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

Percentage Accurate: 68.3% → 99.8%

Time: 41.5s

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: 68.3% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.5%

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

   1. associate-+r+ 71.5%

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

   2. *-commutative 71.5%

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

   3. frac-times 87.6%

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

   4. associate-*l/ 83.5%

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

   5. times-frac 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 90.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 9.8 \cdot 10^{-232}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \frac{y + x}{y}}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+173}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 9.8e-232)
   (/ (/ y (+ y x)) (+ x (+ y (+ y 1.0))))
   (if (<= y 2.7e-25)
     (/ x (* (+ y x) (/ (+ y x) y)))
     (if (<= y 4.3e+173)
       (* (/ y (* (+ y x) (+ y x))) (/ x (+ 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 (y <= 9.8e-232) {
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)));
	} else if (y <= 2.7e-25) {
		tmp = x / ((y + x) * ((y + x) / y));
	} else if (y <= 4.3e+173) {
		tmp = (y / ((y + x) * (y + x))) * (x / (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 (y <= 9.8d-232) then
        tmp = (y / (y + x)) / (x + (y + (y + 1.0d0)))
    else if (y <= 2.7d-25) then
        tmp = x / ((y + x) * ((y + x) / y))
    else if (y <= 4.3d+173) then
        tmp = (y / ((y + x) * (y + x))) * (x / (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 (y <= 9.8e-232) {
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)));
	} else if (y <= 2.7e-25) {
		tmp = x / ((y + x) * ((y + x) / y));
	} else if (y <= 4.3e+173) {
		tmp = (y / ((y + x) * (y + x))) * (x / (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 y <= 9.8e-232:
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)))
	elif y <= 2.7e-25:
		tmp = x / ((y + x) * ((y + x) / y))
	elif y <= 4.3e+173:
		tmp = (y / ((y + x) * (y + x))) * (x / (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 (y <= 9.8e-232)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + Float64(y + Float64(y + 1.0))));
	elseif (y <= 2.7e-25)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(Float64(y + x) / y)));
	elseif (y <= 4.3e+173)
		tmp = Float64(Float64(y / Float64(Float64(y + x) * Float64(y + x))) * Float64(x / 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 (y <= 9.8e-232)
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)));
	elseif (y <= 2.7e-25)
		tmp = x / ((y + x) * ((y + x) / y));
	elseif (y <= 4.3e+173)
		tmp = (y / ((y + x) * (y + x))) * (x / (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[y, 9.8e-232], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.7e-25], N[(x / N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.3e+173], N[(N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + 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 4 regimes

2. if y < 9.8000000000000006e-232

   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. +-commutative 70.2%

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

      2. +-commutative 70.2%

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

      3. +-commutative 70.2%

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

      4. associate-*l/ 81.5%

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

      5. *-commutative 81.5%

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

      6. *-commutative 81.5%

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

      7. distribute-rgt1-in 54.2%

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

      8. fma-define 81.5%

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

      9. +-commutative 81.5%

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

      10. +-commutative 81.5%

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

      11. cube-unmult 81.5%

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

      12. +-commutative 81.5%

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

   3. Simplified 81.5%

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

      1. associate-*r/ 70.3%

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

      2. fma-undefine 50.7%

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

      3. cube-mult 50.7%

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

      4. distribute-rgt1-in 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 87.1%

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

      8. *-commutative 87.1%

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

      9. associate-/r* 99.8%

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

      10. clear-num 99.8%

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

      11. frac-times 99.1%

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

      12. *-un-lft-identity 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in x around -inf 58.6%

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

      1. mul-1-neg 58.6%

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

      2. unsub-neg 58.6%

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

      3. neg-mul-1 58.6%

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

      4. +-commutative 58.6%

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

      5. unsub-neg 58.6%

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

      6. distribute-lft-in 58.6%

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

      7. metadata-eval 58.6%

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

      8. neg-mul-1 58.6%

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

      9. unsub-neg 58.6%

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

   9. Simplified 58.6%

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

   if 9.8000000000000006e-232 < y < 2.70000000000000016e-25

   1. Initial program 78.6%

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

      1. associate-+r+ 78.6%

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

      2. *-commutative 78.6%

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

      3. frac-times 90.0%

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

      4. associate-*l/ 80.9%

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

      5. times-frac 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. clear-num 99.9%

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

      2. frac-times 99.8%

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

      3. *-un-lft-identity 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 91.8%

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

      1. +-commutative 91.8%

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

   9. Simplified 91.8%

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

   10. Taylor expanded in y around 0 91.8%

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

   if 2.70000000000000016e-25 < y < 4.30000000000000025e173

   1. Initial program 73.9%

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

      1. *-commutative 73.9%

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

      2. +-commutative 73.9%

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

      3. +-commutative 73.9%

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

      4. +-commutative 73.9%

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

      5. times-frac 95.4%

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

      6. +-commutative 95.4%

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

      7. +-commutative 95.4%

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

      8. +-commutative 95.4%

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

      9. associate-+l+ 95.4%

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

   3. Simplified 95.4%

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

   5. Taylor expanded in x around 0 82.2%

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

      1. +-commutative 82.2%

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

   7. Simplified 82.2%

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

   if 4.30000000000000025e173 < y

   1. Initial program 64.3%

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

      1. associate-+r+ 64.3%

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

      2. *-commutative 64.3%

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

      3. frac-times 76.5%

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

      4. associate-*l/ 76.5%

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

      5. times-frac 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. clear-num 99.8%

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

      2. frac-times 99.8%

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

      3. *-un-lft-identity 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around inf 94.3%

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

      1. *-commutative 94.2%

         \[\leadsto \frac{\frac{x}{y + 1}}{y + \color{blue}{x \cdot 2}} \]

   9. Simplified 94.3%

      \[\leadsto \frac{\frac{x}{y + \left(x + 1\right)}}{\color{blue}{y + x \cdot 2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.8 \cdot 10^{-232}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \frac{y + x}{y}}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+173}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.04 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-124}:\\ \;\;\;\;\frac{x}{x + \left(y + 1\right)} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{\left(y + x\right) \cdot \frac{y + x}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.04e+155)
   (/ (/ y (+ y x)) (+ x (+ y (+ y 1.0))))
   (if (<= x -5e-124)
     (* (/ x (+ x (+ y 1.0))) (/ y (* (+ y x) (+ y x))))
     (/ (/ x (+ y 1.0)) (* (+ y x) (/ (+ y x) y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.04e+155) {
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)));
	} else if (x <= -5e-124) {
		tmp = (x / (x + (y + 1.0))) * (y / ((y + x) * (y + x)));
	} else {
		tmp = (x / (y + 1.0)) / ((y + x) * ((y + x) / y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.04d+155)) then
        tmp = (y / (y + x)) / (x + (y + (y + 1.0d0)))
    else if (x <= (-5d-124)) then
        tmp = (x / (x + (y + 1.0d0))) * (y / ((y + x) * (y + x)))
    else
        tmp = (x / (y + 1.0d0)) / ((y + x) * ((y + x) / y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.04e+155) {
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)));
	} else if (x <= -5e-124) {
		tmp = (x / (x + (y + 1.0))) * (y / ((y + x) * (y + x)));
	} else {
		tmp = (x / (y + 1.0)) / ((y + x) * ((y + x) / y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.04e+155:
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)))
	elif x <= -5e-124:
		tmp = (x / (x + (y + 1.0))) * (y / ((y + x) * (y + x)))
	else:
		tmp = (x / (y + 1.0)) / ((y + x) * ((y + x) / y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.04e+155)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + Float64(y + Float64(y + 1.0))));
	elseif (x <= -5e-124)
		tmp = Float64(Float64(x / Float64(x + Float64(y + 1.0))) * Float64(y / Float64(Float64(y + x) * Float64(y + x))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(Float64(y + x) * Float64(Float64(y + x) / y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.04e+155)
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)));
	elseif (x <= -5e-124)
		tmp = (x / (x + (y + 1.0))) * (y / ((y + x) * (y + x)));
	else
		tmp = (x / (y + 1.0)) / ((y + x) * ((y + x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.04e+155], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5e-124], N[(N[(x / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.03999999999999996e155

