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

Percentage Accurate: 68.8% → 99.8%

Time: 21.2s

Alternatives: 21

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. associate-+r+ 68.8%

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

   2. *-commutative 68.8%

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

   3. frac-times 89.4%

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

   4. associate-*l/ 83.2%

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

   5. times-frac 99.8%

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

   6. +-commutative 99.8%

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

   7. +-commutative 99.8%

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

   8. associate-+l+ 99.8%

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

   9. +-commutative 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 96.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{x}{t\_0}}{y + x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -0.0062:\\ \;\;\;\;y \cdot \frac{x}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -5e+74)
     (* (/ (/ x t_0) (+ y x)) (/ y x))
     (if (<= x -0.0062)
       (* y (/ x (* (+ y x) (* (+ y x) t_0))))
       (/ (* (/ y (+ y x)) (/ x (+ y 1.0))) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -5e+74) {
		tmp = ((x / t_0) / (y + x)) * (y / x);
	} else if (x <= -0.0062) {
		tmp = y * (x / ((y + x) * ((y + x) * t_0)));
	} else {
		tmp = ((y / (y + x)) * (x / (y + 1.0))) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (x <= (-5d+74)) then
        tmp = ((x / t_0) / (y + x)) * (y / x)
    else if (x <= (-0.0062d0)) then
        tmp = y * (x / ((y + x) * ((y + x) * t_0)))
    else
        tmp = ((y / (y + x)) * (x / (y + 1.0d0))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -5e+74) {
		tmp = ((x / t_0) / (y + x)) * (y / x);
	} else if (x <= -0.0062) {
		tmp = y * (x / ((y + x) * ((y + x) * t_0)));
	} else {
		tmp = ((y / (y + x)) * (x / (y + 1.0))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -5e+74:
		tmp = ((x / t_0) / (y + x)) * (y / x)
	elif x <= -0.0062:
		tmp = y * (x / ((y + x) * ((y + x) * t_0)))
	else:
		tmp = ((y / (y + x)) * (x / (y + 1.0))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -5e+74)
		tmp = Float64(Float64(Float64(x / t_0) / Float64(y + x)) * Float64(y / x));
	elseif (x <= -0.0062)
		tmp = Float64(y * Float64(x / Float64(Float64(y + x) * Float64(Float64(y + x) * t_0))));
	else
		tmp = Float64(Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(y + 1.0))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -5e+74)
		tmp = ((x / t_0) / (y + x)) * (y / x);
	elseif (x <= -0.0062)
		tmp = y * (x / ((y + x) * ((y + x) * t_0)));
	else
		tmp = ((y / (y + x)) * (x / (y + 1.0))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5e+74], N[(N[(N[(x / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -0.0062], N[(y * N[(x / N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.99999999999999963e74

   1. Initial program 40.6%

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

      1. associate-+r+ 40.6%

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

      2. *-commutative 40.6%

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

      3. frac-times 81.3%

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

      4. associate-*l/ 81.3%

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

      5. times-frac 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

      9. +-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0 89.0%

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

   if -4.99999999999999963e74 < x < -0.00619999999999999978

   1. Initial program 81.7%

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

      1. times-frac 95.4%

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

      2. +-commutative 95.4%

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

      3. +-commutative 95.4%

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

      4. +-commutative 95.4%

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

      5. times-frac 81.7%

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

      6. associate-*l/ 95.2%

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

      7. *-commutative 95.2%

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

      8. *-commutative 95.2%

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

      9. distribute-rgt1-in 81.7%

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

      10. fma-def 95.4%

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

      11. +-commutative 95.4%

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

      12. +-commutative 95.4%

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

      13. cube-unmult 95.4%

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

      14. +-commutative 95.4%

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

   3. Simplified 95.4%

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

      1. fma-udef 81.8%

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

      2. cube-mult 81.7%

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

      3. distribute-rgt1-in 95.2%

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

      4. associate-+r+ 95.2%

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

      5. *-commutative 95.2%

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

      6. associate-*l* 95.3%

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

      7. +-commutative 95.3%

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

      8. +-commutative 95.3%

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

      9. +-commutative 95.3%

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

      10. associate-+l+ 95.3%

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

   6. Applied egg-rr 95.3%

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

   if -0.00619999999999999978 < x

   1. Initial program 73.7%

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

      1. times-frac 90.6%

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

      2. +-commutative 90.6%

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

      3. +-commutative 90.6%

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

      4. +-commutative 90.6%

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

      5. times-frac 73.7%

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

      6. associate-*l/ 85.3%

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

      7. *-commutative 85.3%

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

      8. *-commutative 85.3%

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

      9. distribute-rgt1-in 77.9%

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

      10. fma-def 85.3%

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

      11. +-commutative 85.3%

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

      12. +-commutative 85.3%

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

      13. cube-unmult 85.3%

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

      14. +-commutative 85.3%

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

   3. Simplified 85.3%

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

      1. associate-*r/ 73.7%

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

      2. fma-udef 67.3%

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

      3. cube-mult 67.4%

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

      4. distribute-rgt1-in 73.7%

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

      5. associate-+r+ 73.7%

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

      6. *-commutative 73.7%

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

      7. frac-times 90.6%

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

      8. associate-/r* 99.8%

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

      9. associate-*l/ 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 83.5%

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

      1. +-commutative 83.5%

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

   9. Simplified 83.5%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -0.0062:\\ \;\;\;\;y \cdot \frac{x}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+164}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-156}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5e+164)
   (* (/ y (+ y x)) (/ 1.0 x))
   (if (<= x -8.2e-156)
     (* (/ y (* (+ y x) (+ y x))) (/ x (+ x 1.0)))
     (/ (/ x (+ y (+ x 1.0))) (+ y (* x 2.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5e+164) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -8.2e-156) {
		tmp = (y / ((y + x) * (y + x))) * (x / (x + 1.0));
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + (x * 2.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5d+164)) then
        tmp = (y / (y + x)) * (1.0d0 / x)
    else if (x <= (-8.2d-156)) then
        tmp = (y / ((y + x) * (y + x))) * (x / (x + 1.0d0))
    else
        tmp = (x / (y + (x + 1.0d0))) / (y + (x * 2.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -5e+164) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -8.2e-156) {
		tmp = (y / ((y + x) * (y + x))) * (x / (x + 1.0));
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + (x * 2.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -5e+164:
		tmp = (y / (y + x)) * (1.0 / x)
	elif x <= -8.2e-156:
		tmp = (y / ((y + x) * (y + x))) * (x / (x + 1.0))
	else:
		tmp = (x / (y + (x + 1.0))) / (y + (x * 2.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5e+164)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / x));
	elseif (x <= -8.2e-156)
		tmp = Float64(Float64(y / Float64(Float64(y + x) * Float64(y + x))) * Float64(x / Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) / Float64(y + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5e+164)
		tmp = (y / (y + x)) * (1.0 / x);
	elseif (x <= -8.2e-156)
		tmp = (y / ((y + x) * (y + x))) * (x / (x + 1.0));
	else
		tmp = (x / (y + (x + 1.0))) / (y + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -5e+164], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8.2e-156], N[(N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999995e164

   1. Initial program 42.1%

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

      1. associate-+r+ 42.1%

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

      2. *-commutative 42.1%

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

      3. frac-times 72.2%

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

      4. associate-*l/ 72.2%

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

      5. times-frac 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

      9. +-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf 87.0%

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

   if -4.9999999999999995e164 < x < -8.2000000000000004e-156

   1. Initial program 79.1%

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

      1. associate-/r* 85.1%

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

      2. *-commutative 85.1%

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

      3. +-commutative 85.1%

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

      4. +-commutative 85.1%

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

      5. associate-*l/ 98.2%

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

      6. +-commutative 98.2%

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

      7. associate-*r/ 98.2%

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

      8. remove-double-neg 98.2%

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

      9. +-commutative 98.2%

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

      10. +-commutative 98.2%

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

      11. remove-double-neg 98.2%

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

      12. +-commutative 98.2%

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

      13. associate-+l+ 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in y around 0 79.1%

