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

Percentage Accurate: 69.3% → 99.2%

Time: 15.5s

Alternatives: 19

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.3% 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: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.3%

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

   1. associate-*l* 70.3%

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

   2. +-commutative 70.3%

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

   3. +-commutative 70.3%

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

   4. +-commutative 70.3%

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

   5. associate-*l* 70.3%

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

   6. *-commutative 70.3%

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

   7. associate-*r/ 85.0%

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

   8. *-commutative 85.0%

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

   9. distribute-rgt1-in 66.3%

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

   10. fma-def 85.0%

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

   11. +-commutative 85.0%

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

   12. +-commutative 85.0%

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

   13. cube-unmult 85.0%

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

   14. +-commutative 85.0%

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

3. Simplified 85.0%

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

   1. 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. *-commutative 70.3%

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

   3. fma-udef 54.9%

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

   4. cube-mult 54.9%

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

   5. distribute-rgt1-in 70.3%

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

   6. *-commutative 70.3%

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

   7. +-commutative 70.3%

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

   8. associate-+r+ 70.3%

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

   9. frac-times 88.8%

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

   10. *-commutative 88.8%

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

   11. clear-num 88.7%

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

   12. associate-/r* 99.7%

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

   13. frac-times 99.2%

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

   14. *-un-lft-identity 99.2%

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

   15. associate-+r+ 99.2%

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

   16. +-commutative 99.2%

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

   17. associate-+l+ 99.2%

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

5. Applied egg-rr 99.2%

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

6. Final simplification 99.2%

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

Alternative 2: 91.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-305}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{\frac{x}{\left(x + 1\right) + \frac{x}{y}}}{x + y}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+151}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 7.2e-305)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 2.2e-146)
     (/ (/ x (+ (+ x 1.0) (/ x y))) (+ x y))
     (if (<= y 3e+151)
       (* (/ y (* (+ x y) (+ x y))) (/ x (+ y 1.0)))
       (/ (/ x (+ x y)) (+ y (+ x (+ x 1.0))))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 7.2e-305) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 2.2e-146) {
		tmp = (x / ((x + 1.0) + (x / y))) / (x + y);
	} else if (y <= 3e+151) {
		tmp = (y / ((x + y) * (x + y))) * (x / (y + 1.0));
	} else {
		tmp = (x / (x + y)) / (y + (x + (x + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 7.2d-305) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 2.2d-146) then
        tmp = (x / ((x + 1.0d0) + (x / y))) / (x + y)
    else if (y <= 3d+151) then
        tmp = (y / ((x + y) * (x + y))) * (x / (y + 1.0d0))
    else
        tmp = (x / (x + y)) / (y + (x + (x + 1.0d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 7.2e-305) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 2.2e-146) {
		tmp = (x / ((x + 1.0) + (x / y))) / (x + y);
	} else if (y <= 3e+151) {
		tmp = (y / ((x + y) * (x + y))) * (x / (y + 1.0));
	} else {
		tmp = (x / (x + y)) / (y + (x + (x + 1.0)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 7.2e-305:
		tmp = (y / x) / (x + 1.0)
	elif y <= 2.2e-146:
		tmp = (x / ((x + 1.0) + (x / y))) / (x + y)
	elif y <= 3e+151:
		tmp = (y / ((x + y) * (x + y))) * (x / (y + 1.0))
	else:
		tmp = (x / (x + y)) / (y + (x + (x + 1.0)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 7.2e-305)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 2.2e-146)
		tmp = Float64(Float64(x / Float64(Float64(x + 1.0) + Float64(x / y))) / Float64(x + y));
	elseif (y <= 3e+151)
		tmp = Float64(Float64(y / Float64(Float64(x + y) * Float64(x + y))) * Float64(x / Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(y + Float64(x + Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7.2e-305)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 2.2e-146)
		tmp = (x / ((x + 1.0) + (x / y))) / (x + y);
	elseif (y <= 3e+151)
		tmp = (y / ((x + y) * (x + y))) * (x / (y + 1.0));
	else
		tmp = (x / (x + y)) / (y + (x + (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 7.2e-305], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e-146], N[(N[(x / N[(N[(x + 1.0), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3e+151], N[(N[(y / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 7.20000000000000007e-305

   1. Initial program 68.3%

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

      1. times-frac 89.3%

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

      2. +-commutative 89.3%

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

      3. associate-+l+ 89.3%

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

   3. Simplified 89.3%

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

   4. Taylor expanded in y around 0 49.7%

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

      1. associate-/r* 51.0%

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

      2. +-commutative 51.0%

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

   6. Simplified 51.0%

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

   if 7.20000000000000007e-305 < y < 2.2e-146

   1. Initial program 56.6%

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

      1. associate-*l* 56.6%

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

      2. +-commutative 56.6%

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

      3. +-commutative 56.6%

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

      4. +-commutative 56.6%

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

      5. associate-*l* 56.6%

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

      6. *-commutative 56.6%

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

      7. associate-*r/ 77.6%

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

      8. *-commutative 77.6%

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

      9. distribute-rgt1-in 72.1%

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

      10. fma-def 77.6%

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

      11. +-commutative 77.6%

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

      12. +-commutative 77.6%

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

      13. cube-unmult 77.6%

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

      14. +-commutative 77.6%

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

   3. Simplified 77.6%

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

      1. associate-*r/ 56.6%

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

      2. *-commutative 56.6%

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

      3. fma-udef 51.0%

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

      4. cube-mult 51.0%

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

      5. distribute-rgt1-in 56.6%

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

      6. *-commutative 56.6%

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

      7. +-commutative 56.6%

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

      8. associate-+r+ 56.6%

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

      9. frac-times 77.7%

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

      10. *-commutative 77.7%

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

      11. clear-num 77.6%

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

      12. associate-/r* 99.7%

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

      13. frac-times 99.7%

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

      14. *-un-lft-identity 99.7%

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

      15. associate-+r+ 99.7%

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

      16. +-commutative 99.7%

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

      17. associate-+l+ 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x around 0 90.3%

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

   7. Taylor expanded in y around 0 90.4%

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

      1. associate-+r+ 90.4%

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

      2. +-commutative 90.4%

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

   9. Simplified 90.4%

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

       1. expm1-log1p-u 90.4%

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

       2. expm1-udef 46.4%

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

       3. associate-/l/ 46.4%

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

       4. +-commutative 46.4%

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

   11. Applied egg-rr 46.4%

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

       1. expm1-def 97.9%

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

       2. expm1-log1p 97.9%

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

       3. associate-/r* 90.4%

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

       4. +-commutative 90.4%

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

   13. Simplified 90.4%

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

   if 2.2e-146 < y < 2.9999999999999999e151

   1. Initial program 90.3%

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

      1. associate-*l* 90.3%

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

      2. +-commutative 90.3%

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

      3. +-commutative 90.3%

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

      4. +-commutative 90.3%

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

      5. *-commutative 90.3%

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

      6. associate-*l* 90.3%

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

      7. times-frac 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 85.2%

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

      1. +-commutative 85.2%

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

   6. Simplified 85.2%

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

   if 2.9999999999999999e151 < y

   1. Initial program 55.2%

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

      1. associate-*l* 55.2%

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

      2. +-commutative 55.2%

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

      3. +-commutative 55.2%

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

      4. +-commutative 55.2%

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

      5. associate-*l* 55.2%

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

      6. *-commutative 55.2%

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

      7. associate-*r/ 78.7%

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

      8. *-commutative 78.7%

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

      9. distribute-rgt1-in 78.3%

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

      10. fma-def 78.7%

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

      11. +-commutative 78.7%

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

      12. +-commutative 78.7%

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

      13. cube-unmult 78.7%

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

      14. +-commutative 78.7%

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

   3. Simplified 78.7%

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

      1. associate-*r/ 55.2%

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

      2. *-commutative 55.2%

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

      3. fma-udef 55.2%

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

      4. cube-mult 55.2%

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

      5. distribute-rgt1-in 55.2%

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

      6. *-commutative 55.2%

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

      7. +-commutative 55.2%

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

      8. associate-+r+ 55.2%

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

      9. frac-times 78.7%

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

      10. *-commutative 78.7%

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

      11. clear-num 78.7%

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

      12. associate-/r* 100.0%

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

      13. frac-times 97.8%

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

      14. *-un-lft-identity 97.8%

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

      15. associate-+r+ 97.8%

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

      16. +-commutative 97.8%

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

      17. associate-+l+ 97.8%

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

   5. Applied egg-rr 97.8%

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

   6. Taylor expanded in y around -inf 85.0%

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

      1. mul-1-neg 85.0%

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

      2. unsub-neg 85.0%

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

      3. neg-mul-1 85.0%

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

      4. +-commutative 85.0%

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

      5. unsub-neg 85.0%

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

      6. distribute-lft-in 85.0%

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

      7. metadata-eval 85.0%

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

      8. neg-mul-1 85.0%

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

      9. unsub-neg 85.0%

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

   8. Simplified 85.0%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-305}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{\frac{x}{\left(x + 1\right) + \frac{x}{y}}}{x + y}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+151}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \]