   1. Initial program 53.8%

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

      1. +-commutative 53.8%

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

      2. +-commutative 53.8%

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

      3. +-commutative 53.8%

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

      4. associate-*l/ 79.4%

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

      5. *-commutative 79.4%

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

      6. *-commutative 79.4%

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

      7. distribute-rgt1-in 3.4%

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

      8. fma-define 79.4%

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

      9. +-commutative 79.4%

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

      10. +-commutative 79.4%

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

      11. cube-unmult 79.4%

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

      12. +-commutative 79.4%

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

   3. Simplified 79.4%

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

      1. associate-*r/ 53.8%

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

      2. fma-undefine 0.0%

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

      3. cube-mult 0.0%

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

      4. distribute-rgt1-in 53.8%

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

      5. associate-+r+ 53.8%

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

      6. *-commutative 53.8%

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

      7. frac-times 79.4%

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

      8. *-commutative 79.4%

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

      9. associate-/r* 99.9%

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

      10. clear-num 99.8%

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

      11. frac-times 97.8%

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

      12. *-un-lft-identity 97.8%

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

   6. Applied egg-rr 97.8%

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

   7. Taylor expanded in x around -inf 83.0%

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

      1. mul-1-neg 83.0%

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

      2. unsub-neg 83.0%

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

      3. neg-mul-1 83.0%

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

      4. +-commutative 83.0%

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

      5. unsub-neg 83.0%

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

      6. distribute-lft-in 83.0%

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

      7. metadata-eval 83.0%

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

      8. neg-mul-1 83.0%

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

      9. unsub-neg 83.0%

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

   9. Simplified 83.0%

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

   if -1.03999999999999996e155 < x < -5.0000000000000003e-124

   1. Initial program 76.5%

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

      1. *-commutative 76.5%

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

      2. +-commutative 76.5%

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

      3. +-commutative 76.5%

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

      4. +-commutative 76.5%

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

      5. times-frac 95.6%

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

      6. +-commutative 95.6%

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

      7. +-commutative 95.6%

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

      8. +-commutative 95.6%

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

      9. associate-+l+ 95.6%

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

   3. Simplified 95.6%

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

   if -5.0000000000000003e-124 < x

   1. Initial program 72.3%

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

      1. associate-+r+ 72.3%

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

      2. *-commutative 72.3%

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

      3. frac-times 85.6%

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

      4. 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 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. clear-num 99.5%

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

      2. frac-times 99.0%

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

      3. *-un-lft-identity 99.0%

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

      4. associate-+r+ 99.0%

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

      5. +-commutative 99.0%

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

      6. associate-+l+ 99.0%

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

      7. +-commutative 99.0%

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

      8. +-commutative 99.0%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in x around 0 90.7%

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

      1. +-commutative 90.7%

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

   9. Simplified 90.7%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.04 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-124}:\\ \;\;\;\;\frac{x}{x + \left(y + 1\right)} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{\left(y + x\right) \cdot \frac{y + x}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+155}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -950000:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-108}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.15e+155)
   (* (/ y (+ y x)) (/ 1.0 x))
   (if (<= x -950000.0)
     (/ y (* (+ y x) (+ y x)))
     (if (<= x -9.5e-108) (/ y (* x (+ x 1.0))) (/ (/ x (+ y (+ x 1.0))) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.15e+155) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -950000.0) {
		tmp = y / ((y + x) * (y + x));
	} else if (x <= -9.5e-108) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + (x + 1.0))) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.15d+155)) then
        tmp = (y / (y + x)) * (1.0d0 / x)
    else if (x <= (-950000.0d0)) then
        tmp = y / ((y + x) * (y + x))
    else if (x <= (-9.5d-108)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / (y + (x + 1.0d0))) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.15e+155) {
		tmp = (y / (y + x)) * (1.0 / x);
	} else if (x <= -950000.0) {
		tmp = y / ((y + x) * (y + x));
	} else if (x <= -9.5e-108) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + (x + 1.0))) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.15e+155:
		tmp = (y / (y + x)) * (1.0 / x)
	elif x <= -950000.0:
		tmp = y / ((y + x) * (y + x))
	elif x <= -9.5e-108:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / (y + (x + 1.0))) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.15e+155)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(1.0 / x));
	elseif (x <= -950000.0)
		tmp = Float64(y / Float64(Float64(y + x) * Float64(y + x)));
	elseif (x <= -9.5e-108)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.15e+155)
		tmp = (y / (y + x)) * (1.0 / x);
	elseif (x <= -950000.0)
		tmp = y / ((y + x) * (y + x));
	elseif (x <= -9.5e-108)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / (y + (x + 1.0))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.15e+155], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -950000.0], N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -9.5e-108], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.14999999999999999e155

   1. Initial program 53.8%

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

      1. associate-+r+ 53.8%

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

      2. *-commutative 53.8%

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

      3. frac-times 79.4%

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

      4. associate-*l/ 79.4%

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

      5. times-frac 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf 82.8%

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

   if -1.14999999999999999e155 < x < -9.5e5

   1. Initial program 53.5%

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

      1. *-commutative 53.5%

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

      2. +-commutative 53.5%

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

      3. +-commutative 53.5%

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

      4. +-commutative 53.5%

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

      5. times-frac 88.8%

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

      6. +-commutative 88.8%

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

      7. +-commutative 88.8%

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

      8. +-commutative 88.8%

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

      9. associate-+l+ 88.8%

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

   3. Simplified 88.8%

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

   5. Taylor expanded in x around inf 73.8%

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

   if -9.5e5 < x < -9.5000000000000005e-108

   1. Initial program 91.4%

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

      1. *-commutative 91.4%

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

      2. +-commutative 91.4%

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

      3. +-commutative 91.4%

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

      4. +-commutative 91.4%

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

      5. times-frac 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

      9. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 54.3%

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

   if -9.5000000000000005e-108 < x

   1. Initial program 72.6%

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

      1. associate-+r+ 72.6%

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

      2. *-commutative 72.6%

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

      3. frac-times 86.1%

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

      4. associate-*l/ 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 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. clear-num 99.5%

