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

Percentage Accurate: 68.9% → 99.3%

Time: 16.8s

Alternatives: 26

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: 68.9% 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: 99.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.1%

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

   1. associate-*l* 72.1%

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

   2. +-commutative 72.1%

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

   3. +-commutative 72.1%

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

   4. +-commutative 72.1%

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

   5. associate-*l* 72.1%

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

   6. *-commutative 72.1%

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

   7. associate-*r/ 81.4%

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

   8. *-commutative 81.4%

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

   9. distribute-rgt1-in 62.4%

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

   10. fma-def 81.4%

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

   11. +-commutative 81.4%

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

   12. +-commutative 81.4%

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

   13. cube-unmult 81.4%

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

   14. +-commutative 81.4%

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

3. Simplified 81.4%

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

   1. associate-*r/ 72.2%

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

   2. *-commutative 72.2%

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

   3. fma-udef 55.3%

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

   4. cube-mult 55.3%

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

   5. distribute-rgt1-in 72.1%

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

   6. *-commutative 72.1%

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

   7. +-commutative 72.1%

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

   8. associate-+r+ 72.1%

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

   9. frac-times 88.2%

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

   10. *-commutative 88.2%

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

   11. clear-num 88.2%

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

   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 99.7%

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

   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 y around -inf 95.1%

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

   1. associate-+r+ 95.1%

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

   2. mul-1-neg 95.1%

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

   3. unsub-neg 95.1%

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

   4. neg-mul-1 95.1%

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

   5. distribute-lft-in 95.1%

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

   6. metadata-eval 95.1%

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

   7. neg-mul-1 95.1%

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

   8. +-commutative 95.1%

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

   9. +-commutative 95.1%

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

   10. unsub-neg 95.1%

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

   11. +-commutative 95.1%

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

   12. unsub-neg 95.1%

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

   13. +-commutative 95.1%

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

   14. *-commutative 95.1%

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

   15. associate-*r/ 99.8%

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

   16. *-commutative 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 95.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{+146}:\\ \;\;\;\;\frac{1}{x \cdot \frac{t_0}{y}}\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{\left(x + y\right) \cdot \frac{y + 1}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -1.9e+146)
     (/ 1.0 (* x (/ t_0 y)))
     (if (<= x -4.5e-17)
       (* (/ x (* (+ x y) (+ x y))) (/ y t_0))
       (/ (/ y (+ x y)) (* (+ x y) (/ (+ y 1.0) x)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.9e+146) {
		tmp = 1.0 / (x * (t_0 / y));
	} else if (x <= -4.5e-17) {
		tmp = (x / ((x + y) * (x + y))) * (y / t_0);
	} else {
		tmp = (y / (x + y)) / ((x + y) * ((y + 1.0) / x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (x <= (-1.9d+146)) then
        tmp = 1.0d0 / (x * (t_0 / y))
    else if (x <= (-4.5d-17)) then
        tmp = (x / ((x + y) * (x + y))) * (y / t_0)
    else
        tmp = (y / (x + y)) / ((x + y) * ((y + 1.0d0) / x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.9e+146) {
		tmp = 1.0 / (x * (t_0 / y));
	} else if (x <= -4.5e-17) {
		tmp = (x / ((x + y) * (x + y))) * (y / t_0);
	} else {
		tmp = (y / (x + y)) / ((x + y) * ((y + 1.0) / x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -1.9e+146:
		tmp = 1.0 / (x * (t_0 / y))
	elif x <= -4.5e-17:
		tmp = (x / ((x + y) * (x + y))) * (y / t_0)
	else:
		tmp = (y / (x + y)) / ((x + y) * ((y + 1.0) / x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1.9e+146)
		tmp = Float64(1.0 / Float64(x * Float64(t_0 / y)));
	elseif (x <= -4.5e-17)
		tmp = Float64(Float64(x / Float64(Float64(x + y) * Float64(x + y))) * Float64(y / t_0));
	else
		tmp = Float64(Float64(y / Float64(x + y)) / Float64(Float64(x + y) * Float64(Float64(y + 1.0) / x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -1.9e+146)
		tmp = 1.0 / (x * (t_0 / y));
	elseif (x <= -4.5e-17)
		tmp = (x / ((x + y) * (x + y))) * (y / t_0);
	else
		tmp = (y / (x + y)) / ((x + y) * ((y + 1.0) / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.9e+146], N[(1.0 / N[(x * N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.5e-17], N[(N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.8999999999999999e146