Alternative 3: 97.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6.9 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;x \leq -1.16 \cdot 10^{-116}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(x + y\right) \cdot \frac{y + 1}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -6.9e+155)
   (/ (/ y (+ x y)) (+ x (+ y (+ y 1.0))))
   (if (<= x -1.16e-116)
     (* (/ x (* (+ x y) (+ x y))) (/ y (+ y (+ x 1.0))))
     (/ (/ x (+ x y)) (* (+ x y) (/ (+ y 1.0) y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -6.9e+155) {
		tmp = (y / (x + y)) / (x + (y + (y + 1.0)));
	} else if (x <= -1.16e-116) {
		tmp = (x / ((x + y) * (x + y))) * (y / (y + (x + 1.0)));
	} else {
		tmp = (x / (x + y)) / ((x + y) * ((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 <= (-6.9d+155)) then
        tmp = (y / (x + y)) / (x + (y + (y + 1.0d0)))
    else if (x <= (-1.16d-116)) then
        tmp = (x / ((x + y) * (x + y))) * (y / (y + (x + 1.0d0)))
    else
        tmp = (x / (x + y)) / ((x + y) * ((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 <= -6.9e+155) {
		tmp = (y / (x + y)) / (x + (y + (y + 1.0)));
	} else if (x <= -1.16e-116) {
		tmp = (x / ((x + y) * (x + y))) * (y / (y + (x + 1.0)));
	} else {
		tmp = (x / (x + y)) / ((x + y) * ((y + 1.0) / y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -6.9e+155:
		tmp = (y / (x + y)) / (x + (y + (y + 1.0)))
	elif x <= -1.16e-116:
		tmp = (x / ((x + y) * (x + y))) * (y / (y + (x + 1.0)))
	else:
		tmp = (x / (x + y)) / ((x + y) * ((y + 1.0) / y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -6.9e+155)
		tmp = Float64(Float64(y / Float64(x + y)) / Float64(x + Float64(y + Float64(y + 1.0))));
	elseif (x <= -1.16e-116)
		tmp = Float64(Float64(x / Float64(Float64(x + y) * Float64(x + y))) * Float64(y / Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(Float64(x + y) * Float64(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 <= -6.9e+155)
		tmp = (y / (x + y)) / (x + (y + (y + 1.0)));
	elseif (x <= -1.16e-116)
		tmp = (x / ((x + y) * (x + y))) * (y / (y + (x + 1.0)));
	else
		tmp = (x / (x + y)) / ((x + y) * ((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, -6.9e+155], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.16e-116], N[(N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.9000000000000004e155

   1. Initial program 74.3%

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

      1. associate-*l* 74.3%

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

      2. +-commutative 74.3%

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

      3. +-commutative 74.3%

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

      4. +-commutative 74.3%

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

      5. associate-*l* 74.3%

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

      6. *-commutative 74.3%

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

      7. associate-*r/ 83.0%

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

      8. *-commutative 83.0%

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

      9. distribute-rgt1-in 0.6%

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

      10. fma-def 83.0%

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

      11. +-commutative 83.0%

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

      12. +-commutative 83.0%

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

      13. cube-unmult 83.0%

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

      14. +-commutative 83.0%

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

   3. Simplified 83.0%

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

      1. associate-*r/ 74.3%

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

      2. fma-udef 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 74.3%

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

      5. *-commutative 74.3%

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

      6. +-commutative 74.3%

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

      7. associate-+r+ 74.3%

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

      8. frac-times 83.0%

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

      9. *-commutative 83.0%

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

      10. associate-/r* 99.9%

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

      11. clear-num 99.9%

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

      12. frac-times 99.9%

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

      13. *-un-lft-identity 99.9%

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

      14. associate-+r+ 99.9%

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

      15. +-commutative 99.9%

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

      16. associate-+l+ 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around -inf 92.9%

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

      1. mul-1-neg 92.9%

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

      2. unsub-neg 92.9%

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

      3. neg-mul-1 92.9%

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

      4. +-commutative 92.9%

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

      5. unsub-neg 92.9%

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

      6. distribute-lft-in 92.9%

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

      7. metadata-eval 92.9%

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

      8. neg-mul-1 92.9%

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

      9. unsub-neg 92.9%

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

   8. Simplified 92.9%

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

   if -6.9000000000000004e155 < x < -1.16e-116

   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. Step-by-step derivation

      1. times-frac 95.1%

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

      2. +-commutative 95.1%

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

      3. associate-+l+ 95.1%

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

   3. Simplified 95.1%

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

   if -1.16e-116 < x

   1. Initial program 68.7%

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

      1. associate-*l* 68.7%

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

      2. +-commutative 68.7%

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

      3. +-commutative 68.7%

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

      4. +-commutative 68.7%

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

      5. associate-*l* 68.7%

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

      6. *-commutative 68.7%

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

      7. associate-*r/ 83.9%

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

      8. *-commutative 83.9%

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

      9. distribute-rgt1-in 73.0%

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

      10. fma-def 83.9%

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

      11. +-commutative 83.9%

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

      12. +-commutative 83.9%

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

      13. cube-unmult 83.9%

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

      14. +-commutative 83.9%

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

   3. Simplified 83.9%

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

      1. associate-*r/ 68.7%

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

      2. *-commutative 68.7%

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

      3. fma-udef 61.6%

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

      4. cube-mult 61.5%

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

      5. distribute-rgt1-in 68.7%

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

      6. *-commutative 68.7%

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

      7. +-commutative 68.7%

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

      8. associate-+r+ 68.7%

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

      9. frac-times 87.4%

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

      10. *-commutative 87.4%

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

      11. clear-num 87.4%

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

      12. associate-/r* 99.7%

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

      13. frac-times 99.0%

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

      14. *-un-lft-identity 99.0%

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

      15. associate-+r+ 99.0%

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

      16. +-commutative 99.0%

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

      17. associate-+l+ 99.0%

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

   5. Applied egg-rr 99.0%

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

   6. Taylor expanded in x around 0 79.3%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.9 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;x \leq -1.16 \cdot 10^{-116}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(x + y\right) \cdot \frac{y + 1}{y}}\\ \end{array} \]

Alternative 4: 97.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-63}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{y + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(x + y\right) \cdot \frac{y + 1}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -6.8e+155)
   (/ (/ y (+ x y)) (+ x (+ y (+ y 1.0))))
   (if (<= x -1.55e-63)
     (* (/ y (* (+ x y) (+ x y))) (/ x (+ y (+ x 1.0))))
     (/ (/ x (+ x y)) (* (+ x y) (/ (+ y 1.0) y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -6.8e+155) {
		tmp = (y / (x + y)) / (x + (y + (y + 1.0)));
	} else if (x <= -1.55e-63) {
		tmp = (y / ((x + y) * (x + y))) * (x / (y + (x + 1.0)));
	} else {
		tmp = (x / (x + y)) / ((x + y) * ((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 <= (-6.8d+155)) then
        tmp = (y / (x + y)) / (x + (y + (y + 1.0d0)))
    else if (x <= (-1.55d-63)) then
        tmp = (y / ((x + y) * (x + y))) * (x / (y + (x + 1.0d0)))
    else
        tmp = (x / (x + y)) / ((x + y) * ((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 <= -6.8e+155) {
		tmp = (y / (x + y)) / (x + (y + (y + 1.0)));
	} else if (x <= -1.55e-63) {
		tmp = (y / ((x + y) * (x + y))) * (x / (y + (x + 1.0)));
	} else {
		tmp = (x / (x + y)) / ((x + y) * ((y + 1.0) / y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -6.8e+155:
		tmp = (y / (x + y)) / (x + (y + (y + 1.0)))
	elif x <= -1.55e-63:
		tmp = (y / ((x + y) * (x + y))) * (x / (y + (x + 1.0)))
	else:
		tmp = (x / (x + y)) / ((x + y) * ((y + 1.0) / y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -6.8e+155)
		tmp = Float64(Float64(y / Float64(x + y)) / Float64(x + Float64(y + Float64(y + 1.0))));
	elseif (x <= -1.55e-63)
		tmp = Float64(Float64(y / Float64(Float64(x + y) * Float64(x + y))) * Float64(x / Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(Float64(x + y) * Float64(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 <= -6.8e+155)
		tmp = (y / (x + y)) / (x + (y + (y + 1.0)));
	elseif (x <= -1.55e-63)
		tmp = (y / ((x + y) * (x + y))) * (x / (y + (x + 1.0)));
	else
		tmp = (x / (x + y)) / ((x + y) * ((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, -6.8e+155], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.55e-63], N[(N[(y / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.8000000000000002e155