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

      1. +-commutative 79.1%

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

   7. Simplified 79.1%

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

   if -8.2000000000000004e-156 < x

   1. Initial program 69.1%

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

      1. associate-+r+ 69.1%

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

      2. *-commutative 69.1%

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

      3. frac-times 88.6%

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

      4. associate-*l/ 79.2%

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

      5. times-frac 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

      9. +-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. clear-num 99.8%

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

      2. frac-times 99.5%

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

      3. *-un-lft-identity 99.5%

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

      4. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in y around inf 55.1%

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

      1. *-commutative 55.1%

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

   9. Simplified 55.1%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+164}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-156}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+164}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -4.3 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(x + \left(y + \left(y + 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5.2e+164)
   (* (/ y (+ y x)) (/ 1.0 x))
   (if (<= x -4.3e-138)
     (/ y (* (+ y x) (+ x (+ y (+ y 1.0)))))
     (/ (/ x (+ y (+ x 1.0))) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5.2e+164) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -4.3e-138) {
		tmp = y / ((y + x) * (x + (y + (y + 1.0))));
	} else {
		tmp = (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 <= (-5.2d+164)) then
        tmp = (y / (y + x)) * (1.0d0 / x)
    else if (x <= (-4.3d-138)) then
        tmp = y / ((y + x) * (x + (y + (y + 1.0d0))))
    else
        tmp = (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 <= -5.2e+164) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -4.3e-138) {
		tmp = y / ((y + x) * (x + (y + (y + 1.0))));
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -5.2e+164:
		tmp = (y / (y + x)) * (1.0 / x)
	elif x <= -4.3e-138:
		tmp = y / ((y + x) * (x + (y + (y + 1.0))))
	else:
		tmp = (x / (y + (x + 1.0))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5.2e+164)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / x));
	elseif (x <= -4.3e-138)
		tmp = Float64(y / Float64(Float64(y + x) * Float64(x + Float64(y + Float64(y + 1.0)))));
	else
		tmp = Float64(Float64(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 <= -5.2e+164)
		tmp = (y / (y + x)) * (1.0 / x);
	elseif (x <= -4.3e-138)
		tmp = y / ((y + x) * (x + (y + (y + 1.0))));
	else
		tmp = (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, -5.2e+164], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.3e-138], N[(y / N[(N[(y + x), $MachinePrecision] * N[(x + N[(y + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.1999999999999998e164

   1. Initial program 42.1%

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

      1. associate-+r+ 42.1%

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

      2. *-commutative 42.1%

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

      3. frac-times 72.2%

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

      4. associate-*l/ 72.2%

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

      5. times-frac 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

      9. +-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf 87.0%

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

   if -5.1999999999999998e164 < x < -4.3e-138

   1. Initial program 77.2%

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

      1. associate-+r+ 77.2%

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

      2. *-commutative 77.2%

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

      3. frac-times 98.1%

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

      4. associate-*l/ 96.2%

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

      5. times-frac 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

      9. +-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. clear-num 98.6%

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

      2. frac-times 94.5%

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

      3. *-rgt-identity 94.5%

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

      4. div-inv 94.5%

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

      5. associate-+r+ 94.5%

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

      6. +-commutative 94.5%

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

      7. clear-num 94.6%

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

      8. +-commutative 94.6%

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

      9. associate-+r+ 94.6%

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

      10. +-commutative 94.6%

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

   6. Applied egg-rr 94.6%

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

   7. Taylor expanded in x around -inf 75.3%

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

      1. mul-1-neg 75.3%

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

      2. unsub-neg 75.3%

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

      3. neg-mul-1 75.3%

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

      4. distribute-lft-in 75.3%

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

      5. metadata-eval 75.3%

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

      6. neg-mul-1 75.3%

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

      7. +-commutative 75.3%

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

      8. unsub-neg 75.3%

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

      9. unsub-neg 75.3%

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

   9. Simplified 75.3%

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

   if -4.3e-138 < x

   1. Initial program 70.2%

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

      1. associate-+r+ 70.2%

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

      2. *-commutative 70.2%

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

      3. frac-times 89.0%

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

      4. associate-*l/ 79.9%

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

      5. times-frac 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

      9. +-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf 56.5%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+164}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -4.3 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(x + \left(y + \left(y + 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+164}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(x + \left(y + \left(y + 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5e+164)
   (* (/ y (+ y x)) (/ 1.0 x))
   (if (<= x -3.5e-138)
     (/ y (* (+ y x) (+ x (+ y (+ y 1.0)))))
     (/ (/ x (+ y (+ x 1.0))) (+ y (* x 2.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5e+164) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -3.5e-138) {
		tmp = y / ((y + x) * (x + (y + (y + 1.0))));
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + (x * 2.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5d+164)) then
        tmp = (y / (y + x)) * (1.0d0 / x)
    else if (x <= (-3.5d-138)) then
        tmp = y / ((y + x) * (x + (y + (y + 1.0d0))))
    else
        tmp = (x / (y + (x + 1.0d0))) / (y + (x * 2.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -5e+164) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -3.5e-138) {
		tmp = y / ((y + x) * (x + (y + (y + 1.0))));
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + (x * 2.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -5e+164:
		tmp = (y / (y + x)) * (1.0 / x)
	elif x <= -3.5e-138:
		tmp = y / ((y + x) * (x + (y + (y + 1.0))))
	else:
		tmp = (x / (y + (x + 1.0))) / (y + (x * 2.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5e+164)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / x));
	elseif (x <= -3.5e-138)
		tmp = Float64(y / Float64(Float64(y + x) * Float64(x + Float64(y + Float64(y + 1.0)))));
	else
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) / Float64(y + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5e+164)
		tmp = (y / (y + x)) * (1.0 / x);
	elseif (x <= -3.5e-138)
		tmp = y / ((y + x) * (x + (y + (y + 1.0))));
	else
		tmp = (x / (y + (x + 1.0))) / (y + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -5e+164], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.5e-138], N[(y / N[(N[(y + x), $MachinePrecision] * N[(x + N[(y + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999995e164

