99.7%

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

      8. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around inf 61.0%

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

   if -2.1500000000000001e-38 < x < -1.2e-85 or -4.1e-153 < x

   1. Initial program 66.0%

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

      1. associate-/l* 77.6%

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

      2. associate-+l+ 77.6%

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

   3. Simplified 77.6%

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

   5. Taylor expanded in x around 0 60.9%

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

   if -1.2e-85 < x < -4.1e-153

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

      2. associate-*l* 79.9%

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

      3. times-frac 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+r+ 99.9%

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

      7. +-commutative 99.9%

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

      8. associate-+l+ 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. associate-*l/ 99.8%

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

      2. associate-/r* 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-+r+ 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+l+ 99.7%

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

      8. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around 0 29.0%

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

   8. Taylor expanded in x around 0 29.0%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{+57}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -35000000:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{-38}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-85} \lor \neg \left(x \leq -4.1 \cdot 10^{-153}\right):\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{+57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -15200000:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-38}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-153}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* x (+ x 1.0)))))
   (if (<= x -1.5e+57)
     t_0
     (if (<= x -15200000.0)
       (/ (/ x y) (+ y x))
       (if (<= x -1.7e-38)
         t_0
         (if (<= x -1.2e-85)
           (/ x (* y (+ y 1.0)))
           (if (<= x -4.1e-153) (/ y (+ y x)) (/ (/ x y) (+ y 1.0)))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x * (x + 1.0));
	double tmp;
	if (x <= -1.5e+57) {
		tmp = t_0;
	} else if (x <= -15200000.0) {
		tmp = (x / y) / (y + x);
	} else if (x <= -1.7e-38) {
		tmp = t_0;
	} else if (x <= -1.2e-85) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -4.1e-153) {
		tmp = y / (y + x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (x * (x + 1.0d0))
    if (x <= (-1.5d+57)) then
        tmp = t_0
    else if (x <= (-15200000.0d0)) then
        tmp = (x / y) / (y + x)
    else if (x <= (-1.7d-38)) then
        tmp = t_0
    else if (x <= (-1.2d-85)) then
        tmp = x / (y * (y + 1.0d0))
    else if (x <= (-4.1d-153)) then
        tmp = y / (y + x)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (x * (x + 1.0));
	double tmp;
	if (x <= -1.5e+57) {
		tmp = t_0;
	} else if (x <= -15200000.0) {
		tmp = (x / y) / (y + x);
	} else if (x <= -1.7e-38) {
		tmp = t_0;
	} else if (x <= -1.2e-85) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -4.1e-153) {
		tmp = y / (y + x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (x * (x + 1.0))
	tmp = 0
	if x <= -1.5e+57:
		tmp = t_0
	elif x <= -15200000.0:
		tmp = (x / y) / (y + x)
	elif x <= -1.7e-38:
		tmp = t_0
	elif x <= -1.2e-85:
		tmp = x / (y * (y + 1.0))
	elif x <= -4.1e-153:
		tmp = y / (y + x)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(x * Float64(x + 1.0)))
	tmp = 0.0
	if (x <= -1.5e+57)
		tmp = t_0;
	elseif (x <= -15200000.0)
		tmp = Float64(Float64(x / y) / Float64(y + x));
	elseif (x <= -1.7e-38)
		tmp = t_0;
	elseif (x <= -1.2e-85)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	elseif (x <= -4.1e-153)
		tmp = Float64(y / Float64(y + x));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (x * (x + 1.0));
	tmp = 0.0;
	if (x <= -1.5e+57)
		tmp = t_0;
	elseif (x <= -15200000.0)
		tmp = (x / y) / (y + x);
	elseif (x <= -1.7e-38)
		tmp = t_0;
	elseif (x <= -1.2e-85)
		tmp = x / (y * (y + 1.0));
	elseif (x <= -4.1e-153)
		tmp = y / (y + x);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.5e+57], t$95$0, If[LessEqual[x, -15200000.0], N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.7e-38], t$95$0, If[LessEqual[x, -1.2e-85], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.1e-153], N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.5e57 or -1.52e7 < x < -1.7000000000000001e-38

   1. Initial program 58.4%

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

      1. associate-/l* 74.1%

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

      2. associate-+l+ 74.1%

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

   3. Simplified 74.1%

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

   5. Taylor expanded in y around 0 79.7%

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

      1. +-commutative 79.7%

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

   7. Simplified 79.7%

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

   if -1.5e57 < x < -1.52e7

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-*l* 80.7%

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

      3. times-frac 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.7%

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

      2. associate-/r* 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-+r+ 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+l+ 99.7%

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

      8. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around inf 61.0%

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

   if -1.7000000000000001e-38 < x < -1.2e-85

   1. Initial program 93.1%

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

      1. associate-/l* 93.0%

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

      2. associate-+l+ 93.0%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in x around 0 77.2%

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

   if -1.2e-85 < x < -4.1e-153

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

      2. associate-*l* 79.9%

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

      3. times-frac 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+r+ 99.9%

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

      7. +-commutative 99.9%

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

      8. associate-+l+ 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. associate-*l/ 99.8%

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

      2. associate-/r* 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-+r+ 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+l+ 99.7%

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

      8. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around 0 29.0%

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

   8. Taylor expanded in x around 0 29.0%

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

   if -4.1e-153 < x

   1. Initial program 63.7%

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

      1. associate-/l* 76.3%

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

      2. associate-+l+ 76.3%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in x around 0 59.5%

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

      1. associate-/r* 62.2%

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

      2. +-commutative 62.2%

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

   7. Simplified 62.2%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+57}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -15200000:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-38}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-153}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+57}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;x \leq -14500000:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \mathbf{elif}\;x \leq -2.25 \cdot 10^{-38}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-153}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.15e+57)
   (/ (/ y x) (+ x 1.0))
   (if (<= x -14500000.0)
     (/ (/ x y) (+ y x))
     (if (<= x -2.25e-38)
       (/ y (* x (+ x 1.0)))
       (if (<= x -1.2e-85)
         (/ x (* y (+ y 1.0)))
         (if (<= x -4.1e-153) (/ y (+ y x)) (/ (/ x y) (+ y 1.0))))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.15e+57) {
		tmp = (y / x) / (x + 1.0);
	} else if (x <= -14500000.0) {
		tmp = (x / y) / (y + x);
	} else if (x <= -2.25e-38) {
		tmp = y / (x * (x + 1.0));
	} else if (x <= -1.2e-85) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -4.1e-153) {
		tmp = y / (y + x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.15d+57)) then
        tmp = (y / x) / (x + 1.0d0)
    else if (x <= (-14500000.0d0)) then
        tmp = (x / y) / (y + x)
    else if (x <= (-2.25d-38)) then
        tmp = y / (x * (x + 1.0d0))
    else if (x <= (-1.2d-85)) then
        tmp = x / (y * (y + 1.0d0))
    else if (x <= (-4.1d-153)) then
        tmp = y / (y + x)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.15e+57) {
		tmp = (y / x) / (x + 1.0);
	} else if (x <= -14500000.0) {
		tmp = (x / y) / (y + x);
	} else if (x <= -2.25e-38) {
		tmp = y / (x * (x + 1.0));
	} else if (x <= -1.2e-85) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -4.1e-153) {
		tmp = y / (y + x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.15e+57:
		tmp = (y / x) / (x + 1.0)
	elif x <= -14500000.0:
		tmp = (x / y) / (y + x)
	elif x <= -2.25e-38:
		tmp = y / (x * (x + 1.0))
	elif x <= -1.2e-85:
		tmp = x / (y * (y + 1.0))
	elif x <= -4.1e-153:
		tmp = y / (y + x)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.15e+57)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (x <= -14500000.0)
		tmp = Float64(Float64(x / y) / Float64(y + x));
	elseif (x <= -2.25e-38)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (x <= -1.2e-85)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	elseif (x <= -4.1e-153)
		tmp = Float64(y / Float64(y + x));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.15e+57)
		tmp = (y / x) / (x + 1.0);
	elseif (x <= -14500000.0)
		tmp = (x / y) / (y + x);
	elseif (x <= -2.25e-38)
		tmp = y / (x * (x + 1.0));
	elseif (x <= -1.2e-85)
		tmp = x / (y * (y + 1.0));
	elseif (x <= -4.1e-153)
		tmp = y / (y + x);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.15e+57], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -14500000.0], N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.25e-38], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.2e-85], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.1e-153], N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -1.1499999999999999e57

   1. Initial program 52.5%

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

      1. *-commutative 52.5%

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

      2. associate-*l* 52.5%

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

      3. times-frac 87.6%

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

      4. +-commutative 87.6%

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

      5. +-commutative 87.6%

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

      6. associate-+r+ 87.6%

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

      7. +-commutative 87.6%

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

      8. associate-+l+ 87.6%

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

   4. Applied egg-rr 87.6%

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

      1. associate-*l/ 87.6%

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

      2. associate-/r* 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+l+ 99.8%

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

      8. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around 0 82.7%

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

      1. associate-/r* 87.5%

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

   9. Simplified 87.5%

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

   if -1.1499999999999999e57 < x < -1.45e7

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-*l* 80.7%

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

      3. times-frac 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.7%

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

      2. associate-/r* 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-+r+ 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+l+ 99.7%