   1. Initial program 74.3%

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

      1. associate-*l* 74.3%

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

      2. +-commutative 74.3%

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

      3. +-commutative 74.3%

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

      4. +-commutative 74.3%

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

      5. associate-*l* 74.3%

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

      6. *-commutative 74.3%

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

      7. associate-*r/ 83.0%

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

      8. *-commutative 83.0%

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

      9. distribute-rgt1-in 0.6%

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

      10. fma-def 83.0%

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

      11. +-commutative 83.0%

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

      12. +-commutative 83.0%

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

      13. cube-unmult 83.0%

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

      14. +-commutative 83.0%

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

   3. Simplified 83.0%

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

      1. associate-*r/ 74.3%

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

      2. fma-udef 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 74.3%

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

      5. *-commutative 74.3%

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

      6. +-commutative 74.3%

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

      7. associate-+r+ 74.3%

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

      8. frac-times 83.0%

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

      9. *-commutative 83.0%

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

      10. associate-/r* 99.9%

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

      11. clear-num 99.9%

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

      12. frac-times 99.9%

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

      13. *-un-lft-identity 99.9%

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

      14. associate-+r+ 99.9%

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

      15. +-commutative 99.9%

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

      16. associate-+l+ 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around -inf 92.9%

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

      1. mul-1-neg 92.9%

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

      2. unsub-neg 92.9%

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

      3. neg-mul-1 92.9%

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

      4. +-commutative 92.9%

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

      5. unsub-neg 92.9%

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

      6. distribute-lft-in 92.9%

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

      7. metadata-eval 92.9%

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

      8. neg-mul-1 92.9%

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

      9. unsub-neg 92.9%

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

   8. Simplified 92.9%

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

   if -6.8000000000000002e155 < x < -1.54999999999999992e-63

   1. Initial program 73.6%

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

      1. associate-*l* 73.6%

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

      2. +-commutative 73.6%

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

      3. +-commutative 73.6%

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

      4. +-commutative 73.6%

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

      5. *-commutative 73.6%

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

      6. associate-*l* 73.6%

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

      7. times-frac 93.9%

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

      8. +-commutative 93.9%

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

      9. +-commutative 93.9%

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

      10. associate-+l+ 93.9%

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

   3. Simplified 93.9%

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

   if -1.54999999999999992e-63 < x

   1. Initial program 68.8%

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

      1. associate-*l* 68.8%

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

      2. +-commutative 68.8%

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

      3. +-commutative 68.8%

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

      4. +-commutative 68.8%

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

      5. associate-*l* 68.8%

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

      6. *-commutative 68.8%

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

      7. associate-*r/ 85.0%

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

      8. *-commutative 85.0%

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

      9. distribute-rgt1-in 74.4%

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

      10. fma-def 85.0%

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

      11. +-commutative 85.0%

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

      12. +-commutative 85.0%

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

      13. cube-unmult 85.0%

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

      14. +-commutative 85.0%

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

   3. Simplified 85.0%

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

      1. associate-*r/ 68.9%

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

      2. *-commutative 68.9%

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

      3. fma-udef 61.6%

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

      4. cube-mult 61.6%

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

      5. distribute-rgt1-in 68.8%

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

      6. *-commutative 68.8%

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

      7. +-commutative 68.8%

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

      8. associate-+r+ 68.8%

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

      9. frac-times 88.3%

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

      10. *-commutative 88.3%

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

      11. clear-num 88.3%

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

      12. associate-/r* 99.7%

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

      13. frac-times 99.0%

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

      14. *-un-lft-identity 99.0%

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

      15. associate-+r+ 99.0%

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

      16. +-commutative 99.0%

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

      17. associate-+l+ 99.0%

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

   5. Applied egg-rr 99.0%

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

   6. Taylor expanded in x around 0 80.7%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-63}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{y + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(x + y\right) \cdot \frac{y + 1}{y}}\\ \end{array} \]

Alternative 5: 86.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;y \leq 3.75 \cdot 10^{-305}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{-25}:\\ \;\;\;\;\frac{t_0}{\left(x + 1\right) + \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))))
   (if (<= y 3.75e-305)
     (/ (/ y x) (+ x 1.0))
     (if (<= y 7.4e-25) (/ t_0 (+ (+ x 1.0) (/ x y))) (/ t_0 (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= 3.75e-305) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 7.4e-25) {
		tmp = t_0 / ((x + 1.0) + (x / y));
	} else {
		tmp = t_0 / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (x + y)
    if (y <= 3.75d-305) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 7.4d-25) then
        tmp = t_0 / ((x + 1.0d0) + (x / y))
    else
        tmp = t_0 / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= 3.75e-305) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 7.4e-25) {
		tmp = t_0 / ((x + 1.0) + (x / y));
	} else {
		tmp = t_0 / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (x + y)
	tmp = 0
	if y <= 3.75e-305:
		tmp = (y / x) / (x + 1.0)
	elif y <= 7.4e-25:
		tmp = t_0 / ((x + 1.0) + (x / y))
	else:
		tmp = t_0 / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	tmp = 0.0
	if (y <= 3.75e-305)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 7.4e-25)
		tmp = Float64(t_0 / Float64(Float64(x + 1.0) + Float64(x / y)));
	else
		tmp = Float64(t_0 / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	tmp = 0.0;
	if (y <= 3.75e-305)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 7.4e-25)
		tmp = t_0 / ((x + 1.0) + (x / y));
	else
		tmp = t_0 / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 3.75e-305], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.4e-25], N[(t$95$0 / N[(N[(x + 1.0), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.7500000000000001e-305

   1. Initial program 68.3%

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

      1. times-frac 89.3%

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

      2. +-commutative 89.3%

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

      3. associate-+l+ 89.3%

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

   3. Simplified 89.3%

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

   4. Taylor expanded in y around 0 49.7%

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

      1. associate-/r* 51.0%

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

      2. +-commutative 51.0%

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

   6. Simplified 51.0%

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

   if 3.7500000000000001e-305 < y < 7.40000000000000017e-25

   1. Initial program 74.3%

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

      1. associate-*l* 74.3%

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

      2. +-commutative 74.3%

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

      3. +-commutative 74.3%

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

      4. +-commutative 74.3%

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

      5. associate-*l* 74.3%

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

      6. *-commutative 74.3%

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

      7. associate-*r/ 87.5%

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

      8. *-commutative 87.5%

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

      9. distribute-rgt1-in 75.2%

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

      10. fma-def 87.5%

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

      11. +-commutative 87.5%

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

      12. +-commutative 87.5%

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

      13. cube-unmult 87.5%

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

      14. +-commutative 87.5%

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

   3. Simplified 87.5%

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

      1. associate-*r/ 74.3%

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

      2. *-commutative 74.3%

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

      3. fma-udef 62.0%

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

      4. cube-mult 62.0%

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

      5. distribute-rgt1-in 74.3%

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

      6. *-commutative 74.3%

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

      7. +-commutative 74.3%

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

      8. associate-+r+ 74.3%

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

      9. frac-times 87.5%

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

      10. *-commutative 87.5%

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

      11. clear-num 87.4%

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

      12. associate-/r* 99.7%

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

      13. frac-times 99.7%

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

      14. *-un-lft-identity 99.7%

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

      15. associate-+r+ 99.7%

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

      16. +-commutative 99.7%

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

      17. associate-+l+ 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x around 0 78.8%

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

   7. Taylor expanded in y around 0 78.8%

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

      1. associate-+r+ 78.8%

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

      2. +-commutative 78.8%

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

   9. Simplified 78.8%

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

   if 7.40000000000000017e-25 < y

   1. Initial program 70.1%

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

      1. associate-*l* 70.1%

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

      2. +-commutative 70.1%

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

      3. +-commutative 70.1%

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

      4. +-commutative 70.1%

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

      5. associate-*l* 70.1%

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

      6. *-commutative 70.1%

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

      7. associate-*r/ 80.8%

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

      8. *-commutative 80.8%

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

      9. distribute-rgt1-in 77.7%

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

      10. fma-def 80.8%

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

      11. +-commutative 80.8%

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

      12. +-commutative 80.8%

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

      13. cube-unmult 80.8%

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

      14. +-commutative 80.8%

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

   3. Simplified 80.8%

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

      1. associate-*r/ 70.1%

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

      2. *-commutative 70.1%

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

      3. fma-udef 67.1%

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

      4. cube-mult 67.1%

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

      5. distribute-rgt1-in 70.1%

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

      6. *-commutative 70.1%

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

      7. +-commutative 70.1%

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

      8. associate-+r+ 70.1%

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

      9. frac-times 89.1%

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

      10. *-commutative 89.1%

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

      11. clear-num 89.1%

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

      12. associate-/r* 99.8%

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

      13. frac-times 98.7%

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

      14. *-un-lft-identity 98.7%

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

      15. associate-+r+ 98.7%

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

      16. +-commutative 98.7%

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

      17. associate-+l+ 98.7%

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

   5. Applied egg-rr 98.7%

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

   6. Taylor expanded in x around 0 75.0%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.75 \cdot 10^{-305}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(x + 1\right) + \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]