   1. Initial program 42.1%

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

      1. associate-+r+ 42.1%

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

      2. *-commutative 42.1%

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

      3. frac-times 72.2%

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

      4. associate-*l/ 72.2%

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

      5. times-frac 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

      9. +-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf 87.0%

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

   if -4.9999999999999995e164 < x < -3.4999999999999999e-138

   1. Initial program 77.2%

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

      1. associate-+r+ 77.2%

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

      2. *-commutative 77.2%

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

      3. frac-times 98.1%

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

      4. associate-*l/ 96.2%

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

      5. times-frac 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

      9. +-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. clear-num 98.6%

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

      2. frac-times 94.5%

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

      3. *-rgt-identity 94.5%

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

      4. div-inv 94.5%

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

      5. associate-+r+ 94.5%

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

      6. +-commutative 94.5%

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

      7. clear-num 94.6%

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

      8. +-commutative 94.6%

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

      9. associate-+r+ 94.6%

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

      10. +-commutative 94.6%

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

   6. Applied egg-rr 94.6%

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

   7. Taylor expanded in x around -inf 75.3%

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

      1. mul-1-neg 75.3%

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

      2. unsub-neg 75.3%

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

      3. neg-mul-1 75.3%

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

      4. distribute-lft-in 75.3%

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

      5. metadata-eval 75.3%

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

      6. neg-mul-1 75.3%

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

      7. +-commutative 75.3%

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

      8. unsub-neg 75.3%

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

      9. unsub-neg 75.3%

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

   9. Simplified 75.3%

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

   if -3.4999999999999999e-138 < x

   1. Initial program 70.2%

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

      1. associate-+r+ 70.2%

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

      2. *-commutative 70.2%

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

      3. frac-times 89.0%

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

      4. associate-*l/ 79.9%

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

      5. times-frac 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

      9. +-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. clear-num 99.8%

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

      2. frac-times 99.5%

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

      3. *-un-lft-identity 99.5%

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

      4. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in y around inf 56.8%

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

      1. *-commutative 56.8%

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

   9. Simplified 56.8%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+164}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(x + \left(y + \left(y + 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+164}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;x \leq -2500000000:\\ \;\;\;\;\frac{y}{x \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -6e+164)
   (/ (/ y x) (+ x 1.0))
   (if (<= x -2500000000.0)
     (/ y (* x (+ y x)))
     (if (<= x -5.05e-138) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -6e+164) {
		tmp = (y / x) / (x + 1.0);
	} else if (x <= -2500000000.0) {
		tmp = y / (x * (y + x));
	} else if (x <= -5.05e-138) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-6d+164)) then
        tmp = (y / x) / (x + 1.0d0)
    else if (x <= (-2500000000.0d0)) then
        tmp = y / (x * (y + x))
    else if (x <= (-5.05d-138)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -6e+164) {
		tmp = (y / x) / (x + 1.0);
	} else if (x <= -2500000000.0) {
		tmp = y / (x * (y + x));
	} else if (x <= -5.05e-138) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -6e+164:
		tmp = (y / x) / (x + 1.0)
	elif x <= -2500000000.0:
		tmp = y / (x * (y + x))
	elif x <= -5.05e-138:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -6e+164)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (x <= -2500000000.0)
		tmp = Float64(y / Float64(x * Float64(y + x)));
	elseif (x <= -5.05e-138)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6e+164)
		tmp = (y / x) / (x + 1.0);
	elseif (x <= -2500000000.0)
		tmp = y / (x * (y + x));
	elseif (x <= -5.05e-138)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -6e+164], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2500000000.0], N[(y / N[(x * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.05e-138], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.00000000000000001e164

   1. Initial program 42.1%

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

      1. associate-/r* 42.1%

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

      2. *-commutative 42.1%

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

      3. +-commutative 42.1%

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

      4. +-commutative 42.1%

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

      5. associate-*l/ 72.2%

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

      6. +-commutative 72.2%

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

      7. associate-*r/ 72.2%

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

      8. remove-double-neg 72.2%

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

      9. +-commutative 72.2%

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

      10. +-commutative 72.2%

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

      11. remove-double-neg 72.2%

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

      12. +-commutative 72.2%

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

      13. associate-+l+ 72.2%

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

   3. Simplified 72.2%

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

   5. Taylor expanded in y around 0 72.2%

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

      1. associate-/r* 86.7%

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

      2. +-commutative 86.7%

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

   7. Simplified 86.7%

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

   if -6.00000000000000001e164 < x < -2.5e9

   1. Initial program 57.6%

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

      1. associate-+r+ 57.6%

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

      2. *-commutative 57.6%

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

      3. frac-times 96.7%

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

      4. associate-*l/ 96.7%

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

      5. times-frac 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

      9. +-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. clear-num 99.6%

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

      2. frac-times 93.6%

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

      3. *-rgt-identity 93.6%

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

      4. div-inv 93.6%

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

      5. associate-+r+ 93.6%

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

      6. +-commutative 93.6%

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

      7. clear-num 93.7%

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

      8. +-commutative 93.7%

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

      9. associate-+r+ 93.7%

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

      10. +-commutative 93.7%

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

   6. Applied egg-rr 93.7%

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

   7. Taylor expanded in x around inf 68.6%

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

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

   1. Initial program 93.9%

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

      1. associate-/r* 95.7%

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

      2. *-commutative 95.7%

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

      3. +-commutative 95.7%

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

      4. +-commutative 95.7%

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

      5. associate-*l/ 99.3%

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

      6. +-commutative 99.3%

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

      7. associate-*r/ 99.3%

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

      8. remove-double-neg 99.3%

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

      9. +-commutative 99.3%

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

      10. +-commutative 99.3%

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

      11. remove-double-neg 99.3%

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

      12. +-commutative 99.3%

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

      13. associate-+l+ 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around 0 53.1%

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

   if -5.0499999999999997e-138 < x

   1. Initial program 70.2%

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

      1. associate-/r* 75.3%

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

      2. *-commutative 75.3%

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

      3. +-commutative 75.3%

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

      4. +-commutative 75.3%

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

      5. associate-*l/ 89.0%

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

      6. +-commutative 89.0%

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

      7. associate-*r/ 89.0%

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

      8. remove-double-neg 89.0%

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

      9. +-commutative 89.0%

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

      10. +-commutative 89.0%

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

      11. remove-double-neg 89.0%

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

      12. +-commutative 89.0%

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

      13. associate-+l+ 89.0%

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

   3. Simplified 89.0%

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

   5. Taylor expanded in x around 0 53.5%

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

      1. +-commutative 53.5%

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

   7. Simplified 53.5%

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

      1. associate-/r* 55.7%

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

      2. div-inv 55.7%

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

   9. Applied egg-rr 55.7%

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

       1. un-div-inv 55.7%

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

   11. Applied egg-rr 55.7%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+164}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;x \leq -2500000000:\\ \;\;\;\;\frac{y}{x \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+164}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -2500000000:\\ \;\;\;\;\frac{y}{x \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5e+164)
   (* (/ y (+ y x)) (/ 1.0 x))
   (if (<= x -2500000000.0)
     (/ y (* x (+ y x)))
     (if (<= x -5.05e-138) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5e+164) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -2500000000.0) {
		tmp = y / (x * (y + x));
	} else if (x <= -5.05e-138) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5d+164)) then
        tmp = (y / (y + x)) * (1.0d0 / x)
    else if (x <= (-2500000000.0d0)) then
        tmp = y / (x * (y + x))
    else if (x <= (-5.05d-138)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -5e+164) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -2500000000.0) {
		tmp = y / (x * (y + x));
	} else if (x <= -5.05e-138) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -5e+164:
		tmp = (y / (y + x)) * (1.0 / x)
	elif x <= -2500000000.0:
		tmp = y / (x * (y + x))
	elif x <= -5.05e-138:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5e+164)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / x));
	elseif (x <= -2500000000.0)
		tmp = Float64(y / Float64(x * Float64(y + x)));
	elseif (x <= -5.05e-138)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5e+164)
		tmp = (y / (y + x)) * (1.0 / x);
	elseif (x <= -2500000000.0)
		tmp = y / (x * (y + x));
	elseif (x <= -5.05e-138)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -5e+164], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2500000000.0], N[(y / N[(x * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.05e-138], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.9999999999999995e164