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

      8. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around inf 61.0%

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

   if -1.45e7 < x < -2.25000000000000004e-38

   1. Initial program 99.8%

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

      1. associate-/l* 100.0%

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

      2. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 58.5%

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

      1. +-commutative 58.5%

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

   7. Simplified 58.5%

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

   if -2.25000000000000004e-38 < x < -1.2e-85

   1. Initial program 93.1%

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

      1. associate-/l* 93.0%

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

      2. associate-+l+ 93.0%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in x around 0 77.2%

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

   if -1.2e-85 < x < -4.1e-153

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

      2. associate-*l* 79.9%

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

      3. times-frac 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+r+ 99.9%

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

      7. +-commutative 99.9%

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

      8. associate-+l+ 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. associate-*l/ 99.8%

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

      2. associate-/r* 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-+r+ 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+l+ 99.7%

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

      8. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around 0 29.0%

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

   8. Taylor expanded in x around 0 29.0%

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

   if -4.1e-153 < x

   1. Initial program 63.7%

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

      1. associate-/l* 76.3%

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

      2. associate-+l+ 76.3%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in x around 0 59.5%

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

      1. associate-/r* 62.2%

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

      2. +-commutative 62.2%

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

   7. Simplified 62.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+57}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;x \leq -14500000:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \mathbf{elif}\;x \leq -2.25 \cdot 10^{-38}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-153}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.46 \cdot 10^{+57}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -20500000:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \mathbf{elif}\;x \leq -2.25 \cdot 10^{-38}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-153}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.46e+57)
   (/ (/ y x) (+ y x))
   (if (<= x -20500000.0)
     (/ (/ x y) (+ y x))
     (if (<= x -2.25e-38)
       (/ y (* x (+ x 1.0)))
       (if (<= x -1.25e-85)
         (/ x (* y (+ y 1.0)))
         (if (<= x -4.1e-153) (/ y (+ y x)) (/ (/ x y) (+ y 1.0))))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.46e+57) {
		tmp = (y / x) / (y + x);
	} else if (x <= -20500000.0) {
		tmp = (x / y) / (y + x);
	} else if (x <= -2.25e-38) {
		tmp = y / (x * (x + 1.0));
	} else if (x <= -1.25e-85) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -4.1e-153) {
		tmp = y / (y + x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.46d+57)) then
        tmp = (y / x) / (y + x)
    else if (x <= (-20500000.0d0)) then
        tmp = (x / y) / (y + x)
    else if (x <= (-2.25d-38)) then
        tmp = y / (x * (x + 1.0d0))
    else if (x <= (-1.25d-85)) then
        tmp = x / (y * (y + 1.0d0))
    else if (x <= (-4.1d-153)) then
        tmp = y / (y + x)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.46e+57) {
		tmp = (y / x) / (y + x);
	} else if (x <= -20500000.0) {
		tmp = (x / y) / (y + x);
	} else if (x <= -2.25e-38) {
		tmp = y / (x * (x + 1.0));
	} else if (x <= -1.25e-85) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -4.1e-153) {
		tmp = y / (y + x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.46e+57:
		tmp = (y / x) / (y + x)
	elif x <= -20500000.0:
		tmp = (x / y) / (y + x)
	elif x <= -2.25e-38:
		tmp = y / (x * (x + 1.0))
	elif x <= -1.25e-85:
		tmp = x / (y * (y + 1.0))
	elif x <= -4.1e-153:
		tmp = y / (y + x)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.46e+57)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -20500000.0)
		tmp = Float64(Float64(x / y) / Float64(y + x));
	elseif (x <= -2.25e-38)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (x <= -1.25e-85)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	elseif (x <= -4.1e-153)
		tmp = Float64(y / Float64(y + x));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.46e+57)
		tmp = (y / x) / (y + x);
	elseif (x <= -20500000.0)
		tmp = (x / y) / (y + x);
	elseif (x <= -2.25e-38)
		tmp = y / (x * (x + 1.0));
	elseif (x <= -1.25e-85)
		tmp = x / (y * (y + 1.0));
	elseif (x <= -4.1e-153)
		tmp = y / (y + x);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.46e+57], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -20500000.0], N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.25e-38], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.25e-85], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.1e-153], N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -1.46e57

   1. Initial program 52.5%

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

      1. *-commutative 52.5%

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

      2. associate-*l* 52.5%

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

      3. times-frac 87.6%

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

      4. +-commutative 87.6%

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

      5. +-commutative 87.6%

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

      6. associate-+r+ 87.6%

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

      7. +-commutative 87.6%

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

      8. associate-+l+ 87.6%

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

   4. Applied egg-rr 87.6%

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

      1. associate-*l/ 87.6%

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

      2. associate-/r* 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+l+ 99.8%

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

      8. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf 87.9%

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

   if -1.46e57 < x < -2.05e7

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-*l* 80.7%

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

      3. times-frac 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.7%

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

      2. associate-/r* 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-+r+ 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+l+ 99.7%

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

      8. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around inf 61.0%

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

   if -2.05e7 < x < -2.25000000000000004e-38

   1. Initial program 99.8%

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

      1. associate-/l* 100.0%

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

      2. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 58.5%

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

      1. +-commutative 58.5%

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

   7. Simplified 58.5%

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

   if -2.25000000000000004e-38 < x < -1.25e-85

   1. Initial program 93.1%

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

      1. associate-/l* 93.0%

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

      2. associate-+l+ 93.0%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in x around 0 77.2%

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

   if -1.25e-85 < x < -4.1e-153

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

      2. associate-*l* 79.9%

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

      3. times-frac 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+r+ 99.9%

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

      7. +-commutative 99.9%

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

      8. associate-+l+ 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. associate-*l/ 99.8%

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

      2. associate-/r* 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-+r+ 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+l+ 99.7%

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

      8. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around 0 29.0%

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

   8. Taylor expanded in x around 0 29.0%

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

   if -4.1e-153 < x

   1. Initial program 63.7%

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

      1. associate-/l* 76.3%

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

      2. associate-+l+ 76.3%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in x around 0 59.5%

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

      1. associate-/r* 62.2%

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

      2. +-commutative 62.2%

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

   7. Simplified 62.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.46 \cdot 10^{+57}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -20500000:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \mathbf{elif}\;x \leq -2.25 \cdot 10^{-38}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-153}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+56}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -15800000:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \mathbf{elif}\;x \leq -1.02 \cdot 10^{-38}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x + 1}{y}}\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-153}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -8.5e+56)
   (/ (/ y x) (+ y x))
   (if (<= x -15800000.0)
     (/ (/ x y) (+ y x))
     (if (<= x -1.02e-38)
       (/ 1.0 (* x (/ (+ x 1.0) y)))
       (if (<= x -1.65e-85)
         (/ x (* y (+ y 1.0)))
         (if (<= x -4.1e-153) (/ y (+ y x)) (/ (/ x y) (+ y 1.0))))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -8.5e+56) {
		tmp = (y / x) / (y + x);
	} else if (x <= -15800000.0) {
		tmp = (x / y) / (y + x);
	} else if (x <= -1.02e-38) {
		tmp = 1.0 / (x * ((x + 1.0) / y));
	} else if (x <= -1.65e-85) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -4.1e-153) {
		tmp = y / (y + x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-8.5d+56)) then
        tmp = (y / x) / (y + x)
    else if (x <= (-15800000.0d0)) then
        tmp = (x / y) / (y + x)
    else if (x <= (-1.02d-38)) then
        tmp = 1.0d0 / (x * ((x + 1.0d0) / y))
    else if (x <= (-1.65d-85)) then
        tmp = x / (y * (y + 1.0d0))
    else if (x <= (-4.1d-153)) then
        tmp = y / (y + x)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -8.5e+56) {
		tmp = (y / x) / (y + x);
	} else if (x <= -15800000.0) {
		tmp = (x / y) / (y + x);
	} else if (x <= -1.02e-38) {
		tmp = 1.0 / (x * ((x + 1.0) / y));
	} else if (x <= -1.65e-85) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -4.1e-153) {
		tmp = y / (y + x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -8.5e+56:
		tmp = (y / x) / (y + x)
	elif x <= -15800000.0:
		tmp = (x / y) / (y + x)
	elif x <= -1.02e-38:
		tmp = 1.0 / (x * ((x + 1.0) / y))
	elif x <= -1.65e-85:
		tmp = x / (y * (y + 1.0))
	elif x <= -4.1e-153:
		tmp = y / (y + x)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -8.5e+56)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -15800000.0)
		tmp = Float64(Float64(x / y) / Float64(y + x));
	elseif (x <= -1.02e-38)
		tmp = Float64(1.0 / Float64(x * Float64(Float64(x + 1.0) / y)));
	elseif (x <= -1.65e-85)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	elseif (x <= -4.1e-153)
		tmp = Float64(y / Float64(y + x));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8.5e+56)
		tmp = (y / x) / (y + x);
	elseif (x <= -15800000.0)
		tmp = (x / y) / (y + x);
	elseif (x <= -1.02e-38)
		tmp = 1.0 / (x * ((x + 1.0) / y));
	elseif (x <= -1.65e-85)
		tmp = x / (y * (y + 1.0));
	elseif (x <= -4.1e-153)
		tmp = y / (y + x);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -8.5e+56], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -15800000.0], N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.02e-38], N[(1.0 / N[(x * N[(N[(x + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.65e-85], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.1e-153], N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -8.4999999999999998e56