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

      2. frac-times 99.1%

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

      3. *-un-lft-identity 99.1%

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

      4. associate-+r+ 99.1%

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

      5. +-commutative 99.1%

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

      6. associate-+l+ 99.1%

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

      7. +-commutative 99.1%

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

      8. +-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in y around inf 61.6%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+155}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -950000:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-108}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-229}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \frac{y + x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.8e-229)
   (/ (/ y (+ y x)) (+ x 1.0))
   (if (<= y 1.1e-11)
     (/ x (* (+ y x) (/ (+ y x) y)))
     (/ (/ x (+ y (+ x 1.0))) (+ y (* x 2.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.8e-229) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else if (y <= 1.1e-11) {
		tmp = x / ((y + x) * ((y + x) / y));
	} 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 (y <= 3.8d-229) then
        tmp = (y / (y + x)) / (x + 1.0d0)
    else if (y <= 1.1d-11) then
        tmp = x / ((y + x) * ((y + x) / y))
    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 (y <= 3.8e-229) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else if (y <= 1.1e-11) {
		tmp = x / ((y + x) * ((y + x) / y));
	} 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 y <= 3.8e-229:
		tmp = (y / (y + x)) / (x + 1.0)
	elif y <= 1.1e-11:
		tmp = x / ((y + x) * ((y + x) / y))
	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 (y <= 3.8e-229)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + 1.0));
	elseif (y <= 1.1e-11)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(Float64(y + x) / y)));
	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 (y <= 3.8e-229)
		tmp = (y / (y + x)) / (x + 1.0);
	elseif (y <= 1.1e-11)
		tmp = x / ((y + x) * ((y + x) / y));
	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[y, 3.8e-229], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e-11], N[(x / N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] / y), $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 y < 3.8000000000000002e-229

   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. +-commutative 70.2%

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

      2. +-commutative 70.2%

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

      3. +-commutative 70.2%

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

      4. associate-*l/ 81.5%

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

      5. *-commutative 81.5%

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

      6. *-commutative 81.5%

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

      7. distribute-rgt1-in 54.2%

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

      8. fma-define 81.5%

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

      9. +-commutative 81.5%

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

      10. +-commutative 81.5%

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

      11. cube-unmult 81.5%

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

      12. +-commutative 81.5%

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

   3. Simplified 81.5%

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

      1. associate-*r/ 70.3%

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

      2. fma-undefine 50.7%

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

      3. cube-mult 50.7%

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

      4. distribute-rgt1-in 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 87.1%

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

      8. *-commutative 87.1%

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

      9. associate-/r* 99.8%

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

      10. clear-num 99.8%

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

      11. frac-times 99.1%

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

      12. *-un-lft-identity 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in y around 0 57.8%

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

      1. +-commutative 57.8%

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

   9. Simplified 57.8%

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

   if 3.8000000000000002e-229 < y < 1.1000000000000001e-11

   1. Initial program 78.9%

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

      1. associate-+r+ 78.9%

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

      2. *-commutative 78.9%

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

      3. frac-times 91.0%

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

      4. associate-*l/ 82.9%

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

      5. times-frac 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. clear-num 99.9%

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

      2. frac-times 99.8%

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

      3. *-un-lft-identity 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 88.7%

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

      1. +-commutative 88.7%

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

   9. Simplified 88.7%

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

   10. Taylor expanded in y around 0 88.6%

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

   if 1.1000000000000001e-11 < y

   1. Initial program 68.9%

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

      1. associate-+r+ 68.9%

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

      2. *-commutative 68.9%

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

      3. frac-times 86.3%

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

      4. associate-*l/ 86.3%

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

      5. times-frac 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. clear-num 99.8%

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

      2. frac-times 99.7%

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

      3. *-un-lft-identity 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+l+ 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around inf 82.8%

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

      1. *-commutative 82.6%

         \[\leadsto \frac{\frac{x}{y + 1}}{y + \color{blue}{x \cdot 2}} \]

   9. Simplified 82.8%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-229}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \frac{y + x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{-232}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \frac{y + x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 5.6e-232)
   (/ (/ y (+ y x)) (+ x (+ y (+ y 1.0))))
   (if (<= y 9.5e-11)
     (/ x (* (+ y x) (/ (+ y x) y)))
     (/ (/ x (+ y (+ x 1.0))) (+ y (* x 2.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 5.6e-232) {
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)));
	} else if (y <= 9.5e-11) {
		tmp = x / ((y + x) * ((y + x) / y));
	} 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 (y <= 5.6d-232) then
        tmp = (y / (y + x)) / (x + (y + (y + 1.0d0)))
    else if (y <= 9.5d-11) then
        tmp = x / ((y + x) * ((y + x) / y))
    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 (y <= 5.6e-232) {
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)));
	} else if (y <= 9.5e-11) {
		tmp = x / ((y + x) * ((y + x) / y));
	} 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 y <= 5.6e-232:
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)))
	elif y <= 9.5e-11:
		tmp = x / ((y + x) * ((y + x) / y))
	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 (y <= 5.6e-232)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + Float64(y + Float64(y + 1.0))));
	elseif (y <= 9.5e-11)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(Float64(y + x) / y)));
	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 (y <= 5.6e-232)
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)));
	elseif (y <= 9.5e-11)
		tmp = x / ((y + x) * ((y + x) / y));
	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[y, 5.6e-232], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.5e-11], N[(x / N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] / y), $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 y < 5.59999999999999985e-232

   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. +-commutative 70.2%

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

      2. +-commutative 70.2%

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

      3. +-commutative 70.2%

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

      4. associate-*l/ 81.5%

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

      5. *-commutative 81.5%

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

      6. *-commutative 81.5%

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

      7. distribute-rgt1-in 54.2%

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

      8. fma-define 81.5%

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

      9. +-commutative 81.5%

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

      10. +-commutative 81.5%

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

      11. cube-unmult 81.5%

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

      12. +-commutative 81.5%

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

   3. Simplified 81.5%

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

      1. associate-*r/ 70.3%

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

      2. fma-undefine 50.7%

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

      3. cube-mult 50.7%

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

      4. distribute-rgt1-in 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 87.1%

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

      8. *-commutative 87.1%

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

      9. associate-/r* 99.8%

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

      10. clear-num 99.8%

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

      11. frac-times 99.1%

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

      12. *-un-lft-identity 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in x around -inf 58.6%

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

      1. mul-1-neg 58.6%

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

      2. unsub-neg 58.6%

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

      3. neg-mul-1 58.6%

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

      4. +-commutative 58.6%

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

      5. unsub-neg 58.6%

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

      6. distribute-lft-in 58.6%

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

      7. metadata-eval 58.6%

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

      8. neg-mul-1 58.6%

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

      9. unsub-neg 58.6%

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

   9. Simplified 58.6%

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

   if 5.59999999999999985e-232 < y < 9.49999999999999951e-11

   1. Initial program 79.3%

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

      1. associate-+r+ 79.3%

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

      2. *-commutative 79.3%

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

      3. frac-times 91.2%

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

      4. associate-*l/ 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 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. clear-num 99.9%

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

      2. frac-times 99.8%

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

      3. *-un-lft-identity 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 88.9%

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

      1. +-commutative 88.9%

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

   9. Simplified 88.9%

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

   10. Taylor expanded in y around 0 88.9%

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

   if 9.49999999999999951e-11 < y

   1. Initial program 68.5%

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

      1. associate-+r+ 68.5%

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

      2. *-commutative 68.5%

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

      3. frac-times 86.1%

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

      4. associate-*l/ 86.1%

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

      5. times-frac 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. clear-num 99.8%

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

      2. frac-times 99.7%

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

      3. *-un-lft-identity 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+l+ 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around inf 83.9%

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

      1. *-commutative 83.7%

         \[\leadsto \frac{\frac{x}{y + 1}}{y + \color{blue}{x \cdot 2}} \]