   1. Initial program 42.1%

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

      1. associate-+r+ 42.1%

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

      2. *-commutative 42.1%

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

      3. frac-times 72.2%

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

      4. associate-*l/ 72.2%

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

      5. times-frac 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

      9. +-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf 87.0%

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

   if -4.9999999999999995e164 < x < -2.5e9

   1. Initial program 57.6%

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

      1. associate-+r+ 57.6%

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

      2. *-commutative 57.6%

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

      3. frac-times 96.7%

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

      4. associate-*l/ 96.7%

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

      5. times-frac 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

      9. +-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. clear-num 99.6%

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

      2. frac-times 93.6%

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

      3. *-rgt-identity 93.6%

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

      4. div-inv 93.6%

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

      5. associate-+r+ 93.6%

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

      6. +-commutative 93.6%

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

      7. clear-num 93.7%

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

      8. +-commutative 93.7%

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

      9. associate-+r+ 93.7%

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

      10. +-commutative 93.7%

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

   6. Applied egg-rr 93.7%

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

   7. Taylor expanded in x around inf 68.6%

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

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

   1. Initial program 93.9%

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

      1. associate-/r* 95.7%

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

      2. *-commutative 95.7%

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

      3. +-commutative 95.7%

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

      4. +-commutative 95.7%

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

      5. associate-*l/ 99.3%

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

      6. +-commutative 99.3%

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

      7. associate-*r/ 99.3%

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

      8. remove-double-neg 99.3%

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

      9. +-commutative 99.3%

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

      10. +-commutative 99.3%

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

      11. remove-double-neg 99.3%

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

      12. +-commutative 99.3%

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

      13. associate-+l+ 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around 0 53.1%

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

   if -5.0499999999999997e-138 < x

   1. Initial program 70.2%

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

      1. associate-/r* 75.3%

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

      2. *-commutative 75.3%

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

      3. +-commutative 75.3%

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

      4. +-commutative 75.3%

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

      5. associate-*l/ 89.0%

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

      6. +-commutative 89.0%

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

      7. associate-*r/ 89.0%

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

      8. remove-double-neg 89.0%

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

      9. +-commutative 89.0%

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

      10. +-commutative 89.0%

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

      11. remove-double-neg 89.0%

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

      12. +-commutative 89.0%

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

      13. associate-+l+ 89.0%

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

   3. Simplified 89.0%

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

   5. Taylor expanded in x around 0 53.5%

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

      1. +-commutative 53.5%

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

   7. Simplified 53.5%

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

      1. associate-/r* 55.7%

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

      2. div-inv 55.7%

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

   9. Applied egg-rr 55.7%

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

       1. un-div-inv 55.7%

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

   11. Applied egg-rr 55.7%

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

4. Final simplification 60.4%

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

Alternative 8: 95.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;y \leq 1.8 \cdot 10^{+155}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t\_0}}{y + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= y 1.8e+155)
     (* (/ x (+ y x)) (/ y (* (+ y x) t_0)))
     (/ (/ x t_0) (+ y (* x 2.0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= 1.8e+155) {
		tmp = (x / (y + x)) * (y / ((y + x) * t_0));
	} else {
		tmp = (x / t_0) / (y + (x * 2.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (y <= 1.8d+155) then
        tmp = (x / (y + x)) * (y / ((y + x) * t_0))
    else
        tmp = (x / t_0) / (y + (x * 2.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= 1.8e+155) {
		tmp = (x / (y + x)) * (y / ((y + x) * t_0));
	} else {
		tmp = (x / t_0) / (y + (x * 2.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if y <= 1.8e+155:
		tmp = (x / (y + x)) * (y / ((y + x) * t_0))
	else:
		tmp = (x / t_0) / (y + (x * 2.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (y <= 1.8e+155)
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(Float64(y + x) * t_0)));
	else
		tmp = Float64(Float64(x / t_0) / Float64(y + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (y <= 1.8e+155)
		tmp = (x / (y + x)) * (y / ((y + x) * t_0));
	else
		tmp = (x / t_0) / (y + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.8e+155], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / t$95$0), $MachinePrecision] / N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.80000000000000004e155

   1. Initial program 69.1%

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

      1. associate-+r+ 69.1%

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

      2. associate-*l* 69.1%

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

      3. times-frac 95.2%

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

      4. +-commutative 95.2%

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

      5. +-commutative 95.2%

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

      6. +-commutative 95.2%

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

      7. associate-+l+ 95.2%

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

   4. Applied egg-rr 95.2%

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

   if 1.80000000000000004e155 < y

   1. Initial program 67.1%

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

      1. associate-+r+ 67.1%

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

      2. *-commutative 67.1%

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

      3. frac-times 82.3%

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

      4. associate-*l/ 82.3%

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

      5. times-frac 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. associate-+l+ 99.9%

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

      9. +-commutative 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. clear-num 99.9%

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

      2. frac-times 99.9%

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

      3. *-un-lft-identity 99.9%

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

      4. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around inf 97.5%

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

      1. *-commutative 97.5%

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

   9. Simplified 97.5%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{+155}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5e+164) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -5.05e-138) {
		tmp = y / ((y + x) * (x + 1.0));
	} else {
		tmp = (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 <= (-5d+164)) then
        tmp = (y / (y + x)) * (1.0d0 / x)
    else if (x <= (-5.05d-138)) then
        tmp = y / ((y + x) * (x + 1.0d0))
    else
        tmp = (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 <= -5e+164) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -5.05e-138) {
		tmp = y / ((y + x) * (x + 1.0));
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + x);
	}
	return tmp;
}

Python

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

Julia

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

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999995e164

   1. Initial program 42.1%

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

      1. associate-+r+ 42.1%

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

      2. *-commutative 42.1%

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

      3. frac-times 72.2%

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

      4. associate-*l/ 72.2%

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

      5. times-frac 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

      9. +-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf 87.0%

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

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

   1. Initial program 77.2%

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

      1. associate-+r+ 77.2%

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

      2. *-commutative 77.2%

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

      3. frac-times 98.1%

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

      4. associate-*l/ 96.2%

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

      5. times-frac 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

      9. +-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. clear-num 98.6%

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

      2. frac-times 94.5%

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

      3. *-rgt-identity 94.5%

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

      4. div-inv 94.5%

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

      5. associate-+r+ 94.5%

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

      6. +-commutative 94.5%

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

      7. clear-num 94.6%

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

      8. +-commutative 94.6%

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

      9. associate-+r+ 94.6%

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

      10. +-commutative 94.6%

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

   6. Applied egg-rr 94.6%

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

   7. Taylor expanded in y around 0 60.5%

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

      1. +-commutative 60.5%

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

   9. Simplified 60.5%

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

   if -5.0499999999999997e-138 < x

   1. Initial program 70.2%

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

      1. associate-+r+ 70.2%

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

      2. *-commutative 70.2%

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

      3. frac-times 89.0%

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

      4. associate-*l/ 79.9%

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

      5. times-frac 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

      9. +-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf 56.5%

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

4. Final simplification 61.0%

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

Alternative 10: 87.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y + \left(x + 1\right)}\\ \mathbf{if}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{t\_0}{y + x} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{y + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y (+ x 1.0)))))
   (if (<= x -5.05e-138) (* (/ t_0 (+ y x)) (/ y x)) (/ t_0 (+ y (* x 2.0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y + (x + 1.0));
	double tmp;
	if (x <= -5.05e-138) {
		tmp = (t_0 / (y + x)) * (y / x);
	} else {
		tmp = t_0 / (y + (x * 2.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y + (x + 1.0d0))
    if (x <= (-5.05d-138)) then
        tmp = (t_0 / (y + x)) * (y / x)
    else
        tmp = t_0 / (y + (x * 2.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y + (x + 1.0));
	double tmp;
	if (x <= -5.05e-138) {
		tmp = (t_0 / (y + x)) * (y / x);
	} else {
		tmp = t_0 / (y + (x * 2.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y + (x + 1.0))
	tmp = 0
	if x <= -5.05e-138:
		tmp = (t_0 / (y + x)) * (y / x)
	else:
		tmp = t_0 / (y + (x * 2.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y + Float64(x + 1.0)))
	tmp = 0.0
	if (x <= -5.05e-138)
		tmp = Float64(Float64(t_0 / Float64(y + x)) * Float64(y / x));
	else
		tmp = Float64(t_0 / Float64(y + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y + (x + 1.0));
	tmp = 0.0;
	if (x <= -5.05e-138)
		tmp = (t_0 / (y + x)) * (y / x);
	else
		tmp = t_0 / (y + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.05e-138], N[(N[(t$95$0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.0499999999999997e-138