Alternative 6: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;y \leq 7.2 \cdot 10^{-305}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{t_0}{\left(x + 1\right) + \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))))
   (if (<= y 7.2e-305)
     (/ (/ y x) (+ x 1.0))
     (if (<= y 8.2e-12)
       (/ t_0 (+ (+ x 1.0) (/ x y)))
       (/ t_0 (+ y (+ x (+ x 1.0))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= 7.2e-305) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 8.2e-12) {
		tmp = t_0 / ((x + 1.0) + (x / y));
	} else {
		tmp = t_0 / (y + (x + (x + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (x + y)
    if (y <= 7.2d-305) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 8.2d-12) then
        tmp = t_0 / ((x + 1.0d0) + (x / y))
    else
        tmp = t_0 / (y + (x + (x + 1.0d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= 7.2e-305) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 8.2e-12) {
		tmp = t_0 / ((x + 1.0) + (x / y));
	} else {
		tmp = t_0 / (y + (x + (x + 1.0)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (x + y)
	tmp = 0
	if y <= 7.2e-305:
		tmp = (y / x) / (x + 1.0)
	elif y <= 8.2e-12:
		tmp = t_0 / ((x + 1.0) + (x / y))
	else:
		tmp = t_0 / (y + (x + (x + 1.0)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	tmp = 0.0
	if (y <= 7.2e-305)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 8.2e-12)
		tmp = Float64(t_0 / Float64(Float64(x + 1.0) + Float64(x / y)));
	else
		tmp = Float64(t_0 / Float64(y + Float64(x + Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	tmp = 0.0;
	if (y <= 7.2e-305)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 8.2e-12)
		tmp = t_0 / ((x + 1.0) + (x / y));
	else
		tmp = t_0 / (y + (x + (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 7.2e-305], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.2e-12], N[(t$95$0 / N[(N[(x + 1.0), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(y + N[(x + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.20000000000000007e-305

   1. Initial program 68.3%

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

      1. times-frac 89.3%

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

      2. +-commutative 89.3%

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

      3. associate-+l+ 89.3%

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

   3. Simplified 89.3%

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

   4. Taylor expanded in y around 0 49.7%

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

      1. associate-/r* 51.0%

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

      2. +-commutative 51.0%

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

   6. Simplified 51.0%

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

   if 7.20000000000000007e-305 < y < 8.19999999999999979e-12

   1. Initial program 75.4%

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

      1. associate-*l* 75.4%

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

      2. +-commutative 75.4%

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

      3. +-commutative 75.4%

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

      4. +-commutative 75.4%

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

      5. associate-*l* 75.4%

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

      6. *-commutative 75.4%

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

      7. associate-*r/ 88.0%

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

      8. *-commutative 88.0%

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

      9. distribute-rgt1-in 74.8%

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

      10. fma-def 88.0%

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

      11. +-commutative 88.0%

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

      12. +-commutative 88.0%

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

      13. cube-unmult 88.0%

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

      14. +-commutative 88.0%

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

   3. Simplified 88.0%

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

      1. associate-*r/ 75.4%

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

      2. *-commutative 75.4%

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

      3. fma-udef 62.2%

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

      4. cube-mult 62.2%

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

      5. distribute-rgt1-in 75.4%

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

      6. *-commutative 75.4%

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

      7. +-commutative 75.4%

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

      8. associate-+r+ 75.4%

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

      9. frac-times 88.0%

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

      10. *-commutative 88.0%

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

      11. clear-num 87.9%

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

      12. associate-/r* 99.7%

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

      13. frac-times 99.7%

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

      14. *-un-lft-identity 99.7%

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

      15. associate-+r+ 99.7%

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

      16. +-commutative 99.7%

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

      17. associate-+l+ 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x around 0 78.3%

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

   7. Taylor expanded in y around 0 78.4%

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

      1. associate-+r+ 78.4%

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

      2. +-commutative 78.4%

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

   9. Simplified 78.4%

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

   if 8.19999999999999979e-12 < y

   1. Initial program 68.7%

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

      1. associate-*l* 68.7%

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

      2. +-commutative 68.7%

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

      3. +-commutative 68.7%

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

      4. +-commutative 68.7%

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

      5. associate-*l* 68.7%

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

      6. *-commutative 68.7%

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

      7. associate-*r/ 80.0%

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

      8. *-commutative 80.0%

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

      9. distribute-rgt1-in 78.2%

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

      10. fma-def 80.0%

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

      11. +-commutative 80.0%

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

      12. +-commutative 80.0%

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

      13. cube-unmult 80.0%

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

      14. +-commutative 80.0%

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

   3. Simplified 80.0%

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

      1. associate-*r/ 68.7%

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

      2. *-commutative 68.7%

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

      3. fma-udef 67.1%

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

      4. cube-mult 67.1%

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

      5. distribute-rgt1-in 68.7%

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

      6. *-commutative 68.7%

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

      7. +-commutative 68.7%

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

      8. associate-+r+ 68.7%

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

      9. frac-times 88.6%

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

      10. *-commutative 88.6%

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

      11. clear-num 88.6%

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

      12. associate-/r* 99.8%

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

      13. frac-times 98.7%

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

      14. *-un-lft-identity 98.7%

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

      15. associate-+r+ 98.7%

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

      16. +-commutative 98.7%

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

      17. associate-+l+ 98.7%

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

   5. Applied egg-rr 98.7%

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

   6. Taylor expanded in y around -inf 77.4%

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

      1. mul-1-neg 77.4%

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

      2. unsub-neg 77.4%

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

      3. neg-mul-1 77.4%

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

      4. +-commutative 77.4%

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

      5. unsub-neg 77.4%

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

      6. distribute-lft-in 77.4%

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

      7. metadata-eval 77.4%

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

      8. neg-mul-1 77.4%

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

      9. unsub-neg 77.4%

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

   8. Simplified 77.4%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-305}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(x + 1\right) + \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \]

Alternative 7: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-305}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{x}{\left(x + 1\right) + \frac{x}{y}}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4.3e-305)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 1.3e-11)
     (/ (/ x (+ (+ x 1.0) (/ x y))) (+ x y))
     (/ (/ x (+ x y)) (+ y (+ x (+ x 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.3e-305) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 1.3e-11) {
		tmp = (x / ((x + 1.0) + (x / y))) / (x + y);
	} else {
		tmp = (x / (x + y)) / (y + (x + (x + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.3d-305) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 1.3d-11) then
        tmp = (x / ((x + 1.0d0) + (x / y))) / (x + y)
    else
        tmp = (x / (x + y)) / (y + (x + (x + 1.0d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 4.3e-305) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 1.3e-11) {
		tmp = (x / ((x + 1.0) + (x / y))) / (x + y);
	} else {
		tmp = (x / (x + y)) / (y + (x + (x + 1.0)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 4.3e-305:
		tmp = (y / x) / (x + 1.0)
	elif y <= 1.3e-11:
		tmp = (x / ((x + 1.0) + (x / y))) / (x + y)
	else:
		tmp = (x / (x + y)) / (y + (x + (x + 1.0)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4.3e-305)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 1.3e-11)
		tmp = Float64(Float64(x / Float64(Float64(x + 1.0) + Float64(x / y))) / Float64(x + y));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(y + Float64(x + Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.3e-305)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 1.3e-11)
		tmp = (x / ((x + 1.0) + (x / y))) / (x + y);
	else
		tmp = (x / (x + y)) / (y + (x + (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 4.3e-305], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e-11], N[(N[(x / N[(N[(x + 1.0), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.3000000000000002e-305

   1. Initial program 68.3%

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

      1. times-frac 89.3%

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

      2. +-commutative 89.3%

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

      3. associate-+l+ 89.3%

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

   3. Simplified 89.3%

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

   4. Taylor expanded in y around 0 49.7%

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

      1. associate-/r* 51.0%

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

      2. +-commutative 51.0%

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

   6. Simplified 51.0%

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

   if 4.3000000000000002e-305 < y < 1.3e-11

   1. Initial program 75.4%

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

      1. associate-*l* 75.4%

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

      2. +-commutative 75.4%

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

      3. +-commutative 75.4%

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

      4. +-commutative 75.4%

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

      5. associate-*l* 75.4%

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

      6. *-commutative 75.4%

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

      7. associate-*r/ 88.0%

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

      8. *-commutative 88.0%

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

      9. distribute-rgt1-in 74.8%

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

      10. fma-def 88.0%

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

      11. +-commutative 88.0%

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

      12. +-commutative 88.0%

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

      13. cube-unmult 88.0%

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

      14. +-commutative 88.0%

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

   3. Simplified 88.0%

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

      1. associate-*r/ 75.4%

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

      2. *-commutative 75.4%

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

      3. fma-udef 62.2%

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

      4. cube-mult 62.2%

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

      5. distribute-rgt1-in 75.4%

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

      6. *-commutative 75.4%

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

      7. +-commutative 75.4%

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

      8. associate-+r+ 75.4%

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

      9. frac-times 88.0%

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

      10. *-commutative 88.0%

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

      11. clear-num 87.9%

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

      12. associate-/r* 99.7%

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

      13. frac-times 99.7%

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

      14. *-un-lft-identity 99.7%

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

      15. associate-+r+ 99.7%

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

      16. +-commutative 99.7%

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

      17. associate-+l+ 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x around 0 78.3%