   9. Simplified 83.9%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{-232}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \frac{y + x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-108}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{-171}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (* (/ 1.0 x) (/ y x))
   (if (<= x -2.2e-108)
     (- (/ y x) y)
     (if (<= x 2.45e-171) (/ x y) (* (/ x y) (/ 1.0 y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -2.2e-108) {
		tmp = (y / x) - y;
	} else if (x <= 2.45e-171) {
		tmp = x / y;
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (1.0d0 / x) * (y / x)
    else if (x <= (-2.2d-108)) then
        tmp = (y / x) - y
    else if (x <= 2.45d-171) then
        tmp = x / y
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -2.2e-108) {
		tmp = (y / x) - y;
	} else if (x <= 2.45e-171) {
		tmp = x / y;
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (1.0 / x) * (y / x)
	elif x <= -2.2e-108:
		tmp = (y / x) - y
	elif x <= 2.45e-171:
		tmp = x / y
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(1.0 / x) * Float64(y / x));
	elseif (x <= -2.2e-108)
		tmp = Float64(Float64(y / x) - y);
	elseif (x <= 2.45e-171)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (1.0 / x) * (y / x);
	elseif (x <= -2.2e-108)
		tmp = (y / x) - y;
	elseif (x <= 2.45e-171)
		tmp = x / y;
	else
		tmp = (x / y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.0], N[(N[(1.0 / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.2e-108], N[(N[(y / x), $MachinePrecision] - y), $MachinePrecision], If[LessEqual[x, 2.45e-171], N[(x / y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1

   1. Initial program 55.4%

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

      1. associate-+r+ 55.4%

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

      2. *-commutative 55.4%

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

      3. frac-times 84.5%

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

      4. associate-*l/ 84.5%

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

      5. times-frac 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf 70.8%

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

   6. Taylor expanded in y around 0 70.5%

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

   if -1 < x < -2.2000000000000001e-108

   1. Initial program 90.9%

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

      1. *-commutative 90.9%

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

      2. +-commutative 90.9%

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

      3. +-commutative 90.9%

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

      4. +-commutative 90.9%

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

      5. times-frac 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

      9. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 51.7%

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

      1. associate-/r* 51.6%

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

      2. +-commutative 51.6%

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

   7. Simplified 51.6%

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

   8. Taylor expanded in x around 0 50.8%

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

      1. neg-mul-1 50.8%

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

      2. +-commutative 50.8%

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

      3. unsub-neg 50.8%

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

   10. Simplified 50.8%

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

   if -2.2000000000000001e-108 < x < 2.44999999999999991e-171

   1. Initial program 70.8%

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

      1. *-commutative 70.8%

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

      2. +-commutative 70.8%

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

      3. +-commutative 70.8%

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

      4. +-commutative 70.8%

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

      5. times-frac 78.5%

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

      6. +-commutative 78.5%

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

      7. +-commutative 78.5%

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

      8. +-commutative 78.5%

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

      9. associate-+l+ 78.5%

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

   3. Simplified 78.5%

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

   5. Taylor expanded in x around 0 85.7%

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

      1. +-commutative 85.7%

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

   7. Simplified 85.7%

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

   8. Taylor expanded in y around 0 71.6%

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

   if 2.44999999999999991e-171 < x

   1. Initial program 74.2%

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

      1. *-commutative 74.2%

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

      2. +-commutative 74.2%

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

      3. +-commutative 74.2%

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

      4. +-commutative 74.2%

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

      5. times-frac 92.7%

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

      6. +-commutative 92.7%

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

      7. +-commutative 92.7%

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

      8. +-commutative 92.7%

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

      9. associate-+l+ 92.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in x around 0 37.0%

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

      1. +-commutative 37.0%

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

   7. Simplified 37.0%

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

      1. associate-/r* 40.1%

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

      2. div-inv 40.1%

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

   9. Applied egg-rr 40.1%

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

   10. Taylor expanded in y around inf 34.4%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-108}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{-171}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{-232}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \frac{y + x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4.2e-232)
   (/ (/ y (+ y x)) (+ x 1.0))
   (if (<= y 3.9e-11)
     (/ x (* (+ y x) (/ (+ y x) y)))
     (/ (/ x (+ y (+ x 1.0))) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.2e-232) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else if (y <= 3.9e-11) {
		tmp = x / ((y + x) * ((y + x) / y));
	} else {
		tmp = (x / (y + (x + 1.0))) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.2d-232) then
        tmp = (y / (y + x)) / (x + 1.0d0)
    else if (y <= 3.9d-11) then
        tmp = x / ((y + x) * ((y + x) / y))
    else
        tmp = (x / (y + (x + 1.0d0))) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 4.2e-232) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else if (y <= 3.9e-11) {
		tmp = x / ((y + x) * ((y + x) / y));
	} else {
		tmp = (x / (y + (x + 1.0))) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 4.2e-232:
		tmp = (y / (y + x)) / (x + 1.0)
	elif y <= 3.9e-11:
		tmp = x / ((y + x) * ((y + x) / y))
	else:
		tmp = (x / (y + (x + 1.0))) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4.2e-232)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + 1.0));
	elseif (y <= 3.9e-11)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(Float64(y + x) / y)));
	else
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.2e-232)
		tmp = (y / (y + x)) / (x + 1.0);
	elseif (y <= 3.9e-11)
		tmp = x / ((y + x) * ((y + x) / y));
	else
		tmp = (x / (y + (x + 1.0))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 4.2e-232], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.9e-11], N[(x / N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.2000000000000001e-232

   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. +-commutative 70.2%

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

      2. +-commutative 70.2%

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

      3. +-commutative 70.2%

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

      4. associate-*l/ 81.5%

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

      5. *-commutative 81.5%

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

      6. *-commutative 81.5%

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

      7. distribute-rgt1-in 54.2%

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

      8. fma-define 81.5%

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

      9. +-commutative 81.5%

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

      10. +-commutative 81.5%

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

      11. cube-unmult 81.5%

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

      12. +-commutative 81.5%

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

   3. Simplified 81.5%

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

      1. associate-*r/ 70.3%

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

      2. fma-undefine 50.7%

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

      3. cube-mult 50.7%

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

      4. distribute-rgt1-in 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 87.1%

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

      8. *-commutative 87.1%

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

      9. associate-/r* 99.8%

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

      10. clear-num 99.8%

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

      11. frac-times 99.1%

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

      12. *-un-lft-identity 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in y around 0 57.8%

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

      1. +-commutative 57.8%

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

   9. Simplified 57.8%

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

   if 4.2000000000000001e-232 < y < 3.9000000000000001e-11

   1. Initial program 78.9%

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

      1. associate-+r+ 78.9%

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

      2. *-commutative 78.9%

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

      3. frac-times 91.0%

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

      4. associate-*l/ 82.9%

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

      5. times-frac 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. clear-num 99.9%

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

      2. frac-times 99.8%

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

      3. *-un-lft-identity 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 88.7%

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

      1. +-commutative 88.7%

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

   9. Simplified 88.7%

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

   10. Taylor expanded in y around 0 88.6%

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

   if 3.9000000000000001e-11 < y

   1. Initial program 68.9%

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

      1. associate-+r+ 68.9%

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

      2. *-commutative 68.9%

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

      3. frac-times 86.3%

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

      4. associate-*l/ 86.3%

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

      5. times-frac 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. clear-num 99.8%