   1. Initial program 66.4%

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

      1. associate-+r+ 66.4%

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

      2. *-commutative 66.4%

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

      3. frac-times 90.1%

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

      4. associate-*l/ 88.8%

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

      5. times-frac 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

      9. +-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around 0 76.7%

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

   if -5.0499999999999997e-138 < x

   1. Initial program 70.2%

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

      1. associate-+r+ 70.2%

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

      2. *-commutative 70.2%

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

      3. frac-times 89.0%

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

      4. associate-*l/ 79.9%

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

      5. times-frac 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

      9. +-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. clear-num 99.8%

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

      2. frac-times 99.5%

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

      3. *-un-lft-identity 99.5%

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

      4. +-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in y around inf 56.8%

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

      1. *-commutative 56.8%

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

   9. Simplified 56.8%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 95.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.15)
		tmp = Float64(Float64(Float64(x / Float64(y + Float64(x + 1.0))) / Float64(y + x)) * Float64(y / x));
	else
		tmp = Float64(Float64(y / Float64(y + x)) * 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 (x <= -1.15)
		tmp = ((x / (y + (x + 1.0))) / (y + x)) * (y / x);
	else
		tmp = (y / (y + x)) * ((x / (y + 1.0)) / (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.1499999999999999

   1. Initial program 54.5%

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

      1. associate-+r+ 54.5%

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

      2. *-commutative 54.5%

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

      3. frac-times 86.0%

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

      4. associate-*l/ 86.0%

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

      5. times-frac 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

      9. +-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around 0 81.7%

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

   if -1.1499999999999999 < x

   1. Initial program 73.7%

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

      1. associate-+r+ 73.7%

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

      2. *-commutative 73.7%

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

      3. frac-times 90.6%

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

      4. associate-*l/ 82.2%

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

      5. times-frac 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

      9. +-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 83.5%

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

      1. +-commutative 83.5%

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

   7. Simplified 83.5%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 95.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -0.08:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -0.08)
   (* (/ (/ x (+ y (+ x 1.0))) (+ y x)) (/ y x))
   (/ (* (/ y (+ y x)) (/ x (+ y 1.0))) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -0.08) {
		tmp = ((x / (y + (x + 1.0))) / (y + x)) * (y / x);
	} else {
		tmp = ((y / (y + x)) * (x / (y + 1.0))) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.08d0)) then
        tmp = ((x / (y + (x + 1.0d0))) / (y + x)) * (y / x)
    else
        tmp = ((y / (y + x)) * (x / (y + 1.0d0))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -0.08) {
		tmp = ((x / (y + (x + 1.0))) / (y + x)) * (y / x);
	} else {
		tmp = ((y / (y + x)) * (x / (y + 1.0))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -0.08:
		tmp = ((x / (y + (x + 1.0))) / (y + x)) * (y / x)
	else:
		tmp = ((y / (y + x)) * (x / (y + 1.0))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -0.08)
		tmp = Float64(Float64(Float64(x / Float64(y + Float64(x + 1.0))) / Float64(y + x)) * Float64(y / x));
	else
		tmp = Float64(Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(y + 1.0))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.08)
		tmp = ((x / (y + (x + 1.0))) / (y + x)) * (y / x);
	else
		tmp = ((y / (y + x)) * (x / (y + 1.0))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -0.08], N[(N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0800000000000000017

   1. Initial program 54.5%

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

      1. associate-+r+ 54.5%

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

      2. *-commutative 54.5%

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

      3. frac-times 86.0%

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

      4. associate-*l/ 86.0%

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

      5. times-frac 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

      9. +-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around 0 81.7%

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

   if -0.0800000000000000017 < x

   1. Initial program 73.7%

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

      1. times-frac 90.6%

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

      2. +-commutative 90.6%

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

      3. +-commutative 90.6%

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

      4. +-commutative 90.6%

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

      5. times-frac 73.7%

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

      6. associate-*l/ 85.3%

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

      7. *-commutative 85.3%

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

      8. *-commutative 85.3%

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

      9. distribute-rgt1-in 77.9%

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

      10. fma-def 85.3%

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

      11. +-commutative 85.3%

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

      12. +-commutative 85.3%

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

      13. cube-unmult 85.3%

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

      14. +-commutative 85.3%

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

   3. Simplified 85.3%

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

      1. associate-*r/ 73.7%

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

      2. fma-udef 67.3%

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

      3. cube-mult 67.4%

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

      4. distribute-rgt1-in 73.7%

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

      5. associate-+r+ 73.7%

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

      6. *-commutative 73.7%

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

      7. frac-times 90.6%

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

      8. associate-/r* 99.8%

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

      9. associate-*l/ 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 83.5%

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

      1. +-commutative 83.5%

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

   9. Simplified 83.5%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.08:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 83.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999995e164

   1. Initial program 42.1%

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

      1. associate-+r+ 42.1%

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

      2. *-commutative 42.1%

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

      3. frac-times 72.2%

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

      4. associate-*l/ 72.2%

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

      5. times-frac 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

      9. +-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf 87.0%

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

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

   1. Initial program 77.2%

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

      1. associate-+r+ 77.2%

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

      2. *-commutative 77.2%

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

      3. frac-times 98.1%

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

      4. associate-*l/ 96.2%

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

      5. times-frac 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

      9. +-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. clear-num 98.6%

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

      2. frac-times 94.5%

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

      3. *-rgt-identity 94.5%

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

      4. div-inv 94.5%

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

      5. associate-+r+ 94.5%

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

      6. +-commutative 94.5%

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

      7. clear-num 94.6%

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

      8. +-commutative 94.6%

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

      9. associate-+r+ 94.6%

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

      10. +-commutative 94.6%

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

   6. Applied egg-rr 94.6%

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

   7. Taylor expanded in y around 0 60.5%

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

      1. +-commutative 60.5%

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

   9. Simplified 60.5%

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

   if -5.0499999999999997e-138 < x

   1. Initial program 70.2%

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

      1. associate-/r* 75.3%

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

      2. *-commutative 75.3%

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

      3. +-commutative 75.3%

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

      4. +-commutative 75.3%

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

      5. associate-*l/ 89.0%

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

      6. +-commutative 89.0%

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

      7. associate-*r/ 89.0%

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

      8. remove-double-neg 89.0%

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

      9. +-commutative 89.0%

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

      10. +-commutative 89.0%

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

      11. remove-double-neg 89.0%

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

      12. +-commutative 89.0%

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

      13. associate-+l+ 89.0%

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

   3. Simplified 89.0%

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

   5. Taylor expanded in x around 0 53.5%

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

      1. +-commutative 53.5%

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

   7. Simplified 53.5%

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

      1. associate-/r* 55.7%

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

      2. div-inv 55.7%

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

   9. Applied egg-rr 55.7%

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

       1. un-div-inv 55.7%

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

   11. Applied egg-rr 55.7%

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

4. Final simplification 60.5%

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

Alternative 14: 83.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5e+164) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -5.05e-138) {
		tmp = y / ((y + x) * (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 (x <= (-5d+164)) then
        tmp = (y / (y + x)) * (1.0d0 / x)
    else if (x <= (-5.05d-138)) then
        tmp = y / ((y + x) * (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 (x <= -5e+164) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -5.05e-138) {
		tmp = y / ((y + x) * (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 x <= -5e+164:
		tmp = (y / (y + x)) * (1.0 / x)
	elif x <= -5.05e-138:
		tmp = y / ((y + x) * (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 (x <= -5e+164)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / x));
	elseif (x <= -5.05e-138)
		tmp = Float64(y / Float64(Float64(y + x) * 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 (x <= -5e+164)
		tmp = (y / (y + x)) * (1.0 / x);
	elseif (x <= -5.05e-138)
		tmp = y / ((y + x) * (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[x, -5e+164], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.05e-138], N[(y / N[(N[(y + x), $MachinePrecision] * N[(x + 1.0), $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 x < -4.9999999999999995e164