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

   7. Taylor expanded in y around 0 78.4%

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

      1. associate-+r+ 78.4%

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

      2. +-commutative 78.4%

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

   9. Simplified 78.4%

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

       1. expm1-log1p-u 78.4%

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

       2. expm1-udef 49.5%

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

       3. associate-/l/ 49.5%

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

       4. +-commutative 49.5%

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

   11. Applied egg-rr 49.5%

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

       1. expm1-def 93.3%

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

       2. expm1-log1p 93.3%

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

       3. associate-/r* 78.4%

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

       4. +-commutative 78.4%

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

   13. Simplified 78.4%

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

   if 1.3e-11 < y

   1. Initial program 68.7%

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

      1. associate-*l* 68.7%

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

      2. +-commutative 68.7%

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

      3. +-commutative 68.7%

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

      4. +-commutative 68.7%

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

      5. associate-*l* 68.7%

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

      6. *-commutative 68.7%

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

      7. associate-*r/ 80.0%

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

      8. *-commutative 80.0%

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

      9. distribute-rgt1-in 78.2%

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

      10. fma-def 80.0%

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

      11. +-commutative 80.0%

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

      12. +-commutative 80.0%

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

      13. cube-unmult 80.0%

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

      14. +-commutative 80.0%

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

   3. Simplified 80.0%

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

      1. associate-*r/ 68.7%

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

      2. *-commutative 68.7%

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

      3. fma-udef 67.1%

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

      4. cube-mult 67.1%

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

      5. distribute-rgt1-in 68.7%

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

      6. *-commutative 68.7%

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

      7. +-commutative 68.7%

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

      8. associate-+r+ 68.7%

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

      9. frac-times 88.6%

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

      10. *-commutative 88.6%

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

      11. clear-num 88.6%

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

      12. associate-/r* 99.8%

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

      13. frac-times 98.7%

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

      14. *-un-lft-identity 98.7%

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

      15. associate-+r+ 98.7%

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

      16. +-commutative 98.7%

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

      17. associate-+l+ 98.7%

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

   5. Applied egg-rr 98.7%

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

   6. Taylor expanded in y around -inf 77.4%

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

      1. mul-1-neg 77.4%

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

      2. unsub-neg 77.4%

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

      3. neg-mul-1 77.4%

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

      4. +-commutative 77.4%

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

      5. unsub-neg 77.4%

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

      6. distribute-lft-in 77.4%

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

      7. metadata-eval 77.4%

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

      8. neg-mul-1 77.4%

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

      9. unsub-neg 77.4%

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

   8. Simplified 77.4%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-305}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{x}{\left(x + 1\right) + \frac{x}{y}}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \]

Alternative 8: 93.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(x + y\right) \cdot \frac{y + 1}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.25e-6)
   (/ (/ y (+ x y)) (+ x (+ y (+ y 1.0))))
   (/ (/ x (+ x y)) (* (+ x y) (/ (+ y 1.0) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.25e-6) {
		tmp = (y / (x + y)) / (x + (y + (y + 1.0)));
	} else {
		tmp = (x / (x + y)) / ((x + y) * ((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 <= (-2.25d-6)) then
        tmp = (y / (x + y)) / (x + (y + (y + 1.0d0)))
    else
        tmp = (x / (x + y)) / ((x + y) * ((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 <= -2.25e-6) {
		tmp = (y / (x + y)) / (x + (y + (y + 1.0)));
	} else {
		tmp = (x / (x + y)) / ((x + y) * ((y + 1.0) / y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.25e-6:
		tmp = (y / (x + y)) / (x + (y + (y + 1.0)))
	else:
		tmp = (x / (x + y)) / ((x + y) * ((y + 1.0) / y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.25e-6)
		tmp = Float64(Float64(y / Float64(x + y)) / Float64(x + Float64(y + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(Float64(x + y) * Float64(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 <= -2.25e-6)
		tmp = (y / (x + y)) / (x + (y + (y + 1.0)));
	else
		tmp = (x / (x + y)) / ((x + y) * ((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, -2.25e-6], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.25000000000000006e-6

   1. Initial program 71.9%

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

      1. associate-*l* 71.9%

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

      2. +-commutative 71.9%

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

      3. +-commutative 71.9%

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

      4. +-commutative 71.9%

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

      5. associate-*l* 71.9%

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

      6. *-commutative 71.9%

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

      7. associate-*r/ 81.0%

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

      8. *-commutative 81.0%

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

      9. distribute-rgt1-in 36.3%

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

      10. fma-def 81.0%

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

      11. +-commutative 81.0%

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

      12. +-commutative 81.0%

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

      13. cube-unmult 81.0%

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

      14. +-commutative 81.0%

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

   3. Simplified 81.0%

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

      1. associate-*r/ 71.9%

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

      2. fma-udef 30.9%

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

      3. cube-mult 30.9%

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

      4. distribute-rgt1-in 71.9%

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

      5. *-commutative 71.9%

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

      6. +-commutative 71.9%

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

      7. associate-+r+ 71.9%

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

      8. frac-times 87.4%

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

      9. *-commutative 87.4%

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

      10. associate-/r* 99.8%

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

      11. clear-num 99.8%

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

      12. frac-times 99.8%

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

      13. *-un-lft-identity 99.8%

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

      14. associate-+r+ 99.8%

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

      15. +-commutative 99.8%

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

      16. associate-+l+ 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around -inf 74.5%

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

      1. mul-1-neg 74.5%

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

      2. unsub-neg 74.5%

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

      3. neg-mul-1 74.5%

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

      4. +-commutative 74.5%

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

      5. unsub-neg 74.5%

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

      6. distribute-lft-in 74.5%

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

      7. metadata-eval 74.5%

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

      8. neg-mul-1 74.5%

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

      9. unsub-neg 74.5%

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

   8. Simplified 74.5%

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

   if -2.25000000000000006e-6 < x

   1. Initial program 69.8%

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

      1. associate-*l* 69.8%

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

      2. +-commutative 69.8%

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

      3. +-commutative 69.8%

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

      4. +-commutative 69.8%

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

      5. associate-*l* 69.8%

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

      6. *-commutative 69.8%

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

      7. associate-*r/ 86.2%

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

      8. *-commutative 86.2%

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

      9. distribute-rgt1-in 75.3%

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

      10. fma-def 86.2%

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

      11. +-commutative 86.2%

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

      12. +-commutative 86.2%

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

      13. cube-unmult 86.2%

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

      14. +-commutative 86.2%

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

   3. Simplified 86.2%

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

      1. associate-*r/ 69.8%

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

      2. *-commutative 69.8%

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

      3. fma-udef 62.1%

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

      4. cube-mult 62.1%

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

      5. distribute-rgt1-in 69.8%

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

      6. *-commutative 69.8%

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

      7. +-commutative 69.8%

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

      8. associate-+r+ 69.8%

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

      9. frac-times 89.2%

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

      10. *-commutative 89.2%

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

      11. clear-num 89.2%

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

      12. associate-/r* 99.7%

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

      13. frac-times 99.1%

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

      14. *-un-lft-identity 99.1%

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

      15. associate-+r+ 99.1%

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

      16. +-commutative 99.1%

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

      17. associate-+l+ 99.1%

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

   5. Applied egg-rr 99.1%

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

   6. Taylor expanded in x around 0 82.0%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(x + y\right) \cdot \frac{y + 1}{y}}\\ \end{array} \]