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

      2. frac-times 99.7%

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

      3. *-un-lft-identity 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+l+ 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around inf 82.6%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{-232}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \frac{y + x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-229}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{elif}\;y \leq 10^{-10}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \frac{y + x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.8e-229)
   (/ (/ y (+ y x)) (+ x 1.0))
   (if (<= y 1e-10)
     (/ x (* (+ y x) (/ (+ y x) y)))
     (/ (/ x (+ y 1.0)) (+ y (* x 2.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.8e-229) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else if (y <= 1e-10) {
		tmp = x / ((y + x) * ((y + x) / y));
	} else {
		tmp = (x / (y + 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 (y <= 3.8d-229) then
        tmp = (y / (y + x)) / (x + 1.0d0)
    else if (y <= 1d-10) then
        tmp = x / ((y + x) * ((y + x) / y))
    else
        tmp = (x / (y + 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 (y <= 3.8e-229) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else if (y <= 1e-10) {
		tmp = x / ((y + x) * ((y + x) / y));
	} else {
		tmp = (x / (y + 1.0)) / (y + (x * 2.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.8e-229:
		tmp = (y / (y + x)) / (x + 1.0)
	elif y <= 1e-10:
		tmp = x / ((y + x) * ((y + x) / y))
	else:
		tmp = (x / (y + 1.0)) / (y + (x * 2.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.8e-229)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + 1.0));
	elseif (y <= 1e-10)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(Float64(y + x) / y)));
	else
		tmp = Float64(Float64(x / Float64(y + 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 (y <= 3.8e-229)
		tmp = (y / (y + x)) / (x + 1.0);
	elseif (y <= 1e-10)
		tmp = x / ((y + x) * ((y + x) / y));
	else
		tmp = (x / (y + 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[y, 3.8e-229], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e-10], N[(x / N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.8000000000000002e-229

   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. +-commutative 70.2%

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

      2. +-commutative 70.2%

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

      3. +-commutative 70.2%

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

      4. associate-*l/ 81.5%

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

      5. *-commutative 81.5%

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

      6. *-commutative 81.5%

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

      7. distribute-rgt1-in 54.2%

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

      8. fma-define 81.5%

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

      9. +-commutative 81.5%

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

      10. +-commutative 81.5%

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

      11. cube-unmult 81.5%

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

      12. +-commutative 81.5%

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

   3. Simplified 81.5%

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

      1. associate-*r/ 70.3%

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

      2. fma-undefine 50.7%

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

      3. cube-mult 50.7%

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

      4. distribute-rgt1-in 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 87.1%

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

      8. *-commutative 87.1%

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

      9. associate-/r* 99.8%

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

      10. clear-num 99.8%

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

      11. frac-times 99.1%

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

      12. *-un-lft-identity 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in y around 0 57.8%

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

      1. +-commutative 57.8%

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

   9. Simplified 57.8%

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

   if 3.8000000000000002e-229 < y < 1.00000000000000004e-10

   1. Initial program 79.3%

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

      1. associate-+r+ 79.3%

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

      2. *-commutative 79.3%

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

      3. frac-times 91.2%

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

      4. associate-*l/ 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 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. clear-num 99.9%

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

      2. frac-times 99.8%

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

      3. *-un-lft-identity 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 88.9%

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

      1. +-commutative 88.9%

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

   9. Simplified 88.9%

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

   10. Taylor expanded in y around 0 88.9%

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

   if 1.00000000000000004e-10 < y

   1. Initial program 68.5%

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

      1. associate-+r+ 68.5%

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

      2. *-commutative 68.5%

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

      3. frac-times 86.1%

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

      4. associate-*l/ 86.1%

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

      5. times-frac 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. clear-num 99.8%

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

      2. frac-times 99.7%

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

      3. *-un-lft-identity 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+l+ 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0 89.1%

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

      1. +-commutative 89.1%

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

   9. Simplified 89.1%

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

   10. Taylor expanded in y around inf 83.7%

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

       1. *-commutative 83.7%

          \[\leadsto \frac{\frac{x}{y + 1}}{y + \color{blue}{x \cdot 2}} \]

   12. Simplified 83.7%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-229}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{elif}\;y \leq 10^{-10}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \frac{y + x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-232}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{\left(y + x\right) \cdot \frac{y + x}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4e-232)
   (/ (/ y (+ y x)) (+ x (+ y (+ y 1.0))))
   (/ (/ x (+ y 1.0)) (* (+ y x) (/ (+ y x) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4e-232) {
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) / ((y + x) * ((y + x) / y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4d-232) then
        tmp = (y / (y + x)) / (x + (y + (y + 1.0d0)))
    else
        tmp = (x / (y + 1.0d0)) / ((y + x) * ((y + x) / y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 4e-232) {
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) / ((y + x) * ((y + x) / y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 4e-232:
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)))
	else:
		tmp = (x / (y + 1.0)) / ((y + x) * ((y + x) / y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4e-232)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + Float64(y + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(Float64(y + x) * Float64(Float64(y + x) / y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4e-232)
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)));
	else
		tmp = (x / (y + 1.0)) / ((y + x) * ((y + x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 4e-232], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.0000000000000001e-232

   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. +-commutative 70.2%

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

      2. +-commutative 70.2%

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

      3. +-commutative 70.2%

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

      4. associate-*l/ 81.5%

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

      5. *-commutative 81.5%

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

      6. *-commutative 81.5%

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

      7. distribute-rgt1-in 54.2%

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

      8. fma-define 81.5%

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

      9. +-commutative 81.5%

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

      10. +-commutative 81.5%

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

      11. cube-unmult 81.5%

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

      12. +-commutative 81.5%

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

   3. Simplified 81.5%

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

      1. associate-*r/ 70.3%

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

      2. fma-undefine 50.7%

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

      3. cube-mult 50.7%

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

      4. distribute-rgt1-in 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 87.1%

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

      8. *-commutative 87.1%

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

      9. associate-/r* 99.8%

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

      10. clear-num 99.8%

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

      11. frac-times 99.1%

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

      12. *-un-lft-identity 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in x around -inf 58.6%

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

      1. mul-1-neg 58.6%

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

      2. unsub-neg 58.6%

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

      3. neg-mul-1 58.6%

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

      4. +-commutative 58.6%

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

      5. unsub-neg 58.6%

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

      6. distribute-lft-in 58.6%

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

      7. metadata-eval 58.6%

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

      8. neg-mul-1 58.6%

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

      9. unsub-neg 58.6%

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

   9. Simplified 58.6%

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

   if 4.0000000000000001e-232 < y

   1. Initial program 72.9%

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

      1. associate-+r+ 72.9%

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

      2. *-commutative 72.9%

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

      3. frac-times 88.2%

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

      4. associate-*l/ 84.9%

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

      5. times-frac 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. clear-num 99.9%

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

      2. frac-times 99.7%

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

      3. *-un-lft-identity 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+l+ 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0 89.0%

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

      1. +-commutative 89.0%

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

   9. Simplified 89.0%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-232}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{\left(y + x\right) \cdot \frac{y + x}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{-118}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1.6:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 8.2e-118)
   (/ y x)
   (if (<= y 1.6) (- (/ x y) x) (* (/ x y) (/ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 8.2e-118) {
		tmp = y / x;
	} else if (y <= 1.6) {
		tmp = (x / y) - x;
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 8.2d-118) then
        tmp = y / x
    else if (y <= 1.6d0) then
        tmp = (x / y) - x
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 8.2e-118) {
		tmp = y / x;
	} else if (y <= 1.6) {
		tmp = (x / y) - x;
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 8.2e-118:
		tmp = y / x
	elif y <= 1.6:
		tmp = (x / y) - x
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 8.2e-118)
		tmp = Float64(y / x);
	elseif (y <= 1.6)
		tmp = Float64(Float64(x / y) - x);
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8.2e-118)
		tmp = y / x;
	elseif (y <= 1.6)
		tmp = (x / y) - x;
	else
		tmp = (x / y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 8.2e-118], N[(y / x), $MachinePrecision], If[LessEqual[y, 1.6], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 8.2000000000000006e-118