   1. Initial program 52.5%

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

      1. *-commutative 52.5%

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

      2. associate-*l* 52.5%

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

      3. times-frac 87.6%

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

      4. +-commutative 87.6%

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

      5. +-commutative 87.6%

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

      6. associate-+r+ 87.6%

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

      7. +-commutative 87.6%

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

      8. associate-+l+ 87.6%

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

   4. Applied egg-rr 87.6%

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

      1. associate-*l/ 87.6%

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

      2. associate-/r* 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+l+ 99.8%

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

      8. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf 87.9%

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

   if -8.4999999999999998e56 < x < -1.58e7

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-*l* 80.7%

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

      3. times-frac 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.7%

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

      2. associate-/r* 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-+r+ 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+l+ 99.7%

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

      8. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around inf 61.0%

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

   if -1.58e7 < x < -1.01999999999999998e-38

   1. Initial program 99.8%

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

      1. associate-/l* 100.0%

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

      2. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

      1. associate-*r/ 99.8%

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

      2. associate-+r+ 99.8%

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

      3. clear-num 99.3%

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

      4. associate-/l* 99.6%

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

      5. pow2 99.6%

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

      6. +-commutative 99.6%

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

      7. associate-+r+ 99.6%

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

      8. +-commutative 99.6%

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

      9. associate-+l+ 99.6%

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

      10. *-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. /-rgt-identity 99.6%

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

      2. /-rgt-identity 99.6%

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

      3. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in y around 0 58.3%

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

       1. associate-/l* 58.3%

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

       2. +-commutative 58.3%

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

   11. Simplified 58.3%

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

   if -1.01999999999999998e-38 < x < -1.64999999999999986e-85

   1. Initial program 93.1%

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

      1. associate-/l* 93.0%

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

      2. associate-+l+ 93.0%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in x around 0 77.2%

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

   if -1.64999999999999986e-85 < x < -4.1e-153

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

      2. associate-*l* 79.9%

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

      3. times-frac 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+r+ 99.9%

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

      7. +-commutative 99.9%

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

      8. associate-+l+ 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. associate-*l/ 99.8%

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

      2. associate-/r* 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-+r+ 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+l+ 99.7%

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

      8. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around 0 29.0%

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

   8. Taylor expanded in x around 0 29.0%

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

   if -4.1e-153 < x

   1. Initial program 63.7%

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

      1. associate-/l* 76.3%

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

      2. associate-+l+ 76.3%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in x around 0 59.5%

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

      1. associate-/r* 62.2%

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

      2. +-commutative 62.2%

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

   7. Simplified 62.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+56}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -15800000:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \mathbf{elif}\;x \leq -1.02 \cdot 10^{-38}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x + 1}{y}}\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-153}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+57}:\\ \;\;\;\;\frac{y \cdot \frac{1}{x}}{y + x}\\ \mathbf{elif}\;x \leq -30000000:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x + 1}{y}}\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-153}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.15e+57)
   (/ (* y (/ 1.0 x)) (+ y x))
   (if (<= x -30000000.0)
     (/ (/ x y) (+ y x))
     (if (<= x -7.5e-39)
       (/ 1.0 (* x (/ (+ x 1.0) y)))
       (if (<= x -1.4e-85)
         (/ x (* y (+ y 1.0)))
         (if (<= x -4.1e-153) (/ y (+ y x)) (/ (/ x y) (+ y 1.0))))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.15e+57) {
		tmp = (y * (1.0 / x)) / (y + x);
	} else if (x <= -30000000.0) {
		tmp = (x / y) / (y + x);
	} else if (x <= -7.5e-39) {
		tmp = 1.0 / (x * ((x + 1.0) / y));
	} else if (x <= -1.4e-85) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -4.1e-153) {
		tmp = y / (y + x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.15d+57)) then
        tmp = (y * (1.0d0 / x)) / (y + x)
    else if (x <= (-30000000.0d0)) then
        tmp = (x / y) / (y + x)
    else if (x <= (-7.5d-39)) then
        tmp = 1.0d0 / (x * ((x + 1.0d0) / y))
    else if (x <= (-1.4d-85)) then
        tmp = x / (y * (y + 1.0d0))
    else if (x <= (-4.1d-153)) then
        tmp = y / (y + x)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.15e+57) {
		tmp = (y * (1.0 / x)) / (y + x);
	} else if (x <= -30000000.0) {
		tmp = (x / y) / (y + x);
	} else if (x <= -7.5e-39) {
		tmp = 1.0 / (x * ((x + 1.0) / y));
	} else if (x <= -1.4e-85) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -4.1e-153) {
		tmp = y / (y + x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.15e+57:
		tmp = (y * (1.0 / x)) / (y + x)
	elif x <= -30000000.0:
		tmp = (x / y) / (y + x)
	elif x <= -7.5e-39:
		tmp = 1.0 / (x * ((x + 1.0) / y))
	elif x <= -1.4e-85:
		tmp = x / (y * (y + 1.0))
	elif x <= -4.1e-153:
		tmp = y / (y + x)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.15e+57)
		tmp = Float64(Float64(y * Float64(1.0 / x)) / Float64(y + x));
	elseif (x <= -30000000.0)
		tmp = Float64(Float64(x / y) / Float64(y + x));
	elseif (x <= -7.5e-39)
		tmp = Float64(1.0 / Float64(x * Float64(Float64(x + 1.0) / y)));
	elseif (x <= -1.4e-85)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	elseif (x <= -4.1e-153)
		tmp = Float64(y / Float64(y + x));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.15e+57)
		tmp = (y * (1.0 / x)) / (y + x);
	elseif (x <= -30000000.0)
		tmp = (x / y) / (y + x);
	elseif (x <= -7.5e-39)
		tmp = 1.0 / (x * ((x + 1.0) / y));
	elseif (x <= -1.4e-85)
		tmp = x / (y * (y + 1.0));
	elseif (x <= -4.1e-153)
		tmp = y / (y + x);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.15e+57], N[(N[(y * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -30000000.0], N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7.5e-39], N[(1.0 / N[(x * N[(N[(x + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.4e-85], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.1e-153], N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -1.1499999999999999e57

   1. Initial program 52.5%

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

      1. *-commutative 52.5%

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

      2. associate-*l* 52.5%

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

      3. times-frac 87.6%

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

      4. +-commutative 87.6%

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

      5. +-commutative 87.6%

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

      6. associate-+r+ 87.6%

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

      7. +-commutative 87.6%

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

      8. associate-+l+ 87.6%

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

   4. Applied egg-rr 87.6%

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

      1. associate-*l/ 87.6%

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

      2. associate-/r* 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+l+ 99.8%

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

      8. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf 87.8%

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

   if -1.1499999999999999e57 < x < -3e7

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-*l* 80.7%

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

      3. times-frac 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.7%

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

      2. associate-/r* 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-+r+ 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+l+ 99.7%

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

      8. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around inf 61.0%

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

   if -3e7 < x < -7.49999999999999971e-39

   1. Initial program 99.8%

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

      1. associate-/l* 100.0%

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

      2. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

      1. associate-*r/ 99.8%

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

      2. associate-+r+ 99.8%

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

      3. clear-num 99.3%

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

      4. associate-/l* 99.6%

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

      5. pow2 99.6%

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

      6. +-commutative 99.6%

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

      7. associate-+r+ 99.6%

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

      8. +-commutative 99.6%

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

      9. associate-+l+ 99.6%

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

      10. *-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. /-rgt-identity 99.6%

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

      2. /-rgt-identity 99.6%

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

      3. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in y around 0 58.3%

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

       1. associate-/l* 58.3%

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

       2. +-commutative 58.3%

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

   11. Simplified 58.3%

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

   if -7.49999999999999971e-39 < x < -1.40000000000000008e-85

   1. Initial program 93.1%

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

      1. associate-/l* 93.0%

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

      2. associate-+l+ 93.0%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in x around 0 77.2%

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

   if -1.40000000000000008e-85 < x < -4.1e-153

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

      2. associate-*l* 79.9%

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

      3. times-frac 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+r+ 99.9%

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

      7. +-commutative 99.9%

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

      8. associate-+l+ 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. associate-*l/ 99.8%

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

      2. associate-/r* 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-+r+ 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+l+ 99.7%

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

      8. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around 0 29.0%

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

   8. Taylor expanded in x around 0 29.0%

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

   if -4.1e-153 < x

   1. Initial program 63.7%

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

      1. associate-/l* 76.3%

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

      2. associate-+l+ 76.3%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in x around 0 59.5%