Alternative 9: 84.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 22500000:\\ \;\;\;\;\frac{x}{y + \left(x + 1\right)} \cdot \frac{1}{y}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+151}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.4e-77)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 22500000.0)
     (* (/ x (+ y (+ x 1.0))) (/ 1.0 y))
     (if (<= y 3e+151) (/ x (* (+ x y) (+ x y))) (/ (/ x (+ x y)) (+ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.4e-77) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 22500000.0) {
		tmp = (x / (y + (x + 1.0))) * (1.0 / y);
	} else if (y <= 3e+151) {
		tmp = x / ((x + y) * (x + y));
	} else {
		tmp = (x / (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 <= 1.4d-77) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 22500000.0d0) then
        tmp = (x / (y + (x + 1.0d0))) * (1.0d0 / y)
    else if (y <= 3d+151) then
        tmp = x / ((x + y) * (x + y))
    else
        tmp = (x / (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 <= 1.4e-77) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 22500000.0) {
		tmp = (x / (y + (x + 1.0))) * (1.0 / y);
	} else if (y <= 3e+151) {
		tmp = x / ((x + y) * (x + y));
	} else {
		tmp = (x / (x + y)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.4e-77:
		tmp = (y / x) / (x + 1.0)
	elif y <= 22500000.0:
		tmp = (x / (y + (x + 1.0))) * (1.0 / y)
	elif y <= 3e+151:
		tmp = x / ((x + y) * (x + y))
	else:
		tmp = (x / (x + y)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.4e-77)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 22500000.0)
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) * Float64(1.0 / y));
	elseif (y <= 3e+151)
		tmp = Float64(x / Float64(Float64(x + y) * Float64(x + y)));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.4e-77)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 22500000.0)
		tmp = (x / (y + (x + 1.0))) * (1.0 / y);
	elseif (y <= 3e+151)
		tmp = x / ((x + y) * (x + y));
	else
		tmp = (x / (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, 1.4e-77], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 22500000.0], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3e+151], N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 1.4e-77

   1. Initial program 68.5%

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

      1. times-frac 87.8%

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

      2. +-commutative 87.8%

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

      3. associate-+l+ 87.8%

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

   3. Simplified 87.8%

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

   4. Taylor expanded in y around 0 60.4%

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

      1. associate-/r* 61.4%

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

      2. +-commutative 61.4%

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

   6. Simplified 61.4%

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

   if 1.4e-77 < y < 2.25e7

   1. Initial program 95.4%

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

      1. associate-*l* 95.4%

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

      2. +-commutative 95.4%

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

      3. +-commutative 95.4%

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

      4. +-commutative 95.4%

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

      5. *-commutative 95.4%

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

      6. associate-*l* 95.4%

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

      7. times-frac 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 42.9%

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

   if 2.25e7 < y < 2.9999999999999999e151

   1. Initial program 80.3%

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

      1. times-frac 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 82.5%

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

   if 2.9999999999999999e151 < y

   1. Initial program 55.2%

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

      1. associate-*l* 55.2%

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

      2. +-commutative 55.2%

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

      3. +-commutative 55.2%

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

      4. +-commutative 55.2%

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

      5. associate-*l* 55.2%

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

      6. *-commutative 55.2%

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

      7. associate-*r/ 78.7%

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

      8. *-commutative 78.7%

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

      9. distribute-rgt1-in 78.3%

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

      10. fma-def 78.7%

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

      11. +-commutative 78.7%

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

      12. +-commutative 78.7%

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

      13. cube-unmult 78.7%

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

      14. +-commutative 78.7%

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

   3. Simplified 78.7%

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

      1. associate-*r/ 55.2%

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

      2. *-commutative 55.2%

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

      3. fma-udef 55.2%

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

      4. cube-mult 55.2%

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

      5. distribute-rgt1-in 55.2%

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

      6. *-commutative 55.2%

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

      7. +-commutative 55.2%

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

      8. associate-+r+ 55.2%

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

      9. frac-times 78.7%

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

      10. associate-*l/ 78.7%

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

      11. associate-/r* 100.0%

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

      12. clear-num 100.0%

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

      13. un-div-inv 100.0%

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

      14. associate-+r+ 100.0%

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

      15. +-commutative 100.0%

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

      16. associate-+l+ 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around inf 84.3%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 22500000:\\ \;\;\;\;\frac{x}{y + \left(x + 1\right)} \cdot \frac{1}{y}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+151}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y}\\ \end{array} \]

Alternative 10: 82.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.55 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + \left(x + 1\right)} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.55e-74)
   (/ (/ y x) (+ x 1.0))
   (* (/ x (+ y (+ x 1.0))) (/ 1.0 y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.55e-74) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + (x + 1.0))) * (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 <= 2.55d-74) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (y + (x + 1.0d0))) * (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 <= 2.55e-74) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + (x + 1.0))) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.55e-74:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (y + (x + 1.0))) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.55e-74)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) * 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 <= 2.55e-74)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (y + (x + 1.0))) * (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, 2.55e-74], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.5499999999999998e-74

   1. Initial program 68.5%

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

      1. times-frac 87.8%

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

      2. +-commutative 87.8%

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

      3. associate-+l+ 87.8%

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

   3. Simplified 87.8%

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

   4. Taylor expanded in y around 0 60.4%

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

      1. associate-/r* 61.4%

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

      2. +-commutative 61.4%

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

   6. Simplified 61.4%

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

   if 2.5499999999999998e-74 < y

   1. Initial program 74.1%

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

      1. associate-*l* 74.1%

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

      2. +-commutative 74.1%

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

      3. +-commutative 74.1%

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

      4. +-commutative 74.1%

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

      5. *-commutative 74.1%

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

      6. associate-*l* 74.1%

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

      7. times-frac 91.0%

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

      8. +-commutative 91.0%

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

      9. +-commutative 91.0%

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

      10. associate-+l+ 91.0%

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

   3. Simplified 91.0%

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

   4. Taylor expanded in y around inf 68.5%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.55 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + \left(x + 1\right)} \cdot \frac{1}{y}\\ \end{array} \]

Alternative 11: 80.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{y}}\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{-100}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

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 (/ x y)))
   (if (<= x -2.15e-100) (- (/ 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 * (x / y));
	} else if (x <= -2.15e-100) {
		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 * (x / y))
    else if (x <= (-2.15d-100)) 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 * (x / y));
	} else if (x <= -2.15e-100) {
		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 * (x / y))
	elif x <= -2.15e-100:
		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(1.0 / Float64(x * Float64(x / y)));
	elseif (x <= -2.15e-100)
		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 * (x / y));
	elseif (x <= -2.15e-100)
		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[(1.0 / N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.15e-100], 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 71.0%

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

      1. times-frac 86.9%

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

      2. +-commutative 86.9%

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

      3. associate-+l+ 86.9%

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

   3. Simplified 86.9%

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

   4. Taylor expanded in x around inf 76.5%

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

      1. *-commutative 76.5%

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

      2. clear-num 76.5%

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

      3. associate-+r+ 76.5%

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

      4. +-commutative 76.5%

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

      5. associate-+r+ 76.5%

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

      6. frac-times 76.4%

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

      7. metadata-eval 76.4%

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

   6. Applied egg-rr 76.4%

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

   7. Taylor expanded in x around inf 73.7%

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

   if -1 < x < -2.14999999999999999e-100

   1. Initial program 84.8%

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

      1. times-frac 99.8%

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

      2. +-commutative 99.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 99.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around 0 57.1%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 57.1%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 57.1%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   6. Simplified 57.1%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   7. Taylor expanded in x around 0 57.1%

      \[\leadsto \color{blue}{-1 \cdot y + \frac{y}{x}} \]
   8. Step-by-step derivation

      1. neg-mul-1 57.1%

         \[\leadsto \color{blue}{\left(-y\right)} + \frac{y}{x} \]

      2. +-commutative 57.1%

         \[\leadsto \color{blue}{\frac{y}{x} + \left(-y\right)} \]

      3. unsub-neg 57.1%

         \[\leadsto \color{blue}{\frac{y}{x} - y} \]

   9. Simplified 57.1%

      \[\leadsto \color{blue}{\frac{y}{x} - y} \]

   if -2.14999999999999999e-100 < x

   1. Initial program 68.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 87.8%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 87.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 87.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 87.8%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around 0 55.2%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 55.2%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   6. Simplified 55.2%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{y}}\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{-100}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]

Alternative 12: 82.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.28 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.28e-73) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ x y)) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.28e-73) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.28d-73) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (x + y)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.28e-73) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.28e-73:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (x + y)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.28e-73)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.28e-73)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (x + y)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.28e-73], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.2799999999999999e-73

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 87.8%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 87.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 87.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 87.8%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around 0 60.4%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 61.4%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 61.4%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   6. Simplified 61.4%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if 1.2799999999999999e-73 < y

   1. Initial program 74.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 74.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 74.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 74.1%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 74.1%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 74.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 74.1%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. associate-*r/ 84.1%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 84.1%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 79.0%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 84.1%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 84.1%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 84.1%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 84.1%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 84.1%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 84.1%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 74.1%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. *-commutative 74.1%

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      3. fma-udef 69.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      4. cube-mult 69.1%

         \[\leadsto \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. distribute-rgt1-in 74.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      6. *-commutative 74.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. +-commutative 74.1%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      8. associate-+r+ 74.1%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      9. frac-times 90.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