   1. Initial program 70.4%

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

      1. associate-+r+ 70.4%

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

      2. *-commutative 70.4%

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

      3. frac-times 86.2%

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

      4. associate-*l/ 80.8%

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

      5. times-frac 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0 60.8%

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

      1. +-commutative 60.8%

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

   7. Simplified 60.8%

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

   8. Taylor expanded in x around 0 38.6%

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

   if 8.2000000000000006e-118 < y < 1.6000000000000001

   1. Initial program 87.4%

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

      1. *-commutative 87.4%

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

      2. +-commutative 87.4%

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

      3. +-commutative 87.4%

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

      4. +-commutative 87.4%

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

      5. times-frac 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

      9. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 47.3%

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

      1. +-commutative 47.3%

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

   7. Simplified 47.3%

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

   8. Taylor expanded in y around 0 41.0%

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

      1. neg-mul-1 41.0%

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

      2. +-commutative 41.0%

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

      3. unsub-neg 41.0%

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

   10. Simplified 41.0%

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

   if 1.6000000000000001 < y

   1. Initial program 66.6%

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

      1. *-commutative 66.6%

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

      2. +-commutative 66.6%

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

      3. +-commutative 66.6%

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

      4. +-commutative 66.6%

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

      5. times-frac 85.3%

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

      6. +-commutative 85.3%

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

      7. +-commutative 85.3%

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

      8. +-commutative 85.3%

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

      9. associate-+l+ 85.3%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in x around 0 75.1%

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

      1. +-commutative 75.1%

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

   7. Simplified 75.1%

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

      1. associate-/r* 82.5%

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

      2. div-inv 82.5%

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

   9. Applied egg-rr 82.5%

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

   10. Taylor expanded in y around inf 81.4%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{-118}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1.6:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-107}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (* (/ 1.0 x) (/ y x))
   (if (<= x -1.7e-107) (- (/ y x) y) (/ x (* y (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -1.7e-107) {
		tmp = (y / x) - y;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (1.0d0 / x) * (y / x)
    else if (x <= (-1.7d-107)) then
        tmp = (y / x) - y
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -1.7e-107) {
		tmp = (y / x) - y;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (1.0 / x) * (y / x)
	elif x <= -1.7e-107:
		tmp = (y / x) - y
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(1.0 / x) * Float64(y / x));
	elseif (x <= -1.7e-107)
		tmp = Float64(Float64(y / x) - y);
	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 <= -1.0)
		tmp = (1.0 / x) * (y / x);
	elseif (x <= -1.7e-107)
		tmp = (y / x) - y;
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.0], N[(N[(1.0 / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.7e-107], N[(N[(y / x), $MachinePrecision] - y), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 55.4%

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

      1. associate-+r+ 55.4%

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

      2. *-commutative 55.4%

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

      3. frac-times 84.5%

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

      4. associate-*l/ 84.5%

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

      5. times-frac 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf 70.8%

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

   6. Taylor expanded in y around 0 70.5%

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

   if -1 < x < -1.69999999999999997e-107

   1. Initial program 90.9%

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

      1. *-commutative 90.9%

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

      2. +-commutative 90.9%

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

      3. +-commutative 90.9%

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

      4. +-commutative 90.9%

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

      5. times-frac 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

      9. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 51.7%

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

      1. associate-/r* 51.6%

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

      2. +-commutative 51.6%

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

   7. Simplified 51.6%

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

   8. Taylor expanded in x around 0 50.8%

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

      1. neg-mul-1 50.8%

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

      2. +-commutative 50.8%

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

      3. unsub-neg 50.8%

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

   10. Simplified 50.8%

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

   if -1.69999999999999997e-107 < x

   1. Initial program 72.6%

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

      1. *-commutative 72.6%

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

      2. +-commutative 72.6%

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

      3. +-commutative 72.6%

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

      4. +-commutative 72.6%

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

      5. times-frac 86.1%

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

      6. +-commutative 86.1%

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

      7. +-commutative 86.1%

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

      8. +-commutative 86.1%

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

      9. associate-+l+ 86.1%

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

   3. Simplified 86.1%

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

   5. Taylor expanded in x around 0 59.7%

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

      1. +-commutative 59.7%

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

   7. Simplified 59.7%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-107}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+115}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1e+115)
   (* (/ 1.0 x) (/ y x))
   (if (<= x -1.8e-107) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1e+115) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -1.8e-107) {
		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 <= (-1d+115)) then
        tmp = (1.0d0 / x) * (y / x)
    else if (x <= (-1.8d-107)) 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 <= -1e+115) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -1.8e-107) {
		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 <= -1e+115:
		tmp = (1.0 / x) * (y / x)
	elif x <= -1.8e-107:
		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 <= -1e+115)
		tmp = Float64(Float64(1.0 / x) * Float64(y / x));
	elseif (x <= -1.8e-107)
		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 <= -1e+115)
		tmp = (1.0 / x) * (y / x);
	elseif (x <= -1.8e-107)
		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, -1e+115], N[(N[(1.0 / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.8e-107], 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 < -1e115

   1. Initial program 50.3%

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

      1. associate-+r+ 50.3%

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

      2. *-commutative 50.3%

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

      3. frac-times 79.9%

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

      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 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf 79.8%

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

   6. Taylor expanded in y around 0 79.5%

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

   if -1e115 < x < -1.79999999999999988e-107

   1. Initial program 82.9%

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

      1. *-commutative 82.9%

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

      2. +-commutative 82.9%

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

      3. +-commutative 82.9%

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

      4. +-commutative 82.9%

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

      5. times-frac 98.0%

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

      6. +-commutative 98.0%

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

      7. +-commutative 98.0%

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

      8. +-commutative 98.0%

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

      9. associate-+l+ 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in y around 0 54.4%

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

   if -1.79999999999999988e-107 < x

   1. Initial program 72.6%

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

      1. *-commutative 72.6%

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

      2. +-commutative 72.6%

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

      3. +-commutative 72.6%

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

      4. +-commutative 72.6%

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

      5. times-frac 86.1%

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

      6. +-commutative 86.1%

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

      7. +-commutative 86.1%

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

      8. +-commutative 86.1%

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

      9. associate-+l+ 86.1%

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

   3. Simplified 86.1%

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

   5. Taylor expanded in x around 0 59.7%

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

      1. +-commutative 59.7%

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

   7. Simplified 59.7%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+115}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 81.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+115}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1e+115)
   (* (/ 1.0 x) (/ y x))
   (if (<= x -1.8e-107) (/ y (* x (+ x 1.0))) (/ (/ x (+ y 1.0)) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1e+115) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -1.8e-107) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1d+115)) then
        tmp = (1.0d0 / x) * (y / x)
    else if (x <= (-1.8d-107)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1e+115) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -1.8e-107) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1e+115:
		tmp = (1.0 / x) * (y / x)
	elif x <= -1.8e-107:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1e+115)
		tmp = Float64(Float64(1.0 / x) * Float64(y / x));
	elseif (x <= -1.8e-107)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1e+115)
		tmp = (1.0 / x) * (y / x);
	elseif (x <= -1.8e-107)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1e+115], N[(N[(1.0 / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.8e-107], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1e115