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

      1. associate-/r* 62.2%

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

      2. +-commutative 62.2%

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

   7. Simplified 62.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+57}:\\ \;\;\;\;\frac{y \cdot \frac{1}{x}}{y + x}\\ \mathbf{elif}\;x \leq -30000000:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x + 1}{y}}\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-153}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+57}:\\ \;\;\;\;\frac{y \cdot \frac{1}{x}}{y + x}\\ \mathbf{elif}\;x \leq -59000000:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \mathbf{elif}\;x \leq -2.25 \cdot 10^{-38}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x + 1}{y}}\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-153}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1e+57)
   (/ (* y (/ 1.0 x)) (+ y x))
   (if (<= x -59000000.0)
     (/ (/ x (+ y 1.0)) (+ y x))
     (if (<= x -2.25e-38)
       (/ 1.0 (* x (/ (+ x 1.0) y)))
       (if (<= x -1.2e-85)
         (/ x (* y (+ y 1.0)))
         (if (<= x -4.1e-153) (/ y (+ y x)) (/ (/ x y) (+ y 1.0))))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1e+57) {
		tmp = (y * (1.0 / x)) / (y + x);
	} else if (x <= -59000000.0) {
		tmp = (x / (y + 1.0)) / (y + x);
	} else if (x <= -2.25e-38) {
		tmp = 1.0 / (x * ((x + 1.0) / y));
	} else if (x <= -1.2e-85) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -4.1e-153) {
		tmp = y / (y + x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1d+57)) then
        tmp = (y * (1.0d0 / x)) / (y + x)
    else if (x <= (-59000000.0d0)) then
        tmp = (x / (y + 1.0d0)) / (y + x)
    else if (x <= (-2.25d-38)) then
        tmp = 1.0d0 / (x * ((x + 1.0d0) / y))
    else if (x <= (-1.2d-85)) then
        tmp = x / (y * (y + 1.0d0))
    else if (x <= (-4.1d-153)) then
        tmp = y / (y + x)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1e+57) {
		tmp = (y * (1.0 / x)) / (y + x);
	} else if (x <= -59000000.0) {
		tmp = (x / (y + 1.0)) / (y + x);
	} else if (x <= -2.25e-38) {
		tmp = 1.0 / (x * ((x + 1.0) / y));
	} else if (x <= -1.2e-85) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -4.1e-153) {
		tmp = y / (y + x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1e+57:
		tmp = (y * (1.0 / x)) / (y + x)
	elif x <= -59000000.0:
		tmp = (x / (y + 1.0)) / (y + x)
	elif x <= -2.25e-38:
		tmp = 1.0 / (x * ((x + 1.0) / y))
	elif x <= -1.2e-85:
		tmp = x / (y * (y + 1.0))
	elif x <= -4.1e-153:
		tmp = y / (y + x)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1e+57)
		tmp = Float64(Float64(y * Float64(1.0 / x)) / Float64(y + x));
	elseif (x <= -59000000.0)
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	elseif (x <= -2.25e-38)
		tmp = Float64(1.0 / Float64(x * Float64(Float64(x + 1.0) / y)));
	elseif (x <= -1.2e-85)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	elseif (x <= -4.1e-153)
		tmp = Float64(y / Float64(y + x));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1e+57)
		tmp = (y * (1.0 / x)) / (y + x);
	elseif (x <= -59000000.0)
		tmp = (x / (y + 1.0)) / (y + x);
	elseif (x <= -2.25e-38)
		tmp = 1.0 / (x * ((x + 1.0) / y));
	elseif (x <= -1.2e-85)
		tmp = x / (y * (y + 1.0));
	elseif (x <= -4.1e-153)
		tmp = y / (y + x);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1e+57], N[(N[(y * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -59000000.0], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.25e-38], N[(1.0 / N[(x * N[(N[(x + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.2e-85], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.1e-153], N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -1.00000000000000005e57

   1. Initial program 52.5%

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

      1. *-commutative 52.5%

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

      2. associate-*l* 52.5%

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

      3. times-frac 87.6%

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

      4. +-commutative 87.6%

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

      5. +-commutative 87.6%

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

      6. associate-+r+ 87.6%

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

      7. +-commutative 87.6%

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

      8. associate-+l+ 87.6%

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

   4. Applied egg-rr 87.6%

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

      1. associate-*l/ 87.6%

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

      2. associate-/r* 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+l+ 99.8%

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

      8. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf 87.8%

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

   if -1.00000000000000005e57 < x < -5.9e7

   1. Initial program 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-*l* 80.7%

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

      3. times-frac 99.7%

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

      4. +-commutative 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.7%

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

      2. associate-/r* 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-+r+ 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+l+ 99.7%

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

      8. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0 61.4%

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

      1. +-commutative 61.4%

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

   9. Simplified 61.4%

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

   if -5.9e7 < x < -2.25000000000000004e-38

   1. Initial program 99.8%

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

      1. associate-/l* 100.0%

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

      2. associate-+l+ 100.0%

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

   3. Simplified 100.0%

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

      1. associate-*r/ 99.8%

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

      2. associate-+r+ 99.8%

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

      3. clear-num 99.3%

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

      4. associate-/l* 99.6%

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

      5. pow2 99.6%

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

      6. +-commutative 99.6%

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

      7. associate-+r+ 99.6%

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

      8. +-commutative 99.6%

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

      9. associate-+l+ 99.6%

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

      10. *-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. /-rgt-identity 99.6%

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

      2. /-rgt-identity 99.6%

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

      3. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in y around 0 58.3%

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

       1. associate-/l* 58.3%

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

       2. +-commutative 58.3%

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

   11. Simplified 58.3%

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

   if -2.25000000000000004e-38 < x < -1.2e-85

   1. Initial program 93.1%

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

      1. associate-/l* 93.0%

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

      2. associate-+l+ 93.0%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in x around 0 77.2%

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

   if -1.2e-85 < x < -4.1e-153

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

      2. associate-*l* 79.9%

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

      3. times-frac 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+r+ 99.9%

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

      7. +-commutative 99.9%

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

      8. associate-+l+ 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. associate-*l/ 99.8%

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

      2. associate-/r* 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-+r+ 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+l+ 99.7%

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

      8. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around 0 29.0%

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

   8. Taylor expanded in x around 0 29.0%

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

   if -4.1e-153 < x

   1. Initial program 63.7%

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

      1. associate-/l* 76.3%

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

      2. associate-+l+ 76.3%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in x around 0 59.5%

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

      1. associate-/r* 62.2%

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

      2. +-commutative 62.2%

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

   7. Simplified 62.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+57}:\\ \;\;\;\;\frac{y \cdot \frac{1}{x}}{y + x}\\ \mathbf{elif}\;x \leq -59000000:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \mathbf{elif}\;x \leq -2.25 \cdot 10^{-38}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x + 1}{y}}\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-153}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.58 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.24 \cdot 10^{-147}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-8}:\\ \;\;\;\;\frac{y \cdot x}{\left(x + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+132}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{\left(y + x\right) + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.58e-158)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 1.24e-147)
     (/ x y)
     (if (<= y 1.45e-8)
       (/ (* y x) (* (+ x 1.0) (* (+ y x) (+ y x))))
       (if (<= y 1.8e+132)
         (/ x (* (+ y x) (+ y (+ x 1.0))))
         (/ (/ x (+ y x)) (+ (+ y x) 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.58e-158) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 1.24e-147) {
		tmp = x / y;
	} else if (y <= 1.45e-8) {
		tmp = (y * x) / ((x + 1.0) * ((y + x) * (y + x)));
	} else if (y <= 1.8e+132) {
		tmp = x / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = (x / (y + x)) / ((y + x) + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.58d-158) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 1.24d-147) then
        tmp = x / y
    else if (y <= 1.45d-8) then
        tmp = (y * x) / ((x + 1.0d0) * ((y + x) * (y + x)))
    else if (y <= 1.8d+132) then
        tmp = x / ((y + x) * (y + (x + 1.0d0)))
    else
        tmp = (x / (y + x)) / ((y + x) + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.58e-158) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 1.24e-147) {
		tmp = x / y;
	} else if (y <= 1.45e-8) {
		tmp = (y * x) / ((x + 1.0) * ((y + x) * (y + x)));
	} else if (y <= 1.8e+132) {
		tmp = x / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = (x / (y + x)) / ((y + x) + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.58e-158:
		tmp = (y / x) / (x + 1.0)
	elif y <= 1.24e-147:
		tmp = x / y
	elif y <= 1.45e-8:
		tmp = (y * x) / ((x + 1.0) * ((y + x) * (y + x)))
	elif y <= 1.8e+132:
		tmp = x / ((y + x) * (y + (x + 1.0)))
	else:
		tmp = (x / (y + x)) / ((y + x) + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.58e-158)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 1.24e-147)
		tmp = Float64(x / y);
	elseif (y <= 1.45e-8)
		tmp = Float64(Float64(y * x) / Float64(Float64(x + 1.0) * Float64(Float64(y + x) * Float64(y + x))));
	elseif (y <= 1.8e+132)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(Float64(y + x) + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.58e-158)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 1.24e-147)
		tmp = x / y;
	elseif (y <= 1.45e-8)
		tmp = (y * x) / ((x + 1.0) * ((y + x) * (y + x)));
	elseif (y <= 1.8e+132)
		tmp = x / ((y + x) * (y + (x + 1.0)));
	else
		tmp = (x / (y + x)) / ((y + x) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.58e-158], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.24e-147], N[(x / y), $MachinePrecision], If[LessEqual[y, 1.45e-8], N[(N[(y * x), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e+132], 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] / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < 1.57999999999999994e-158