      10. *-commutative 90.9%

          \[\leadsto \color{blue}{\frac{y}{y + \left(x + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      11. clear-num 90.9%

          \[\leadsto \color{blue}{\frac{1}{\frac{y + \left(x + 1\right)}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      12. associate-/r* 99.8%

          \[\leadsto \frac{1}{\frac{y + \left(x + 1\right)}{y}} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      13. frac-times 98.9%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{y + \left(x + 1\right)}{y} \cdot \left(x + y\right)}} \]

      14. *-un-lft-identity 98.9%

          \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{\frac{y + \left(x + 1\right)}{y} \cdot \left(x + y\right)} \]

      15. associate-+r+ 98.9%

          \[\leadsto \frac{\frac{x}{x + y}}{\frac{\color{blue}{\left(y + x\right) + 1}}{y} \cdot \left(x + y\right)} \]

      16. +-commutative 98.9%

          \[\leadsto \frac{\frac{x}{x + y}}{\frac{\color{blue}{\left(x + y\right)} + 1}{y} \cdot \left(x + y\right)} \]

      17. associate-+l+ 98.9%

          \[\leadsto \frac{\frac{x}{x + y}}{\frac{\color{blue}{x + \left(y + 1\right)}}{y} \cdot \left(x + y\right)} \]

   5. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}} \]

   6. Taylor expanded in x around 0 68.5%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{1 + y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.28 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]

Alternative 13: 67.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{y}}\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-134}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ 1.0 (* x (/ x y)))
   (if (<= x -5.4e-134) (- (/ y x) y) (/ x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0 / (x * (x / y));
	} else if (x <= -5.4e-134) {
		tmp = (y / x) - y;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 1.0d0 / (x * (x / y))
    else if (x <= (-5.4d-134)) then
        tmp = (y / x) - y
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0 / (x * (x / y));
	} else if (x <= -5.4e-134) {
		tmp = (y / x) - y;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0 / (x * (x / y))
	elif x <= -5.4e-134:
		tmp = (y / x) - y
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(1.0 / Float64(x * Float64(x / y)));
	elseif (x <= -5.4e-134)
		tmp = Float64(Float64(y / x) - y);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 1.0 / (x * (x / y));
	elseif (x <= -5.4e-134)
		tmp = (y / x) - y;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.0], N[(1.0 / N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.4e-134], N[(N[(y / x), $MachinePrecision] - y), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 71.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 86.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 86.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 86.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 86.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around inf 76.5%

      \[\leadsto \color{blue}{\frac{1}{x}} \cdot \frac{y}{y + \left(x + 1\right)} \]
   5. Step-by-step derivation

      1. *-commutative 76.5%

         \[\leadsto \color{blue}{\frac{y}{y + \left(x + 1\right)} \cdot \frac{1}{x}} \]

      2. clear-num 76.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{y + \left(x + 1\right)}{y}}} \cdot \frac{1}{x} \]

      3. associate-+r+ 76.5%

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(y + x\right) + 1}}{y}} \cdot \frac{1}{x} \]

      4. +-commutative 76.5%

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(x + y\right)} + 1}{y}} \cdot \frac{1}{x} \]

      5. associate-+r+ 76.5%

         \[\leadsto \frac{1}{\frac{\color{blue}{x + \left(y + 1\right)}}{y}} \cdot \frac{1}{x} \]

      6. frac-times 76.4%

         \[\leadsto \color{blue}{\frac{1 \cdot 1}{\frac{x + \left(y + 1\right)}{y} \cdot x}} \]

      7. metadata-eval 76.4%

         \[\leadsto \frac{\color{blue}{1}}{\frac{x + \left(y + 1\right)}{y} \cdot x} \]

   6. Applied egg-rr 76.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{x + \left(y + 1\right)}{y} \cdot x}} \]

   7. Taylor expanded in x around inf 73.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{x}{y}} \cdot x} \]

   if -1 < x < -5.3999999999999996e-134

   1. Initial program 78.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.7%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 99.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 99.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around 0 51.3%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 51.3%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 51.3%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   6. Simplified 51.3%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   7. Taylor expanded in x around 0 51.3%

      \[\leadsto \color{blue}{-1 \cdot y + \frac{y}{x}} \]
   8. Step-by-step derivation

      1. neg-mul-1 51.3%

         \[\leadsto \color{blue}{\left(-y\right)} + \frac{y}{x} \]

      2. +-commutative 51.3%

         \[\leadsto \color{blue}{\frac{y}{x} + \left(-y\right)} \]

      3. unsub-neg 51.3%

         \[\leadsto \color{blue}{\frac{y}{x} - y} \]

   9. Simplified 51.3%

      \[\leadsto \color{blue}{\frac{y}{x} - y} \]

   if -5.3999999999999996e-134 < x

   1. Initial program 68.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 68.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 68.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 68.1%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 68.1%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 68.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 68.1%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. associate-*r/ 83.3%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 83.3%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 72.6%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 83.3%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 83.3%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 83.3%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 83.3%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 83.3%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 83.3%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 68.2%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. *-commutative 68.2%

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      3. fma-udef 61.3%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      4. cube-mult 61.3%

         \[\leadsto \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. distribute-rgt1-in 68.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      6. *-commutative 68.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. +-commutative 68.1%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      8. associate-+r+ 68.1%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      9. frac-times 87.0%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

      10. *-commutative 87.0%

          \[\leadsto \color{blue}{\frac{y}{y + \left(x + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      11. clear-num 87.0%

          \[\leadsto \color{blue}{\frac{1}{\frac{y + \left(x + 1\right)}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      12. associate-/r* 99.7%

          \[\leadsto \frac{1}{\frac{y + \left(x + 1\right)}{y}} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      13. frac-times 99.0%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{y + \left(x + 1\right)}{y} \cdot \left(x + y\right)}} \]

      14. *-un-lft-identity 99.0%

          \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{\frac{y + \left(x + 1\right)}{y} \cdot \left(x + y\right)} \]

      15. associate-+r+ 99.0%

          \[\leadsto \frac{\frac{x}{x + y}}{\frac{\color{blue}{\left(y + x\right) + 1}}{y} \cdot \left(x + y\right)} \]

      16. +-commutative 99.0%

          \[\leadsto \frac{\frac{x}{x + y}}{\frac{\color{blue}{\left(x + y\right)} + 1}{y} \cdot \left(x + y\right)} \]

      17. associate-+l+ 99.0%

          \[\leadsto \frac{\frac{x}{x + y}}{\frac{\color{blue}{x + \left(y + 1\right)}}{y} \cdot \left(x + y\right)} \]

   5. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}} \]

   6. Taylor expanded in x around 0 78.5%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{\frac{1 + y}{y}} \cdot \left(x + y\right)} \]

   7. Taylor expanded in y around 0 57.9%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{1 + \left(x + \frac{x}{y}\right)}} \]
   8. Step-by-step derivation

      1. associate-+r+ 57.9%

         \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{\left(1 + x\right) + \frac{x}{y}}} \]

      2. +-commutative 57.9%

         \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{\left(x + 1\right)} + \frac{x}{y}} \]

   9. Simplified 57.9%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{\left(x + 1\right) + \frac{x}{y}}} \]

   10. Taylor expanded in x around 0 35.5%

       \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{y}}\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-134}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 14: 79.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4.6e-75) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.6e-75) {
		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 (y <= 4.6d-75) 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 (y <= 4.6e-75) {
		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 y <= 4.6e-75:
		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 (y <= 4.6e-75)
		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 (y <= 4.6e-75)
		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[y, 4.6e-75], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.6e-75

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 87.8%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 87.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 87.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 87.8%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around 0 60.4%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   if 4.6e-75 < y

   1. Initial program 74.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 90.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 90.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 90.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 90.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around 0 66.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 66.1%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   6. Simplified 66.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]

Alternative 15: 81.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{-73}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.2e-73) (/ y (* x (+ x 1.0))) (/ (/ x (+ y 1.0)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.2e-73) {
		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 (y <= 1.2d-73) 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 (y <= 1.2e-73) {
		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 y <= 1.2e-73:
		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 (y <= 1.2e-73)
		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 (y <= 1.2e-73)
		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[y, 1.2e-73], 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 2 regimes

2. if y < 1.20000000000000003e-73

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 87.8%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 87.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 87.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 87.8%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around 0 60.4%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   if 1.20000000000000003e-73 < y

   1. Initial program 74.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 90.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 90.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 90.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 90.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around 0 66.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 66.1%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   6. Simplified 66.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 66.1%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{y \cdot \left(y + 1\right)} \]

      2. times-frac 68.1%

         \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y + 1}} \]

   8. Applied egg-rr 68.1%

      \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y + 1}} \]
   9. Step-by-step derivation

      1. associate-*l/ 68.2%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y + 1}}{y}} \]

      2. +-commutative 68.2%

         \[\leadsto \frac{1 \cdot \frac{x}{\color{blue}{1 + y}}}{y} \]