   1. Initial program 50.3%

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

      1. associate-+r+ 50.3%

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

      2. *-commutative 50.3%

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

      3. frac-times 79.9%

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

      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 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf 79.8%

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

   6. Taylor expanded in y around 0 79.5%

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

   if -1e115 < x < -1.79999999999999988e-107

   1. Initial program 82.9%

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

      1. *-commutative 82.9%

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

      2. +-commutative 82.9%

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

      3. +-commutative 82.9%

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

      4. +-commutative 82.9%

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

      5. times-frac 98.0%

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

      6. +-commutative 98.0%

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

      7. +-commutative 98.0%

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

      8. +-commutative 98.0%

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

      9. associate-+l+ 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in y around 0 54.4%

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

   if -1.79999999999999988e-107 < x

   1. Initial program 72.6%

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

      1. *-commutative 72.6%

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

      2. +-commutative 72.6%

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

      3. +-commutative 72.6%

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

      4. +-commutative 72.6%

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

      5. times-frac 86.1%

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

      6. +-commutative 86.1%

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

      7. +-commutative 86.1%

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

      8. +-commutative 86.1%

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

      9. associate-+l+ 86.1%

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

   3. Simplified 86.1%

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

   5. Taylor expanded in x around 0 59.7%

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

      1. +-commutative 59.7%

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

   7. Simplified 59.7%

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

      1. *-un-lft-identity 59.7%

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

      2. times-frac 61.4%

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

   9. Applied egg-rr 61.4%

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

       1. associate-*l/ 61.4%

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

       2. *-lft-identity 61.4%

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

   11. Simplified 61.4%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+115}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 81.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.8e-107) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ y (+ x 1.0))) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.8e-107) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + (x + 1.0))) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.8d-107)) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (y + (x + 1.0d0))) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.8e-107) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + (x + 1.0))) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.8e-107:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (y + (x + 1.0))) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.8e-107)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.8e-107)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (y + (x + 1.0))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.8e-107], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.79999999999999988e-107

   1. Initial program 69.4%

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

      1. *-commutative 69.4%

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

      2. +-commutative 69.4%

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

      3. +-commutative 69.4%

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

      4. +-commutative 69.4%

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

      5. times-frac 90.5%

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

      6. +-commutative 90.5%

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

      7. +-commutative 90.5%

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

      8. +-commutative 90.5%

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

      9. associate-+l+ 90.5%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in y around 0 63.9%

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

      1. associate-/r* 64.8%

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

      2. +-commutative 64.8%

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

   7. Simplified 64.8%

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

   if -1.79999999999999988e-107 < x

   1. Initial program 72.6%

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

      1. associate-+r+ 72.6%

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

      2. *-commutative 72.6%

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

      3. frac-times 86.1%

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

      4. associate-*l/ 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 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. clear-num 99.5%

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

      2. frac-times 99.1%

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

      3. *-un-lft-identity 99.1%

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

      4. associate-+r+ 99.1%

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

      5. +-commutative 99.1%

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

      6. associate-+l+ 99.1%

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

      7. +-commutative 99.1%

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

      8. +-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in y around inf 61.6%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 81.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.8e-107) (/ (/ y (+ y x)) (+ x 1.0)) (/ (/ x (+ y (+ x 1.0))) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.8e-107) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = (x / (y + (x + 1.0))) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.8d-107)) then
        tmp = (y / (y + x)) / (x + 1.0d0)
    else
        tmp = (x / (y + (x + 1.0d0))) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.8e-107) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = (x / (y + (x + 1.0))) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.8e-107:
		tmp = (y / (y + x)) / (x + 1.0)
	else:
		tmp = (x / (y + (x + 1.0))) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.8e-107)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.8e-107)
		tmp = (y / (y + x)) / (x + 1.0);
	else
		tmp = (x / (y + (x + 1.0))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.8e-107], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.79999999999999988e-107

   1. Initial program 69.4%

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

      1. +-commutative 69.4%

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

      2. +-commutative 69.4%

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

      3. +-commutative 69.4%

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

      4. associate-*l/ 79.1%

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

      5. *-commutative 79.1%

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

      6. *-commutative 79.1%

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

      7. distribute-rgt1-in 50.4%

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

      8. fma-define 79.1%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      9. +-commutative 79.1%

         \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      10. +-commutative 79.1%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      11. cube-unmult 79.1%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      12. +-commutative 79.1%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 79.1%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 69.4%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-undefine 47.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 47.3%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 69.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. associate-+r+ 69.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 69.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. frac-times 90.5%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

      8. *-commutative 90.5%

         \[\leadsto \color{blue}{\frac{x}{x + \left(y + 1\right)} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      9. associate-/r* 99.8%

         \[\leadsto \frac{x}{x + \left(y + 1\right)} \cdot \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \]

      10. clear-num 99.7%

          \[\leadsto \color{blue}{\frac{1}{\frac{x + \left(y + 1\right)}{x}}} \cdot \frac{\frac{y}{x + y}}{x + y} \]

      11. frac-times 98.6%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{y}{x + y}}{\frac{x + \left(y + 1\right)}{x} \cdot \left(x + y\right)}} \]

      12. *-un-lft-identity 98.6%

          \[\leadsto \frac{\color{blue}{\frac{y}{x + y}}}{\frac{x + \left(y + 1\right)}{x} \cdot \left(x + y\right)} \]

   6. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{\frac{x + \left(y + 1\right)}{x} \cdot \left(x + y\right)}} \]

   7. Taylor expanded in y around 0 65.1%

      \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{1 + x}} \]
   8. Step-by-step derivation

      1. +-commutative 65.1%

         \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{x + 1}} \]

   9. Simplified 65.1%

      \[\leadsto \frac{\frac{y}{x + y}}{\color{blue}{x + 1}} \]

   if -1.79999999999999988e-107 < x

   1. Initial program 72.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-+r+ 72.6%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      2. *-commutative 72.6%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)} \]

      3. frac-times 86.1%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

      4. associate-*l/ 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 99.9%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{\frac{x}{x + \left(y + 1\right)}}{x + y}} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{\frac{x}{x + \left(y + 1\right)}}{x + y}} \]
   5. Step-by-step derivation

      1. clear-num 99.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + y}{y}}} \cdot \frac{\frac{x}{x + \left(y + 1\right)}}{x + y} \]

      2. frac-times 99.1%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + \left(y + 1\right)}}{\frac{x + y}{y} \cdot \left(x + y\right)}} \]

      3. *-un-lft-identity 99.1%

         \[\leadsto \frac{\color{blue}{\frac{x}{x + \left(y + 1\right)}}}{\frac{x + y}{y} \cdot \left(x + y\right)} \]

      4. associate-+r+ 99.1%

         \[\leadsto \frac{\frac{x}{\color{blue}{\left(x + y\right) + 1}}}{\frac{x + y}{y} \cdot \left(x + y\right)} \]

      5. +-commutative 99.1%

         \[\leadsto \frac{\frac{x}{\color{blue}{\left(y + x\right)} + 1}}{\frac{x + y}{y} \cdot \left(x + y\right)} \]

      6. associate-+l+ 99.1%

         \[\leadsto \frac{\frac{x}{\color{blue}{y + \left(x + 1\right)}}}{\frac{x + y}{y} \cdot \left(x + y\right)} \]

      7. +-commutative 99.1%

         \[\leadsto \frac{\frac{x}{y + \left(x + 1\right)}}{\frac{\color{blue}{y + x}}{y} \cdot \left(x + y\right)} \]