   1. Initial program 62.7%

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

      1. *-commutative 62.7%

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

      2. associate-*l* 62.7%

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

      3. times-frac 93.0%

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

      4. +-commutative 93.0%

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

      5. +-commutative 93.0%

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

      6. associate-+r+ 93.0%

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

      7. +-commutative 93.0%

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

      8. associate-+l+ 93.0%

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

   4. Applied egg-rr 93.0%

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

      1. associate-*l/ 93.0%

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

      2. associate-/r* 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+l+ 99.8%

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

      8. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around 0 51.7%

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

      1. associate-/r* 54.1%

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

   9. Simplified 54.1%

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

   if 1.57999999999999994e-158 < y < 1.2400000000000001e-147

   1. Initial program 28.0%

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

      1. associate-/l* 90.1%

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

      2. associate-+l+ 90.1%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in x around 0 76.0%

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

   6. Taylor expanded in y around 0 76.0%

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

   if 1.2400000000000001e-147 < y < 1.4500000000000001e-8

   1. Initial program 85.2%

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

   3. Taylor expanded in y around 0 84.6%

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

      1. +-commutative 84.6%

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

   5. Simplified 84.6%

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

   if 1.4500000000000001e-8 < y < 1.80000000000000008e132

   1. Initial program 64.1%

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

      1. *-commutative 64.1%

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

      2. associate-*l* 64.1%

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

      3. times-frac 96.8%

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

      4. +-commutative 96.8%

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

      5. +-commutative 96.8%

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

      6. associate-+r+ 96.8%

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

      7. +-commutative 96.8%

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

      8. associate-+l+ 96.8%

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

   4. Applied egg-rr 96.8%

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

   5. Taylor expanded in y around inf 72.9%

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

   if 1.80000000000000008e132 < y

   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 88.8%

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

      4. +-commutative 88.8%

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

      5. +-commutative 88.8%

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

      6. associate-+r+ 88.8%

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

      7. +-commutative 88.8%

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

      8. associate-+l+ 88.8%

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

   4. Applied egg-rr 88.8%

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

   5. Taylor expanded in y around inf 88.8%

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

      1. add-exp-log 85.1%

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

      2. associate-/r* 84.4%

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

      3. +-commutative 84.4%

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

      4. log-div 75.0%

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

      5. +-commutative 75.0%

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

      6. associate-+r+ 75.0%

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

      7. +-commutative 75.0%

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

      8. log1p-define 75.0%

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

      9. +-commutative 75.0%

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

   7. Applied egg-rr 75.0%

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

      1. exp-diff 74.9%

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

      2. rem-exp-log 90.1%

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

      3. log1p-undefine 90.1%

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

      4. rem-exp-log 91.7%

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

   9. Simplified 91.7%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.58 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.24 \cdot 10^{-147}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-8}:\\ \;\;\;\;\frac{y \cdot x}{\left(x + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+132}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{\left(y + x\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+58}:\\ \;\;\;\;\frac{y \cdot \frac{1}{x}}{y + x}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-166}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -4.2e+58)
   (/ (* y (/ 1.0 x)) (+ y x))
   (if (<= x -8.2e-166)
     (* x (/ y (* (+ x (+ y 1.0)) (* (+ y x) (+ y x)))))
     (/ (/ x y) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4.2e+58) {
		tmp = (y * (1.0 / x)) / (y + x);
	} else if (x <= -8.2e-166) {
		tmp = x * (y / ((x + (y + 1.0)) * ((y + x) * (y + x))));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.2d+58)) then
        tmp = (y * (1.0d0 / x)) / (y + x)
    else if (x <= (-8.2d-166)) then
        tmp = x * (y / ((x + (y + 1.0d0)) * ((y + x) * (y + x))))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -4.2e+58) {
		tmp = (y * (1.0 / x)) / (y + x);
	} else if (x <= -8.2e-166) {
		tmp = x * (y / ((x + (y + 1.0)) * ((y + x) * (y + x))));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -4.2e+58:
		tmp = (y * (1.0 / x)) / (y + x)
	elif x <= -8.2e-166:
		tmp = x * (y / ((x + (y + 1.0)) * ((y + x) * (y + x))))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -4.2e+58)
		tmp = Float64(Float64(y * Float64(1.0 / x)) / Float64(y + x));
	elseif (x <= -8.2e-166)
		tmp = Float64(x * Float64(y / Float64(Float64(x + Float64(y + 1.0)) * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.2e+58)
		tmp = (y * (1.0 / x)) / (y + x);
	elseif (x <= -8.2e-166)
		tmp = x * (y / ((x + (y + 1.0)) * ((y + x) * (y + x))));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -4.2e+58], N[(N[(y * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8.2e-166], 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], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.20000000000000024e58

   1. Initial program 52.5%

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

      1. *-commutative 52.5%

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

      2. associate-*l* 52.5%

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

      3. times-frac 87.6%

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

      4. +-commutative 87.6%

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

      5. +-commutative 87.6%

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

      6. associate-+r+ 87.6%

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

      7. +-commutative 87.6%

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

      8. associate-+l+ 87.6%

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

   4. Applied egg-rr 87.6%

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

      1. associate-*l/ 87.6%

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

      2. associate-/r* 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+l+ 99.8%

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

      8. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf 87.8%

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

   if -4.20000000000000024e58 < x < -8.1999999999999995e-166

   1. Initial program 88.2%

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

      1. associate-/l* 95.1%

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

      2. associate-+l+ 95.1%

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

   3. Simplified 95.1%

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

   if -8.1999999999999995e-166 < x

   1. Initial program 63.5%

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

      1. associate-/l* 76.2%

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

      2. associate-+l+ 76.2%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in x around 0 59.3%

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

      1. associate-/r* 61.9%

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

      2. +-commutative 61.9%

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

   7. Simplified 61.9%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+58}:\\ \;\;\;\;\frac{y \cdot \frac{1}{x}}{y + x}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-166}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-39}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-85} \lor \neg \left(x \leq -4.1 \cdot 10^{-153}\right):\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -8e-39)
   (/ y (* x (+ x 1.0)))
   (if (or (<= x -1.2e-85) (not (<= x -4.1e-153)))
     (/ x (* y (+ y 1.0)))
     (/ y (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -8e-39) {
		tmp = y / (x * (x + 1.0));
	} else if ((x <= -1.2e-85) || !(x <= -4.1e-153)) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = y / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-8d-39)) then
        tmp = y / (x * (x + 1.0d0))
    else if ((x <= (-1.2d-85)) .or. (.not. (x <= (-4.1d-153)))) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = y / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -8e-39) {
		tmp = y / (x * (x + 1.0));
	} else if ((x <= -1.2e-85) || !(x <= -4.1e-153)) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = y / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -8e-39:
		tmp = y / (x * (x + 1.0))
	elif (x <= -1.2e-85) or not (x <= -4.1e-153):
		tmp = x / (y * (y + 1.0))
	else:
		tmp = y / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -8e-39)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif ((x <= -1.2e-85) || !(x <= -4.1e-153))
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(y / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8e-39)
		tmp = y / (x * (x + 1.0));
	elseif ((x <= -1.2e-85) || ~((x <= -4.1e-153)))
		tmp = x / (y * (y + 1.0));
	else
		tmp = y / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -8e-39], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -1.2e-85], N[Not[LessEqual[x, -4.1e-153]], $MachinePrecision]], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.99999999999999943e-39

   1. Initial program 60.2%

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

      1. associate-/l* 74.6%

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

      2. associate-+l+ 74.6%

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

   3. Simplified 74.6%

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

   5. Taylor expanded in y around 0 76.6%

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

      1. +-commutative 76.6%

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

   7. Simplified 76.6%

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

   if -7.99999999999999943e-39 < x < -1.2e-85 or -4.1e-153 < x

   1. Initial program 66.0%

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

      1. associate-/l* 77.6%

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

      2. associate-+l+ 77.6%

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

   3. Simplified 77.6%

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

   5. Taylor expanded in x around 0 60.9%

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

   if -1.2e-85 < x < -4.1e-153

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

      2. associate-*l* 79.9%

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

      3. times-frac 99.9%

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

      4. +-commutative 99.9%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. +-commutative 99.9%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-+r+ 99.9%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      7. +-commutative 99.9%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      8. associate-+l+ 99.9%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}}{y + x}} \]

      2. associate-/r* 99.7%

         \[\leadsto \frac{y \cdot \color{blue}{\frac{\frac{x}{y + x}}{y + \left(1 + x\right)}}}{y + x} \]

      3. +-commutative 99.7%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{\color{blue}{x + y}}}{y + \left(1 + x\right)}}{y + x} \]

      4. +-commutative 99.7%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{y + \color{blue}{\left(x + 1\right)}}}{y + x} \]

      5. associate-+r+ 99.7%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{\color{blue}{\left(y + x\right) + 1}}}{y + x} \]

      6. +-commutative 99.7%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{\color{blue}{\left(x + y\right)} + 1}}{y + x} \]

      7. associate-+l+ 99.7%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{\color{blue}{x + \left(y + 1\right)}}}{y + x} \]

      8. +-commutative 99.7%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{x + \left(y + 1\right)}}{\color{blue}{x + y}} \]