   1. Initial program 42.1%

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

      1. associate-+r+ 42.1%

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

      2. *-commutative 42.1%

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

      3. frac-times 72.2%

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

      4. associate-*l/ 72.2%

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

      5. times-frac 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

      9. +-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf 87.0%

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

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

   1. Initial program 77.2%

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

      1. associate-+r+ 77.2%

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

      2. *-commutative 77.2%

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

      3. frac-times 98.1%

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

      4. associate-*l/ 96.2%

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

      5. times-frac 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

      9. +-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. clear-num 98.6%

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

      2. frac-times 94.5%

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

      3. *-rgt-identity 94.5%

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

      4. div-inv 94.5%

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

      5. associate-+r+ 94.5%

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

      6. +-commutative 94.5%

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

      7. clear-num 94.6%

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

      8. +-commutative 94.6%

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

      9. associate-+r+ 94.6%

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

      10. +-commutative 94.6%

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

   6. Applied egg-rr 94.6%

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

   7. Taylor expanded in y around 0 60.5%

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

      1. +-commutative 60.5%

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

   9. Simplified 60.5%

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

   if -5.0499999999999997e-138 < x

   1. Initial program 70.2%

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

      1. times-frac 89.1%

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

      2. +-commutative 89.1%

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

      3. +-commutative 89.1%

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

      4. +-commutative 89.1%

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

      5. times-frac 70.2%

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

      6. associate-*l/ 83.4%

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

      7. *-commutative 83.4%

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

      8. *-commutative 83.4%

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

      9. distribute-rgt1-in 75.9%

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

      10. fma-def 83.4%

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

      11. +-commutative 83.4%

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

      12. +-commutative 83.4%

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

      13. cube-unmult 83.4%

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

      14. +-commutative 83.4%

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

   3. Simplified 83.4%

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

      1. associate-*r/ 70.2%

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

      2. fma-udef 64.0%

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

      3. cube-mult 64.0%

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

      4. distribute-rgt1-in 70.2%

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

      5. associate-+r+ 70.2%

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

      6. *-commutative 70.2%

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

      7. frac-times 89.0%

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

      8. associate-/r* 99.8%

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

      9. associate-*l/ 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 55.9%

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

      1. +-commutative 55.9%

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

   9. Simplified 55.9%

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

4. Final simplification 60.6%

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

Alternative 15: 80.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -2.5e9

   1. Initial program 50.0%

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

      1. associate-+r+ 50.0%

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

      2. *-commutative 50.0%

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

      3. frac-times 84.7%

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

      4. associate-*l/ 84.7%

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

      5. times-frac 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

      9. +-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. clear-num 99.1%

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

      2. frac-times 83.1%

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

      3. *-rgt-identity 83.1%

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

      4. div-inv 83.1%

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

      5. associate-+r+ 83.1%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \frac{1}{\frac{x}{\color{blue}{\left(y + 1\right) + x}}}\right)} \]

      6. +-commutative 83.1%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \frac{1}{\frac{x}{\color{blue}{x + \left(y + 1\right)}}}\right)} \]

      7. clear-num 83.1%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \color{blue}{\frac{x + \left(y + 1\right)}{x}}\right)} \]

      8. +-commutative 83.1%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \frac{\color{blue}{\left(y + 1\right) + x}}{x}\right)} \]

      9. associate-+r+ 83.1%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \frac{\color{blue}{y + \left(1 + x\right)}}{x}\right)} \]

      10. +-commutative 83.1%

          \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \frac{y + \color{blue}{\left(x + 1\right)}}{x}\right)} \]

   6. Applied egg-rr 83.1%

      \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \frac{y + \left(x + 1\right)}{x}\right)}} \]

   7. Taylor expanded in x around inf 70.4%

      \[\leadsto \frac{y}{\left(y + x\right) \cdot \color{blue}{x}} \]

   if -2.5e9 < x < -5.0499999999999997e-138

   1. Initial program 93.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 95.7%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. *-commutative 95.7%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 95.7%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      4. +-commutative 95.7%

         \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      5. associate-*l/ 99.3%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 99.3%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 99.3%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 99.3%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 99.3%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 99.3%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 99.3%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 99.3%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 99.3%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 53.1%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   if -5.0499999999999997e-138 < x

   1. Initial program 70.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 75.3%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. *-commutative 75.3%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 75.3%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      4. +-commutative 75.3%

         \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      5. associate-*l/ 89.0%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 89.0%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 89.0%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 89.0%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 89.0%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 89.0%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 89.0%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 89.0%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 89.0%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 89.0%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 53.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 53.5%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   7. Simplified 53.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2500000000:\\ \;\;\;\;\frac{y}{x \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2500000000:\\ \;\;\;\;\frac{y}{x \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2500000000.0)
   (/ y (* x (+ y x)))
   (if (<= x -5.05e-138) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2500000000.0) {
		tmp = y / (x * (y + x));
	} else if (x <= -5.05e-138) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2500000000.0d0)) then
        tmp = y / (x * (y + x))
    else if (x <= (-5.05d-138)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2500000000.0) {
		tmp = y / (x * (y + x));
	} else if (x <= -5.05e-138) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2500000000.0:
		tmp = y / (x * (y + x))
	elif x <= -5.05e-138:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2500000000.0)
		tmp = Float64(y / Float64(x * Float64(y + x)));
	elseif (x <= -5.05e-138)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2500000000.0)
		tmp = y / (x * (y + x));
	elseif (x <= -5.05e-138)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2500000000.0], N[(y / N[(x * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.05e-138], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5e9

   1. Initial program 50.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-+r+ 50.0%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      2. *-commutative 50.0%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)} \]

      3. frac-times 84.7%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

      4. associate-*l/ 84.7%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      5. times-frac 99.7%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{\frac{x}{x + \left(y + 1\right)}}{x + y}} \]

      6. +-commutative 99.7%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{\frac{x}{x + \left(y + 1\right)}}{x + y} \]

      7. +-commutative 99.7%

         \[\leadsto \frac{y}{y + x} \cdot \frac{\frac{x}{\color{blue}{\left(y + 1\right) + x}}}{x + y} \]

      8. associate-+l+ 99.7%

         \[\leadsto \frac{y}{y + x} \cdot \frac{\frac{x}{\color{blue}{y + \left(1 + x\right)}}}{x + y} \]

      9. +-commutative 99.7%

         \[\leadsto \frac{y}{y + x} \cdot \frac{\frac{x}{y + \left(1 + x\right)}}{\color{blue}{y + x}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{\frac{x}{y + \left(1 + x\right)}}{y + x}} \]
   5. Step-by-step derivation

      1. clear-num 99.1%

         \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{\frac{y + x}{\frac{x}{y + \left(1 + x\right)}}}} \]