      8. +-commutative 99.1%

         \[\leadsto \frac{\frac{x}{y + \left(x + 1\right)}}{\frac{y + x}{y} \cdot \color{blue}{\left(y + x\right)}} \]

   6. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + \left(x + 1\right)}}{\frac{y + x}{y} \cdot \left(y + x\right)}} \]

   7. Taylor expanded in y around inf 61.6%

      \[\leadsto \frac{\frac{x}{y + \left(x + 1\right)}}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 81.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.8e-107) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ y 1.0)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.8e-107) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.8d-107)) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.8e-107) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.8e-107:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.8e-107)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.8e-107)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.8e-107], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.79999999999999988e-107

   1. Initial program 69.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 69.4%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 69.4%

         \[\leadsto \frac{y \cdot x}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 69.4%

         \[\leadsto \frac{y \cdot x}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. +-commutative 69.4%

         \[\leadsto \frac{y \cdot x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      5. times-frac 90.5%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      6. +-commutative 90.5%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      7. +-commutative 90.5%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      8. +-commutative 90.5%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      9. associate-+l+ 90.5%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 90.5%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 63.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 64.8%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 64.8%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   7. Simplified 64.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if -1.79999999999999988e-107 < x

   1. Initial program 72.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 72.6%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 72.6%

         \[\leadsto \frac{y \cdot x}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 72.6%

         \[\leadsto \frac{y \cdot x}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. +-commutative 72.6%

         \[\leadsto \frac{y \cdot x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      5. times-frac 86.1%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      6. +-commutative 86.1%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      7. +-commutative 86.1%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      8. +-commutative 86.1%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      9. associate-+l+ 86.1%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 86.1%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 59.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 59.7%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   7. Simplified 59.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 59.7%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{y \cdot \left(y + 1\right)} \]

      2. times-frac 61.4%

         \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y + 1}} \]

   9. Applied egg-rr 61.4%

      \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y + 1}} \]
   10. Step-by-step derivation

       1. associate-*l/ 61.4%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y + 1}}{y}} \]

       2. *-lft-identity 61.4%

          \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{y} \]

   11. Simplified 61.4%

       \[\leadsto \color{blue}{\frac{\frac{x}{y + 1}}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 43.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-108}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -6.6e-108) (/ y x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -6.6e-108) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-6.6d-108)) then
        tmp = y / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -6.6e-108) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -6.6e-108:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -6.6e-108)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.6e-108)
		tmp = y / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -6.6e-108], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.6000000000000004e-108

   1. Initial program 69.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-+r+ 69.4%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      2. *-commutative 69.4%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)} \]

      3. frac-times 90.5%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

      4. associate-*l/ 87.6%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      5. times-frac 99.8%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{\frac{x}{x + \left(y + 1\right)}}{x + y}} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{\frac{x}{x + \left(y + 1\right)}}{x + y}} \]

   5. Taylor expanded in y around 0 63.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 63.9%

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   7. Simplified 63.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]

   8. Taylor expanded in x around 0 35.9%

      \[\leadsto \color{blue}{\frac{y}{x}} \]

   if -6.6000000000000004e-108 < x

   1. Initial program 72.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 72.6%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 72.6%

         \[\leadsto \frac{y \cdot x}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 72.6%

         \[\leadsto \frac{y \cdot x}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. +-commutative 72.6%

         \[\leadsto \frac{y \cdot x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      5. times-frac 86.1%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      6. +-commutative 86.1%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      7. +-commutative 86.1%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      8. +-commutative 86.1%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      9. associate-+l+ 86.1%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 86.1%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 59.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 59.7%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   7. Simplified 59.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   8. Taylor expanded in y around 0 40.9%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-108}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 4.3% 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 71.5%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-+r+ 71.5%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   2. *-commutative 71.5%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)} \]

   3. frac-times 87.6%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   4. associate-*l/ 83.5%

      \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   5. times-frac 99.9%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{\frac{x}{x + \left(y + 1\right)}}{x + y}} \]

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{\frac{x}{x + \left(y + 1\right)}}{x + y}} \]

5. Taylor expanded in x around inf 31.2%

   \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{x}} \]

6. Taylor expanded in y around inf 3.9%

   \[\leadsto \color{blue}{\frac{1}{x}} \]

7. Final simplification 3.9%

   \[\leadsto \frac{1}{x} \]
8. Add Preprocessing

Alternative 20: 26.1% 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 71.5%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. *-commutative 71.5%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. +-commutative 71.5%

      \[\leadsto \frac{y \cdot x}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   3. +-commutative 71.5%

      \[\leadsto \frac{y \cdot x}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   4. +-commutative 71.5%

      \[\leadsto \frac{y \cdot x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

   5. times-frac 87.6%

      \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

   6. +-commutative 87.6%

      \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

   7. +-commutative 87.6%

      \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

   8. +-commutative 87.6%

      \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

   9. associate-+l+ 87.6%

      \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

3. Simplified 87.6%

   \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 50.8%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
6. Step-by-step derivation

   1. +-commutative 50.8%

      \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

7. Simplified 50.8%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

8. Taylor expanded in y around 0 28.8%

   \[\leadsto \color{blue}{\frac{x}{y}} \]

9. Final simplification 28.8%

   \[\leadsto \frac{x}{y} \]
10. Add Preprocessing

Alternative 21: 3.8% accurate, 8.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ -y \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (- y))

C

assert(x < y);
double code(double x, double y) {
	return -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 = -y
end function

Java

assert x < y;
public static double code(double x, double y) {
	return -y;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return -y

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(-y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = -y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := (-y)

Derivation

1. Initial program 71.5%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. *-commutative 71.5%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. +-commutative 71.5%

      \[\leadsto \frac{y \cdot x}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   3. +-commutative 71.5%

      \[\leadsto \frac{y \cdot x}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   4. +-commutative 71.5%

      \[\leadsto \frac{y \cdot x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

   5. times-frac 87.6%

      \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

   6. +-commutative 87.6%

      \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

   7. +-commutative 87.6%

      \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

   8. +-commutative 87.6%

      \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

   9. associate-+l+ 87.6%

      \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

3. Simplified 87.6%

   \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 48.8%

   \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
6. Step-by-step derivation

   1. associate-/r* 49.0%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

   2. +-commutative 49.0%

      \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

7. Simplified 49.0%

   \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

8. Taylor expanded in x around 0 19.9%

   \[\leadsto \color{blue}{-1 \cdot y + \frac{y}{x}} \]
9. Step-by-step derivation

   1. neg-mul-1 19.9%

      \[\leadsto \color{blue}{\left(-y\right)} + \frac{y}{x} \]

   2. +-commutative 19.9%

      \[\leadsto \color{blue}{\frac{y}{x} + \left(-y\right)} \]

   3. unsub-neg 19.9%

      \[\leadsto \color{blue}{\frac{y}{x} - y} \]

10. Simplified 19.9%

    \[\leadsto \color{blue}{\frac{y}{x} - y} \]

11. Taylor expanded in x around inf 3.6%

    \[\leadsto \color{blue}{-1 \cdot y} \]
12. Step-by-step derivation

    1. neg-mul-1 3.6%

       \[\leadsto \color{blue}{-y} \]

13. Simplified 3.6%

    \[\leadsto \color{blue}{-y} \]

14. Final simplification 3.6%

    \[\leadsto -y \]
15. 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 2024034 
(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))))