   6. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{y \cdot \frac{\frac{x}{x + y}}{x + \left(y + 1\right)}}{x + y}} \]

   7. Taylor expanded in y around 0 29.0%

      \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]

   8. Taylor expanded in x around 0 29.0%

      \[\leadsto \frac{\color{blue}{y}}{x + y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-39}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-85} \lor \neg \left(x \leq -4.1 \cdot 10^{-153}\right):\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 93.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{y \cdot \frac{1}{x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -7.5e+58)
   (/ (* y (/ 1.0 x)) (+ y x))
   (* x (/ (/ y (* (+ y x) (+ y (+ x 1.0)))) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -7.5e+58) {
		tmp = (y * (1.0 / x)) / (y + x);
	} else {
		tmp = x * ((y / ((y + x) * (y + (x + 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 <= (-7.5d+58)) then
        tmp = (y * (1.0d0 / x)) / (y + x)
    else
        tmp = x * ((y / ((y + x) * (y + (x + 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 <= -7.5e+58) {
		tmp = (y * (1.0 / x)) / (y + x);
	} else {
		tmp = x * ((y / ((y + x) * (y + (x + 1.0)))) / (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -7.5e+58:
		tmp = (y * (1.0 / x)) / (y + x)
	else:
		tmp = x * ((y / ((y + x) * (y + (x + 1.0)))) / (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -7.5e+58)
		tmp = Float64(Float64(y * Float64(1.0 / x)) / Float64(y + x));
	else
		tmp = Float64(x * Float64(Float64(y / Float64(Float64(y + x) * Float64(y + Float64(x + 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 <= -7.5e+58)
		tmp = (y * (1.0 / x)) / (y + x);
	else
		tmp = x * ((y / ((y + x) * (y + (x + 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, -7.5e+58], N[(N[(y * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.5000000000000001e58

   1. Initial program 52.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 52.5%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. associate-*l* 52.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      3. times-frac 87.6%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. +-commutative 87.6%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. +-commutative 87.6%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-+r+ 87.6%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      7. +-commutative 87.6%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      8. associate-+l+ 87.6%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

   4. Applied egg-rr 87.6%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*l/ 87.6%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}}{y + x}} \]

      2. associate-/r* 99.8%

         \[\leadsto \frac{y \cdot \color{blue}{\frac{\frac{x}{y + x}}{y + \left(1 + x\right)}}}{y + x} \]

      3. +-commutative 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{\color{blue}{x + y}}}{y + \left(1 + x\right)}}{y + x} \]

      4. +-commutative 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{y + \color{blue}{\left(x + 1\right)}}}{y + x} \]

      5. associate-+r+ 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{\color{blue}{\left(y + x\right) + 1}}}{y + x} \]

      6. +-commutative 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{\color{blue}{\left(x + y\right)} + 1}}{y + x} \]

      7. associate-+l+ 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{\color{blue}{x + \left(y + 1\right)}}}{y + x} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{x + \left(y + 1\right)}}{\color{blue}{x + y}} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{y \cdot \frac{\frac{x}{x + y}}{x + \left(y + 1\right)}}{x + y}} \]

   7. Taylor expanded in x around inf 87.8%

      \[\leadsto \frac{y \cdot \color{blue}{\frac{1}{x}}}{x + y} \]

   if -7.5000000000000001e58 < x

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 80.0%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 80.0%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 80.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-un-lft-identity 80.0%

         \[\leadsto x \cdot \frac{\color{blue}{1 \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)} \]

      2. associate-+r+ 80.0%

         \[\leadsto x \cdot \frac{1 \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      3. associate-*l* 80.0%

         \[\leadsto x \cdot \frac{1 \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      4. times-frac 92.1%

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}\right)} \]

      5. +-commutative 92.1%

         \[\leadsto x \cdot \left(\frac{1}{\color{blue}{y + x}} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}\right) \]

      6. +-commutative 92.1%

         \[\leadsto x \cdot \left(\frac{1}{y + x} \cdot \frac{y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)}\right) \]

      7. associate-+r+ 92.1%

         \[\leadsto x \cdot \left(\frac{1}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}}\right) \]

      8. +-commutative 92.1%

         \[\leadsto x \cdot \left(\frac{1}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}}\right) \]

      9. associate-+l+ 92.1%

         \[\leadsto x \cdot \left(\frac{1}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}}\right) \]

   6. Applied egg-rr 92.1%

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 92.2%

         \[\leadsto x \cdot \color{blue}{\frac{1 \cdot \frac{y}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}}{y + x}} \]

      2. *-lft-identity 92.2%

         \[\leadsto x \cdot \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}}}{y + x} \]

      3. +-commutative 92.2%

         \[\leadsto x \cdot \frac{\frac{y}{\left(y + x\right) \cdot \left(y + \color{blue}{\left(x + 1\right)}\right)}}{y + x} \]

   8. Simplified 92.2%

      \[\leadsto x \cdot \color{blue}{\frac{\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}}{y + x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{y \cdot \frac{1}{x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 95.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+143}:\\ \;\;\;\;\frac{y \cdot \frac{1}{x + \left(y + 1\right)}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)} \cdot \frac{y}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.6e+143)
   (/ (* y (/ 1.0 (+ x (+ y 1.0)))) (+ y x))
   (* (/ x (* (+ y x) (+ y (+ x 1.0)))) (/ y (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.6e+143) {
		tmp = (y * (1.0 / (x + (y + 1.0)))) / (y + x);
	} else {
		tmp = (x / ((y + x) * (y + (x + 1.0)))) * (y / (y + x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.6d+143)) then
        tmp = (y * (1.0d0 / (x + (y + 1.0d0)))) / (y + x)
    else
        tmp = (x / ((y + x) * (y + (x + 1.0d0)))) * (y / (y + x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.6e+143) {
		tmp = (y * (1.0 / (x + (y + 1.0)))) / (y + x);
	} else {
		tmp = (x / ((y + x) * (y + (x + 1.0)))) * (y / (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.6e+143:
		tmp = (y * (1.0 / (x + (y + 1.0)))) / (y + x)
	else:
		tmp = (x / ((y + x) * (y + (x + 1.0)))) * (y / (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.6e+143)
		tmp = Float64(Float64(y * Float64(1.0 / Float64(x + Float64(y + 1.0)))) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0)))) * Float64(y / Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.6e+143)
		tmp = (y * (1.0 / (x + (y + 1.0)))) / (y + x);
	else
		tmp = (x / ((y + x) * (y + (x + 1.0)))) * (y / (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.6e+143], N[(N[(y * N[(1.0 / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5999999999999999e143

   1. Initial program 57.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 57.0%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. associate-*l* 57.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      3. times-frac 85.8%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. +-commutative 85.8%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. +-commutative 85.8%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-+r+ 85.8%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      7. +-commutative 85.8%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      8. associate-+l+ 85.8%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

   4. Applied egg-rr 85.8%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*l/ 85.8%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}}{y + x}} \]

      2. associate-/r* 99.9%

         \[\leadsto \frac{y \cdot \color{blue}{\frac{\frac{x}{y + x}}{y + \left(1 + x\right)}}}{y + x} \]

      3. +-commutative 99.9%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{\color{blue}{x + y}}}{y + \left(1 + x\right)}}{y + x} \]

      4. +-commutative 99.9%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{y + \color{blue}{\left(x + 1\right)}}}{y + x} \]

      5. associate-+r+ 99.9%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{\color{blue}{\left(y + x\right) + 1}}}{y + x} \]

      6. +-commutative 99.9%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{\color{blue}{\left(x + y\right)} + 1}}{y + x} \]

      7. associate-+l+ 99.9%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{\color{blue}{x + \left(y + 1\right)}}}{y + x} \]

      8. +-commutative 99.9%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{x + \left(y + 1\right)}}{\color{blue}{x + y}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{y \cdot \frac{\frac{x}{x + y}}{x + \left(y + 1\right)}}{x + y}} \]

   7. Taylor expanded in x around inf 93.2%

      \[\leadsto \frac{y \cdot \frac{\color{blue}{1}}{x + \left(y + 1\right)}}{x + y} \]

   if -2.5999999999999999e143 < x

   1. Initial program 66.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 66.6%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. associate-*l* 66.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      3. times-frac 94.9%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. +-commutative 94.9%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. +-commutative 94.9%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-+r+ 94.9%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      7. +-commutative 94.9%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      8. associate-+l+ 94.9%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

   4. Applied egg-rr 94.9%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+143}:\\ \;\;\;\;\frac{y \cdot \frac{1}{x + \left(y + 1\right)}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)} \cdot \frac{y}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 86.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.58 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+77}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.58e-158)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 3e+77)
     (/ x (* (+ y x) (+ y (+ x 1.0))))
     (/ (/ x (+ y 1.0)) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.58e-158) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 3e+77) {
		tmp = x / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.58d-158) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 3d+77) then
        tmp = x / ((y + x) * (y + (x + 1.0d0)))
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.58e-158) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 3e+77) {
		tmp = x / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.58e-158:
		tmp = (y / x) / (x + 1.0)
	elif y <= 3e+77:
		tmp = x / ((y + x) * (y + (x + 1.0)))
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.58e-158)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 3e+77)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.58e-158)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 3e+77)
		tmp = x / ((y + x) * (y + (x + 1.0)));
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.58e-158], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3e+77], N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.57999999999999994e-158