      2. frac-times 83.1%

         \[\leadsto \color{blue}{\frac{y \cdot 1}{\left(y + x\right) \cdot \frac{y + x}{\frac{x}{y + \left(1 + x\right)}}}} \]

      3. *-rgt-identity 83.1%

         \[\leadsto \frac{\color{blue}{y}}{\left(y + x\right) \cdot \frac{y + x}{\frac{x}{y + \left(1 + x\right)}}} \]

      4. div-inv 83.1%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + x\right) \cdot \frac{1}{\frac{x}{y + \left(1 + x\right)}}\right)}} \]

      5. associate-+r+ 83.1%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \frac{1}{\frac{x}{\color{blue}{\left(y + 1\right) + x}}}\right)} \]

      6. +-commutative 83.1%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \frac{1}{\frac{x}{\color{blue}{x + \left(y + 1\right)}}}\right)} \]

      7. clear-num 83.1%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \color{blue}{\frac{x + \left(y + 1\right)}{x}}\right)} \]

      8. +-commutative 83.1%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \frac{\color{blue}{\left(y + 1\right) + x}}{x}\right)} \]

      9. associate-+r+ 83.1%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \frac{\color{blue}{y + \left(1 + x\right)}}{x}\right)} \]

      10. +-commutative 83.1%

          \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \frac{y + \color{blue}{\left(x + 1\right)}}{x}\right)} \]

   6. Applied egg-rr 83.1%

      \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \frac{y + \left(x + 1\right)}{x}\right)}} \]

   7. Taylor expanded in x around inf 70.4%

      \[\leadsto \frac{y}{\left(y + x\right) \cdot \color{blue}{x}} \]

   if -2.5e9 < x < -5.0499999999999997e-138

   1. Initial program 93.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 95.7%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. *-commutative 95.7%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 95.7%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      4. +-commutative 95.7%

         \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      5. associate-*l/ 99.3%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 99.3%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 99.3%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 99.3%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 99.3%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 99.3%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 99.3%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 99.3%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 99.3%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 53.1%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   if -5.0499999999999997e-138 < x

   1. Initial program 70.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 75.3%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. *-commutative 75.3%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 75.3%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      4. +-commutative 75.3%

         \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      5. associate-*l/ 89.0%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 89.0%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 89.0%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 89.0%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 89.0%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 89.0%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 89.0%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 89.0%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 89.0%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 89.0%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 53.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 53.5%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   7. Simplified 53.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]
   8. Step-by-step derivation

      1. associate-/r* 55.7%

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]

      2. div-inv 55.7%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{1}{y + 1}} \]

   9. Applied egg-rr 55.7%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{1}{y + 1}} \]
   10. Step-by-step derivation

       1. un-div-inv 55.7%

          \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]

   11. Applied egg-rr 55.7%

       \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2500000000:\\ \;\;\;\;\frac{y}{x \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 52.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+110}:\\ \;\;\;\;\frac{y}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -9e+110) (/ y (* y x)) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -9e+110) {
		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 <= (-9d+110)) 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 <= -9e+110) {
		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 <= -9e+110:
		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 <= -9e+110)
		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 (x <= -9e+110)
		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, -9e+110], 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 x < -9.0000000000000005e110

   1. Initial program 38.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-+r+ 38.3%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      2. *-commutative 38.3%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)} \]

      3. frac-times 77.0%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

      4. associate-*l/ 77.0%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      5. times-frac 99.8%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{\frac{x}{x + \left(y + 1\right)}}{x + y}} \]

      6. +-commutative 99.8%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{\frac{x}{x + \left(y + 1\right)}}{x + y} \]

      7. +-commutative 99.8%

         \[\leadsto \frac{y}{y + x} \cdot \frac{\frac{x}{\color{blue}{\left(y + 1\right) + x}}}{x + y} \]

      8. associate-+l+ 99.8%

         \[\leadsto \frac{y}{y + x} \cdot \frac{\frac{x}{\color{blue}{y + \left(1 + x\right)}}}{x + y} \]

      9. +-commutative 99.8%

         \[\leadsto \frac{y}{y + x} \cdot \frac{\frac{x}{y + \left(1 + x\right)}}{\color{blue}{y + x}} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{\frac{x}{y + \left(1 + x\right)}}{y + x}} \]
   5. Step-by-step derivation

      1. clear-num 98.8%

         \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{\frac{y + x}{\frac{x}{y + \left(1 + x\right)}}}} \]

      2. frac-times 77.0%

         \[\leadsto \color{blue}{\frac{y \cdot 1}{\left(y + x\right) \cdot \frac{y + x}{\frac{x}{y + \left(1 + x\right)}}}} \]

      3. *-rgt-identity 77.0%

         \[\leadsto \frac{\color{blue}{y}}{\left(y + x\right) \cdot \frac{y + x}{\frac{x}{y + \left(1 + x\right)}}} \]

      4. div-inv 77.0%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + x\right) \cdot \frac{1}{\frac{x}{y + \left(1 + x\right)}}\right)}} \]

      5. associate-+r+ 77.0%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \frac{1}{\frac{x}{\color{blue}{\left(y + 1\right) + x}}}\right)} \]

      6. +-commutative 77.0%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \frac{1}{\frac{x}{\color{blue}{x + \left(y + 1\right)}}}\right)} \]

      7. clear-num 77.0%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \color{blue}{\frac{x + \left(y + 1\right)}{x}}\right)} \]

      8. +-commutative 77.0%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \frac{\color{blue}{\left(y + 1\right) + x}}{x}\right)} \]

      9. associate-+r+ 77.0%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \frac{\color{blue}{y + \left(1 + x\right)}}{x}\right)} \]

      10. +-commutative 77.0%

          \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \frac{y + \color{blue}{\left(x + 1\right)}}{x}\right)} \]

   6. Applied egg-rr 77.0%

      \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \frac{y + \left(x + 1\right)}{x}\right)}} \]

   7. Taylor expanded in x around inf 77.0%

      \[\leadsto \frac{y}{\left(y + x\right) \cdot \color{blue}{x}} \]

   8. Taylor expanded in y around inf 36.4%

      \[\leadsto \frac{y}{\color{blue}{x \cdot y}} \]

   if -9.0000000000000005e110 < x

   1. Initial program 73.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 78.5%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. *-commutative 78.5%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 78.5%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      4. +-commutative 78.5%

         \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      5. associate-*l/ 91.4%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 91.4%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 91.4%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 91.4%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 91.4%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 91.4%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 91.4%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 91.4%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 91.4%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 91.4%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 52.3%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 52.3%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   7. Simplified 52.3%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+110}:\\ \;\;\;\;\frac{y}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 78.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5.05e-138) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5.05e-138) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.05d-138)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -5.05e-138) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -5.05e-138:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5.05e-138)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.05e-138)
		tmp = y / (x * (x + 1.0));
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -5.05e-138], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.0499999999999997e-138

   1. Initial program 66.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 70.9%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. *-commutative 70.9%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 70.9%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      4. +-commutative 70.9%

         \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      5. associate-*l/ 90.1%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 90.1%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 90.1%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 90.1%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 90.1%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 90.1%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 90.1%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 90.1%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 90.1%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 90.1%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 55.4%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   if -5.0499999999999997e-138 < x

   1. Initial program 70.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 75.3%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. *-commutative 75.3%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 75.3%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      4. +-commutative 75.3%