      3. *-lft-identity 68.2%

         \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{y} \]

   10. Simplified 68.2%

       \[\leadsto \color{blue}{\frac{\frac{x}{1 + y}}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{-73}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]

Alternative 16: 82.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.7e-76) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ y 1.0)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.7e-76) {
		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 (y <= 1.7d-76) 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 (y <= 1.7e-76) {
		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 y <= 1.7e-76:
		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 (y <= 1.7e-76)
		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 (y <= 1.7e-76)
		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[y, 1.7e-76], 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 y < 1.7e-76

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 87.8%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 87.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 87.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 87.8%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around 0 60.4%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 61.4%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 61.4%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   6. Simplified 61.4%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if 1.7e-76 < y

   1. Initial program 74.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 90.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 90.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 90.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 90.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around 0 66.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 66.1%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   6. Simplified 66.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 66.1%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{y \cdot \left(y + 1\right)} \]

      2. times-frac 68.1%

         \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y + 1}} \]

   8. Applied egg-rr 68.1%

      \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y + 1}} \]
   9. Step-by-step derivation

      1. associate-*l/ 68.2%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y + 1}}{y}} \]

      2. +-commutative 68.2%

         \[\leadsto \frac{1 \cdot \frac{x}{\color{blue}{1 + y}}}{y} \]

      3. *-lft-identity 68.2%

         \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{y} \]

   10. Simplified 68.2%

       \[\leadsto \color{blue}{\frac{\frac{x}{1 + y}}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]

Alternative 17: 43.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{-87}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 1.45e-87) (/ y x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.45e-87) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.45d-87) then
        tmp = y / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.45e-87) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.45e-87:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.45e-87)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.45e-87)
		tmp = y / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.45e-87], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.45e-87

   1. Initial program 68.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 87.7%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 87.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 87.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 87.7%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around 0 60.5%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 61.5%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 61.5%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   6. Simplified 61.5%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   7. Taylor expanded in x around 0 40.4%

      \[\leadsto \color{blue}{\frac{y}{x}} \]

   if 1.45e-87 < y

   1. Initial program 74.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 74.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 74.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 74.6%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 74.6%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 74.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 74.6%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. associate-*r/ 84.5%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 84.5%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 79.5%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 84.5%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 84.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 84.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 84.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 84.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 84.5%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 74.6%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. *-commutative 74.6%

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      3. fma-udef 69.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      4. cube-mult 69.8%

         \[\leadsto \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. distribute-rgt1-in 74.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      6. *-commutative 74.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. +-commutative 74.6%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      8. associate-+r+ 74.6%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      9. frac-times 91.1%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

      10. *-commutative 91.1%

          \[\leadsto \color{blue}{\frac{y}{y + \left(x + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      11. clear-num 91.1%

          \[\leadsto \color{blue}{\frac{1}{\frac{y + \left(x + 1\right)}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      12. associate-/r* 99.8%

          \[\leadsto \frac{1}{\frac{y + \left(x + 1\right)}{y}} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      13. frac-times 98.9%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{y + \left(x + 1\right)}{y} \cdot \left(x + y\right)}} \]

      14. *-un-lft-identity 98.9%

          \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{\frac{y + \left(x + 1\right)}{y} \cdot \left(x + y\right)} \]

      15. associate-+r+ 98.9%

          \[\leadsto \frac{\frac{x}{x + y}}{\frac{\color{blue}{\left(y + x\right) + 1}}{y} \cdot \left(x + y\right)} \]

      16. +-commutative 98.9%

          \[\leadsto \frac{\frac{x}{x + y}}{\frac{\color{blue}{\left(x + y\right)} + 1}{y} \cdot \left(x + y\right)} \]

      17. associate-+l+ 98.9%

          \[\leadsto \frac{\frac{x}{x + y}}{\frac{\color{blue}{x + \left(y + 1\right)}}{y} \cdot \left(x + y\right)} \]

   5. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}} \]

   6. Taylor expanded in x around 0 71.2%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{\frac{1 + y}{y}} \cdot \left(x + y\right)} \]

   7. Taylor expanded in y around 0 36.6%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{1 + \left(x + \frac{x}{y}\right)}} \]
   8. Step-by-step derivation

      1. associate-+r+ 36.6%

         \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{\left(1 + x\right) + \frac{x}{y}}} \]

      2. +-commutative 36.6%

         \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{\left(x + 1\right)} + \frac{x}{y}} \]

   9. Simplified 36.6%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{\left(x + 1\right) + \frac{x}{y}}} \]

   10. Taylor expanded in x around 0 32.7%

       \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{-87}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 18: 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 70.3%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. times-frac 88.8%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. +-commutative 88.8%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   3. associate-+l+ 88.8%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

3. Simplified 88.8%

   \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

4. Taylor expanded in x around inf 52.7%

   \[\leadsto \color{blue}{\frac{1}{x}} \cdot \frac{y}{y + \left(x + 1\right)} \]

5. Taylor expanded in y around inf 4.2%

   \[\leadsto \color{blue}{\frac{1}{x}} \]

6. Final simplification 4.2%

   \[\leadsto \frac{1}{x} \]

Alternative 19: 26.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{x}{y} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ x y))

C

assert(x < y);
double code(double x, double y) {
	return x / y;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / y
end function

Java

assert x < y;
public static double code(double x, double y) {
	return x / y;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return x / y

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x / y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x / y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 70.3%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 70.3%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

   2. +-commutative 70.3%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

   3. +-commutative 70.3%

      \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

   4. +-commutative 70.3%

      \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

   5. associate-*l* 70.3%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

   6. *-commutative 70.3%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

   7. associate-*r/ 85.0%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

   8. *-commutative 85.0%

      \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   9. distribute-rgt1-in 66.3%

      \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   10. fma-def 85.0%

       \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

   11. +-commutative 85.0%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   12. +-commutative 85.0%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   13. cube-unmult 85.0%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

   14. +-commutative 85.0%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

3. Simplified 85.0%

   \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
4. Step-by-step derivation

   1. 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. *-commutative 70.3%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

   3. fma-udef 54.9%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   4. cube-mult 54.9%

      \[\leadsto \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   5. distribute-rgt1-in 70.3%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   6. *-commutative 70.3%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   7. +-commutative 70.3%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

   8. associate-+r+ 70.3%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

   9. frac-times 88.8%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   10. *-commutative 88.8%

       \[\leadsto \color{blue}{\frac{y}{y + \left(x + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   11. clear-num 88.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{y + \left(x + 1\right)}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

   12. associate-/r* 99.7%

       \[\leadsto \frac{1}{\frac{y + \left(x + 1\right)}{y}} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

   13. frac-times 99.2%

       \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{y + \left(x + 1\right)}{y} \cdot \left(x + y\right)}} \]

   14. *-un-lft-identity 99.2%

       \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{\frac{y + \left(x + 1\right)}{y} \cdot \left(x + y\right)} \]

   15. associate-+r+ 99.2%

       \[\leadsto \frac{\frac{x}{x + y}}{\frac{\color{blue}{\left(y + x\right) + 1}}{y} \cdot \left(x + y\right)} \]

   16. +-commutative 99.2%

       \[\leadsto \frac{\frac{x}{x + y}}{\frac{\color{blue}{\left(x + y\right)} + 1}{y} \cdot \left(x + y\right)} \]

   17. associate-+l+ 99.2%

       \[\leadsto \frac{\frac{x}{x + y}}{\frac{\color{blue}{x + \left(y + 1\right)}}{y} \cdot \left(x + y\right)} \]

5. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}} \]

6. Taylor expanded in x around 0 76.8%

   \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{\frac{1 + y}{y}} \cdot \left(x + y\right)} \]

7. Taylor expanded in y around 0 55.3%

   \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{1 + \left(x + \frac{x}{y}\right)}} \]
8. Step-by-step derivation

   1. associate-+r+ 55.3%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{\left(1 + x\right) + \frac{x}{y}}} \]

   2. +-commutative 55.3%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{\left(x + 1\right)} + \frac{x}{y}} \]

9. Simplified 55.3%

   \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{\left(x + 1\right) + \frac{x}{y}}} \]

10. Taylor expanded in x around 0 27.1%

    \[\leadsto \color{blue}{\frac{x}{y}} \]

11. Final simplification 27.1%

    \[\leadsto \frac{x}{y} \]

Developer target: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x)))))

C

double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x / ((y + 1.0d0) + x)) / (y + x)) / (1.0d0 / (y / (y + x)))
end function

Java

public static double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Python

def code(x, y):
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)))

Julia

function code(x, y)
	return Float64(Float64(Float64(x / Float64(Float64(y + 1.0) + x)) / Float64(y + x)) / Float64(1.0 / Float64(y / Float64(y + x))))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
end

Wolfram

code[x_, y_] := N[(N[(N[(x / N[(N[(y + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023318 
(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))))