   1. Initial program 62.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 62.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. associate-*l* 62.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      3. times-frac 93.0%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. +-commutative 93.0%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. +-commutative 93.0%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-+r+ 93.0%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      7. +-commutative 93.0%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      8. associate-+l+ 93.0%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

   4. Applied egg-rr 93.0%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*l/ 93.0%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}}{y + x}} \]

      2. associate-/r* 99.8%

         \[\leadsto \frac{y \cdot \color{blue}{\frac{\frac{x}{y + x}}{y + \left(1 + x\right)}}}{y + x} \]

      3. +-commutative 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{\color{blue}{x + y}}}{y + \left(1 + x\right)}}{y + x} \]

      4. +-commutative 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{y + \color{blue}{\left(x + 1\right)}}}{y + x} \]

      5. associate-+r+ 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{\color{blue}{\left(y + x\right) + 1}}}{y + x} \]

      6. +-commutative 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{\color{blue}{\left(x + y\right)} + 1}}{y + x} \]

      7. associate-+l+ 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{\color{blue}{x + \left(y + 1\right)}}}{y + x} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{x + \left(y + 1\right)}}{\color{blue}{x + y}} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{y \cdot \frac{\frac{x}{x + y}}{x + \left(y + 1\right)}}{x + y}} \]

   7. Taylor expanded in y around 0 51.7%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* 54.1%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

   9. Simplified 54.1%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

   if 1.57999999999999994e-158 < y < 2.9999999999999998e77

   1. Initial program 77.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 77.0%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. associate-*l* 76.9%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      3. times-frac 99.7%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. +-commutative 99.7%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. +-commutative 99.7%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-+r+ 99.7%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      7. +-commutative 99.7%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      8. associate-+l+ 99.7%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]

   5. Taylor expanded in y around inf 70.7%

      \[\leadsto \color{blue}{1} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)} \]

   if 2.9999999999999998e77 < y

   1. Initial program 62.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 62.0%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. associate-*l* 62.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      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. associate-*l/ 90.4%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}}{y + x}} \]

      2. associate-/r* 99.8%

         \[\leadsto \frac{y \cdot \color{blue}{\frac{\frac{x}{y + x}}{y + \left(1 + x\right)}}}{y + x} \]

      3. +-commutative 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{\color{blue}{x + y}}}{y + \left(1 + x\right)}}{y + x} \]

      4. +-commutative 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{y + \color{blue}{\left(x + 1\right)}}}{y + x} \]

      5. associate-+r+ 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{\color{blue}{\left(y + x\right) + 1}}}{y + x} \]

      6. +-commutative 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{\color{blue}{\left(x + y\right)} + 1}}{y + x} \]

      7. associate-+l+ 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{\color{blue}{x + \left(y + 1\right)}}}{y + x} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{x + \left(y + 1\right)}}{\color{blue}{x + y}} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{y \cdot \frac{\frac{x}{x + y}}{x + \left(y + 1\right)}}{x + y}} \]

   7. Taylor expanded in x around 0 78.8%

      \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{x + y} \]
   8. Step-by-step derivation

      1. +-commutative 78.8%

         \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{x + y} \]

   9. Simplified 78.8%

      \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{x + y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.58 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+77}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 86.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.58 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+132}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{\left(y + x\right) + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.58e-158)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 1.8e+132)
     (/ x (* (+ y x) (+ y (+ x 1.0))))
     (/ (/ x (+ y x)) (+ (+ y x) 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.58e-158) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 1.8e+132) {
		tmp = x / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = (x / (y + x)) / ((y + x) + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.58d-158) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 1.8d+132) then
        tmp = x / ((y + x) * (y + (x + 1.0d0)))
    else
        tmp = (x / (y + x)) / ((y + x) + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.58e-158) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 1.8e+132) {
		tmp = x / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = (x / (y + x)) / ((y + x) + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.58e-158:
		tmp = (y / x) / (x + 1.0)
	elif y <= 1.8e+132:
		tmp = x / ((y + x) * (y + (x + 1.0)))
	else:
		tmp = (x / (y + x)) / ((y + x) + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.58e-158)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 1.8e+132)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(Float64(y + x) + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.58e-158)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 1.8e+132)
		tmp = x / ((y + x) * (y + (x + 1.0)));
	else
		tmp = (x / (y + x)) / ((y + x) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.58e-158], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e+132], 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] / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.57999999999999994e-158

   1. Initial program 62.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 62.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. associate-*l* 62.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      3. times-frac 93.0%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. +-commutative 93.0%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. +-commutative 93.0%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-+r+ 93.0%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      7. +-commutative 93.0%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      8. associate-+l+ 93.0%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

   4. Applied egg-rr 93.0%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*l/ 93.0%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}}{y + x}} \]

      2. associate-/r* 99.8%

         \[\leadsto \frac{y \cdot \color{blue}{\frac{\frac{x}{y + x}}{y + \left(1 + x\right)}}}{y + x} \]

      3. +-commutative 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{\color{blue}{x + y}}}{y + \left(1 + x\right)}}{y + x} \]

      4. +-commutative 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{y + \color{blue}{\left(x + 1\right)}}}{y + x} \]

      5. associate-+r+ 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{\color{blue}{\left(y + x\right) + 1}}}{y + x} \]

      6. +-commutative 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{\color{blue}{\left(x + y\right)} + 1}}{y + x} \]

      7. associate-+l+ 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{\color{blue}{x + \left(y + 1\right)}}}{y + x} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{x + \left(y + 1\right)}}{\color{blue}{x + y}} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{y \cdot \frac{\frac{x}{x + y}}{x + \left(y + 1\right)}}{x + y}} \]

   7. Taylor expanded in y around 0 51.7%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* 54.1%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

   9. Simplified 54.1%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

   if 1.57999999999999994e-158 < y < 1.80000000000000008e132

   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* 70.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      3. times-frac 98.2%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. +-commutative 98.2%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. +-commutative 98.2%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-+r+ 98.2%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      7. +-commutative 98.2%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      8. associate-+l+ 98.2%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

   4. Applied egg-rr 98.2%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]

   5. Taylor expanded in y around inf 69.5%

      \[\leadsto \color{blue}{1} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)} \]

   if 1.80000000000000008e132 < y

   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 88.8%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. +-commutative 88.8%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. +-commutative 88.8%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-+r+ 88.8%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      7. +-commutative 88.8%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      8. associate-+l+ 88.8%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

   4. Applied egg-rr 88.8%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]

   5. Taylor expanded in y around inf 88.8%

      \[\leadsto \color{blue}{1} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)} \]
   6. Step-by-step derivation

      1. add-exp-log 85.1%

         \[\leadsto 1 \cdot \color{blue}{e^{\log \left(\frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}\right)}} \]

      2. associate-/r* 84.4%

         \[\leadsto 1 \cdot e^{\log \color{blue}{\left(\frac{\frac{x}{y + x}}{y + \left(1 + x\right)}\right)}} \]

      3. +-commutative 84.4%

         \[\leadsto 1 \cdot e^{\log \left(\frac{\frac{x}{y + x}}{y + \color{blue}{\left(x + 1\right)}}\right)} \]

      4. log-div 75.0%

         \[\leadsto 1 \cdot e^{\color{blue}{\log \left(\frac{x}{y + x}\right) - \log \left(y + \left(x + 1\right)\right)}} \]

      5. +-commutative 75.0%

         \[\leadsto 1 \cdot e^{\log \left(\frac{x}{\color{blue}{x + y}}\right) - \log \left(y + \left(x + 1\right)\right)} \]

      6. associate-+r+ 75.0%

         \[\leadsto 1 \cdot e^{\log \left(\frac{x}{x + y}\right) - \log \color{blue}{\left(\left(y + x\right) + 1\right)}} \]

      7. +-commutative 75.0%

         \[\leadsto 1 \cdot e^{\log \left(\frac{x}{x + y}\right) - \log \color{blue}{\left(1 + \left(y + x\right)\right)}} \]

      8. log1p-define 75.0%

         \[\leadsto 1 \cdot e^{\log \left(\frac{x}{x + y}\right) - \color{blue}{\mathsf{log1p}\left(y + x\right)}} \]

      9. +-commutative 75.0%

         \[\leadsto 1 \cdot e^{\log \left(\frac{x}{x + y}\right) - \mathsf{log1p}\left(\color{blue}{x + y}\right)} \]

   7. Applied egg-rr 75.0%

      \[\leadsto 1 \cdot \color{blue}{e^{\log \left(\frac{x}{x + y}\right) - \mathsf{log1p}\left(x + y\right)}} \]
   8. Step-by-step derivation

      1. exp-diff 74.9%

         \[\leadsto 1 \cdot \color{blue}{\frac{e^{\log \left(\frac{x}{x + y}\right)}}{e^{\mathsf{log1p}\left(x + y\right)}}} \]

      2. rem-exp-log 90.1%

         \[\leadsto 1 \cdot \frac{\color{blue}{\frac{x}{x + y}}}{e^{\mathsf{log1p}\left(x + y\right)}} \]