         \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      5. associate-*l/ 89.0%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 89.0%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 89.0%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 89.0%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 89.0%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 89.0%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 89.0%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 89.0%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 89.0%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 89.0%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 53.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 53.5%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   7. Simplified 53.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 34.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+21}:\\ \;\;\;\;\frac{y}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -2.4e+21) (/ y (* y x)) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.4e+21) {
		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 <= (-2.4d+21)) 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 <= -2.4e+21) {
		tmp = y / (y * x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.4e+21:
		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 <= -2.4e+21)
		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 <= -2.4e+21)
		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, -2.4e+21], N[(y / N[(y * x), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4e21

   1. Initial program 48.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-+r+ 48.3%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      2. *-commutative 48.3%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)} \]

      3. frac-times 84.2%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

      4. associate-*l/ 84.1%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      5. times-frac 99.7%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{\frac{x}{x + \left(y + 1\right)}}{x + y}} \]

      6. +-commutative 99.7%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{\frac{x}{x + \left(y + 1\right)}}{x + y} \]

      7. +-commutative 99.7%

         \[\leadsto \frac{y}{y + x} \cdot \frac{\frac{x}{\color{blue}{\left(y + 1\right) + x}}}{x + y} \]

      8. associate-+l+ 99.7%

         \[\leadsto \frac{y}{y + x} \cdot \frac{\frac{x}{\color{blue}{y + \left(1 + x\right)}}}{x + y} \]

      9. +-commutative 99.7%

         \[\leadsto \frac{y}{y + x} \cdot \frac{\frac{x}{y + \left(1 + x\right)}}{\color{blue}{y + x}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{\frac{x}{y + \left(1 + x\right)}}{y + x}} \]
   5. Step-by-step derivation

      1. clear-num 99.1%

         \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{\frac{y + x}{\frac{x}{y + \left(1 + x\right)}}}} \]

      2. frac-times 82.6%

         \[\leadsto \color{blue}{\frac{y \cdot 1}{\left(y + x\right) \cdot \frac{y + x}{\frac{x}{y + \left(1 + x\right)}}}} \]

      3. *-rgt-identity 82.6%

         \[\leadsto \frac{\color{blue}{y}}{\left(y + x\right) \cdot \frac{y + x}{\frac{x}{y + \left(1 + x\right)}}} \]

      4. div-inv 82.5%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + x\right) \cdot \frac{1}{\frac{x}{y + \left(1 + x\right)}}\right)}} \]

      5. associate-+r+ 82.5%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \frac{1}{\frac{x}{\color{blue}{\left(y + 1\right) + x}}}\right)} \]

      6. +-commutative 82.5%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \frac{1}{\frac{x}{\color{blue}{x + \left(y + 1\right)}}}\right)} \]

      7. clear-num 82.6%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \color{blue}{\frac{x + \left(y + 1\right)}{x}}\right)} \]

      8. +-commutative 82.6%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \frac{\color{blue}{\left(y + 1\right) + x}}{x}\right)} \]

      9. associate-+r+ 82.6%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \frac{\color{blue}{y + \left(1 + x\right)}}{x}\right)} \]

      10. +-commutative 82.6%

          \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \frac{y + \color{blue}{\left(x + 1\right)}}{x}\right)} \]

   6. Applied egg-rr 82.6%

      \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \frac{y + \left(x + 1\right)}{x}\right)}} \]

   7. Taylor expanded in x around inf 72.7%

      \[\leadsto \frac{y}{\left(y + x\right) \cdot \color{blue}{x}} \]

   8. Taylor expanded in y around inf 32.5%

      \[\leadsto \frac{y}{\color{blue}{x \cdot y}} \]

   if -2.4e21 < x

   1. Initial program 74.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 79.1%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. *-commutative 79.1%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 79.1%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      4. +-commutative 79.1%

         \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      5. associate-*l/ 91.0%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 91.0%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 90.9%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 90.9%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 90.9%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 90.9%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 90.9%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 90.9%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 90.9%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 90.9%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 52.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 52.7%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   7. Simplified 52.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   8. Taylor expanded in y around 0 32.1%

      \[\leadsto \frac{x}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+21}:\\ \;\;\;\;\frac{y}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 4.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{1}{x} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ 1.0 x))

C

assert(x < y);
double code(double x, double y) {
	return 1.0 / x;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / x
end function

Java

assert x < y;
public static double code(double x, double y) {
	return 1.0 / x;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return 1.0 / x

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(1.0 / x)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 1.0 / x;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(1.0 / x), $MachinePrecision]

Derivation

1. Initial program 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. associate-+r+ 68.8%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   2. *-commutative 68.8%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)} \]

   3. frac-times 89.4%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   4. associate-*l/ 83.2%

      \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   5. times-frac 99.8%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{\frac{x}{x + \left(y + 1\right)}}{x + y}} \]

   6. +-commutative 99.8%

      \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{\frac{x}{x + \left(y + 1\right)}}{x + y} \]

   7. +-commutative 99.8%

      \[\leadsto \frac{y}{y + x} \cdot \frac{\frac{x}{\color{blue}{\left(y + 1\right) + x}}}{x + y} \]

   8. associate-+l+ 99.8%

      \[\leadsto \frac{y}{y + x} \cdot \frac{\frac{x}{\color{blue}{y + \left(1 + x\right)}}}{x + y} \]

   9. +-commutative 99.8%

      \[\leadsto \frac{y}{y + x} \cdot \frac{\frac{x}{y + \left(1 + x\right)}}{\color{blue}{y + x}} \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{\frac{x}{y + \left(1 + x\right)}}{y + x}} \]

5. Taylor expanded in y around inf 50.9%

   \[\leadsto \color{blue}{1} \cdot \frac{\frac{x}{y + \left(1 + x\right)}}{y + x} \]

6. Taylor expanded in x around inf 4.0%

   \[\leadsto 1 \cdot \color{blue}{\frac{1}{x}} \]

7. Taylor expanded in x around 0 4.0%

   \[\leadsto \color{blue}{\frac{1}{x}} \]

8. Final simplification 4.0%

   \[\leadsto \frac{1}{x} \]
9. Add Preprocessing

Alternative 21: 26.5% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{x}{y} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ x y))

C

assert(x < y);
double code(double x, double y) {
	return x / y;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / y
end function

Java

assert x < y;
public static double code(double x, double y) {
	return x / y;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return x / y

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x / y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x / y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 68.8%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 73.7%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

   2. *-commutative 73.7%

      \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

   3. +-commutative 73.7%

      \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

   4. +-commutative 73.7%

      \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

   5. associate-*l/ 89.4%

      \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

   6. +-commutative 89.4%

      \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

   7. associate-*r/ 89.4%

      \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

   8. remove-double-neg 89.4%

      \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

   9. +-commutative 89.4%

      \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

   10. +-commutative 89.4%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

   11. remove-double-neg 89.4%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

   12. +-commutative 89.4%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

   13. associate-+l+ 89.4%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

3. Simplified 89.4%

   \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 47.7%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
6. Step-by-step derivation

   1. +-commutative 47.7%

      \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

7. Simplified 47.7%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

8. Taylor expanded in y around 0 25.6%

   \[\leadsto \frac{x}{\color{blue}{y}} \]

9. Final simplification 25.6%

   \[\leadsto \frac{x}{y} \]
10. Add Preprocessing

Developer target: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x)))))

C

double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x / ((y + 1.0d0) + x)) / (y + x)) / (1.0d0 / (y / (y + x)))
end function

Java

public static double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Python

def code(x, y):
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)))

Julia

function code(x, y)
	return Float64(Float64(Float64(x / Float64(Float64(y + 1.0) + x)) / Float64(y + x)) / Float64(1.0 / Float64(y / Float64(y + x))))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
end

Wolfram

code[x_, y_] := N[(N[(N[(x / N[(N[(y + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024096 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))