      3. log1p-undefine 90.1%

         \[\leadsto 1 \cdot \frac{\frac{x}{x + y}}{e^{\color{blue}{\log \left(1 + \left(x + y\right)\right)}}} \]

      4. rem-exp-log 91.7%

         \[\leadsto 1 \cdot \frac{\frac{x}{x + y}}{\color{blue}{1 + \left(x + y\right)}} \]

   9. Simplified 91.7%

      \[\leadsto 1 \cdot \color{blue}{\frac{\frac{x}{x + y}}{1 + \left(x + y\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.58 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+132}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{\left(y + x\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 66.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{-162}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 8.2e-162) (/ y (+ y x)) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 8.2e-162) {
		tmp = y / (y + x);
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 8.2d-162) then
        tmp = y / (y + x)
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 8.2e-162) {
		tmp = y / (y + x);
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 8.2e-162:
		tmp = y / (y + x)
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 8.2e-162)
		tmp = Float64(y / Float64(y + x));
	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 (y <= 8.2e-162)
		tmp = y / (y + x);
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 8.2e-162], N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.20000000000000039e-162

   1. Initial program 62.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 62.8%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. associate-*l* 62.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      3. times-frac 92.9%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. +-commutative 92.9%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. +-commutative 92.9%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-+r+ 92.9%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      7. +-commutative 92.9%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      8. associate-+l+ 92.9%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

   4. Applied egg-rr 92.9%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*l/ 92.9%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}}{y + x}} \]

      2. associate-/r* 99.8%

         \[\leadsto \frac{y \cdot \color{blue}{\frac{\frac{x}{y + x}}{y + \left(1 + x\right)}}}{y + x} \]

      3. +-commutative 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{\color{blue}{x + y}}}{y + \left(1 + x\right)}}{y + x} \]

      4. +-commutative 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{y + \color{blue}{\left(x + 1\right)}}}{y + x} \]

      5. associate-+r+ 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{\color{blue}{\left(y + x\right) + 1}}}{y + x} \]

      6. +-commutative 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{\color{blue}{\left(x + y\right)} + 1}}{y + x} \]

      7. associate-+l+ 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{\color{blue}{x + \left(y + 1\right)}}}{y + x} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{x + \left(y + 1\right)}}{\color{blue}{x + y}} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{y \cdot \frac{\frac{x}{x + y}}{x + \left(y + 1\right)}}{x + y}} \]

   7. Taylor expanded in y around 0 54.5%

      \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]

   8. Taylor expanded in x around 0 30.0%

      \[\leadsto \frac{\color{blue}{y}}{x + y} \]

   if 8.20000000000000039e-162 < y

   1. Initial program 69.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 80.7%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 80.7%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 80.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 61.4%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{-162}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 35.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-226}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 1.9e-226) (- (/ y x) y) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.9e-226) {
		tmp = (y / x) - y;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.9d-226) then
        tmp = (y / x) - y
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.9e-226) {
		tmp = (y / x) - y;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.9e-226:
		tmp = (y / x) - y
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.9e-226)
		tmp = Float64(Float64(y / x) - y);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.9e-226)
		tmp = (y / x) - y;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.9e-226], N[(N[(y / x), $MachinePrecision] - y), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.89999999999999991e-226

   1. Initial program 61.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 61.8%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. associate-*l* 61.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      3. times-frac 92.1%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. +-commutative 92.1%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. +-commutative 92.1%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-+r+ 92.1%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      7. +-commutative 92.1%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      8. associate-+l+ 92.1%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

   4. Applied egg-rr 92.1%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*l/ 92.1%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}}{y + x}} \]

      2. associate-/r* 99.8%

         \[\leadsto \frac{y \cdot \color{blue}{\frac{\frac{x}{y + x}}{y + \left(1 + x\right)}}}{y + x} \]

      3. +-commutative 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{\color{blue}{x + y}}}{y + \left(1 + x\right)}}{y + x} \]

      4. +-commutative 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{y + \color{blue}{\left(x + 1\right)}}}{y + x} \]

      5. associate-+r+ 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{\color{blue}{\left(y + x\right) + 1}}}{y + x} \]

      6. +-commutative 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{\color{blue}{\left(x + y\right)} + 1}}{y + x} \]

      7. associate-+l+ 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{\color{blue}{x + \left(y + 1\right)}}}{y + x} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{x + \left(y + 1\right)}}{\color{blue}{x + y}} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{y \cdot \frac{\frac{x}{x + y}}{x + \left(y + 1\right)}}{x + y}} \]

   7. Taylor expanded in y around 0 48.5%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* 51.2%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

   9. Simplified 51.2%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

   10. Taylor expanded in x around 0 12.7%

       \[\leadsto \color{blue}{-1 \cdot y + \frac{y}{x}} \]
   11. Step-by-step derivation

       1. +-commutative 12.7%

          \[\leadsto \color{blue}{\frac{y}{x} + -1 \cdot y} \]

       2. neg-mul-1 12.7%

          \[\leadsto \frac{y}{x} + \color{blue}{\left(-y\right)} \]

       3. unsub-neg 12.7%

          \[\leadsto \color{blue}{\frac{y}{x} - y} \]

   12. Simplified 12.7%

       \[\leadsto \color{blue}{\frac{y}{x} - y} \]

   if 1.89999999999999991e-226 < y

   1. Initial program 69.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 80.6%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 80.6%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 80.6%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 56.4%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   6. Taylor expanded in y around 0 32.5%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-226}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 45.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-181}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -4e-181) (/ y (+ y x)) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4e-181) {
		tmp = y / (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 <= (-4d-181)) then
        tmp = y / (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 <= -4e-181) {
		tmp = y / (y + x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -4e-181:
		tmp = y / (y + x)
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -4e-181)
		tmp = Float64(y / 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 <= -4e-181)
		tmp = y / (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, -4e-181], N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.00000000000000019e-181

   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. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 69.3%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. associate-*l* 69.3%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      3. times-frac 93.3%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. +-commutative 93.3%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. +-commutative 93.3%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-+r+ 93.3%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      7. +-commutative 93.3%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      8. associate-+l+ 93.3%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

   4. Applied egg-rr 93.3%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*l/ 93.3%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}}{y + x}} \]

      2. associate-/r* 99.8%

         \[\leadsto \frac{y \cdot \color{blue}{\frac{\frac{x}{y + x}}{y + \left(1 + x\right)}}}{y + x} \]

      3. +-commutative 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{\color{blue}{x + y}}}{y + \left(1 + x\right)}}{y + x} \]

      4. +-commutative 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{y + \color{blue}{\left(x + 1\right)}}}{y + x} \]

      5. associate-+r+ 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{\color{blue}{\left(y + x\right) + 1}}}{y + x} \]

      6. +-commutative 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{\color{blue}{\left(x + y\right)} + 1}}{y + x} \]

      7. associate-+l+ 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{\color{blue}{x + \left(y + 1\right)}}}{y + x} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{y \cdot \frac{\frac{x}{x + y}}{x + \left(y + 1\right)}}{\color{blue}{x + y}} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{y \cdot \frac{\frac{x}{x + y}}{x + \left(y + 1\right)}}{x + y}} \]

   7. Taylor expanded in y around 0 62.0%

      \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]

   8. Taylor expanded in x around 0 27.6%

      \[\leadsto \frac{\color{blue}{y}}{x + y} \]

   if -4.00000000000000019e-181 < x

   1. Initial program 63.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 76.1%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 76.1%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 76.1%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 59.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   6. Taylor expanded in y around 0 38.0%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-181}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 28.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -0.002:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -0.002) (/ 1.0 x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -0.002) {
		tmp = 1.0 / 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 <= (-0.002d0)) then
        tmp = 1.0d0 / 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 <= -0.002) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -0.002:
		tmp = 1.0 / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -0.002)
		tmp = Float64(1.0 / 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 <= -0.002)
		tmp = 1.0 / 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, -0.002], N[(1.0 / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2e-3

   1. Initial program 58.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 58.9%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. associate-*l* 58.9%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      3. times-frac 89.6%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. +-commutative 89.6%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. +-commutative 89.6%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-+r+ 89.6%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      7. +-commutative 89.6%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      8. associate-+l+ 89.6%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

   4. Applied egg-rr 89.6%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]

   5. Taylor expanded in y around inf 52.8%

      \[\leadsto \color{blue}{1} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)} \]

   6. Taylor expanded in x around inf 6.6%

      \[\leadsto 1 \cdot \color{blue}{\frac{1}{x}} \]

   if -2e-3 < x

   1. Initial program 67.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 79.5%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 79.5%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 79.5%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 62.3%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   6. Taylor expanded in y around 0 37.8%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.002:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 27.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{x}{y} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ x y))

C

assert(x < y);
double code(double x, double y) {
	return x / y;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / y
end function

Java

assert x < y;
public static double code(double x, double y) {
	return x / y;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return x / y

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x / y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x / y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 65.4%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l* 78.2%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   2. associate-+l+ 78.2%

      \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

3. Simplified 78.2%

   \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 52.2%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

6. Taylor expanded in y around 0 29.6%

   \[\leadsto \color{blue}{\frac{x}{y}} \]

7. Final simplification 29.6%

   \[\leadsto \frac{x}{y} \]
8. 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 2024039 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))