   1. Initial program 59.8%

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

      1. times-frac 79.9%

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

      2. +-commutative 79.9%

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

      3. associate-+l+ 79.9%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in x around inf 91.3%

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

      1. *-commutative 91.3%

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

      2. clear-num 91.3%

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

      3. associate-+r+ 91.3%

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

      4. +-commutative 91.3%

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

      5. associate-+r+ 91.3%

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

      6. frac-times 91.3%

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

      7. metadata-eval 91.3%

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

      8. associate-+r+ 91.3%

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

      9. +-commutative 91.3%

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

      10. associate-+r+ 91.3%

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

   6. Applied egg-rr 91.3%

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

   if -1.8999999999999999e146 < x < -4.49999999999999978e-17

   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 93.5%

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

      2. +-commutative 93.5%

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

      3. associate-+l+ 93.5%

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

   3. Simplified 93.5%

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

   if -4.49999999999999978e-17 < x

   1. Initial program 74.8%

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

      1. associate-*l* 74.8%

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

      2. +-commutative 74.8%

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

      3. +-commutative 74.8%

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

      4. +-commutative 74.8%

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

      5. *-commutative 74.8%

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

      6. associate-*l* 74.8%

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

      7. times-frac 88.8%

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

      8. +-commutative 88.8%

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

      9. +-commutative 88.8%

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

      10. associate-+l+ 88.8%

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

   3. Simplified 88.8%

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

   4. Taylor expanded in x around 0 83.3%

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

      1. +-commutative 83.3%

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

   6. Simplified 83.3%

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

      1. *-commutative 83.3%

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

      2. clear-num 83.2%

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

      3. associate-/r* 91.4%

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

      4. frac-times 85.8%

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

      5. *-un-lft-identity 85.8%

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

      6. +-commutative 85.8%

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

      7. +-commutative 85.8%

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

   8. Applied egg-rr 85.8%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+146}:\\ \;\;\;\;\frac{1}{x \cdot \frac{y + \left(x + 1\right)}{y}}\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{\left(x + y\right) \cdot \frac{y + 1}{x}}\\ \end{array} \]

Alternative 3: 95.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -8.8 \cdot 10^{+146}:\\ \;\;\;\;\frac{1}{x \cdot \frac{t_0}{y}}\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{t_0} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{\left(x + y\right) \cdot \frac{y + 1}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -8.8e+146)
     (/ 1.0 (* x (/ t_0 y)))
     (if (<= x -3.4e-17)
       (* (/ x t_0) (/ y (* (+ x y) (+ x y))))
       (/ (/ y (+ x y)) (* (+ x y) (/ (+ y 1.0) x)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -8.8e+146) {
		tmp = 1.0 / (x * (t_0 / y));
	} else if (x <= -3.4e-17) {
		tmp = (x / t_0) * (y / ((x + y) * (x + y)));
	} else {
		tmp = (y / (x + y)) / ((x + y) * ((y + 1.0) / x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (x <= (-8.8d+146)) then
        tmp = 1.0d0 / (x * (t_0 / y))
    else if (x <= (-3.4d-17)) then
        tmp = (x / t_0) * (y / ((x + y) * (x + y)))
    else
        tmp = (y / (x + y)) / ((x + y) * ((y + 1.0d0) / x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -8.8e+146) {
		tmp = 1.0 / (x * (t_0 / y));
	} else if (x <= -3.4e-17) {
		tmp = (x / t_0) * (y / ((x + y) * (x + y)));
	} else {
		tmp = (y / (x + y)) / ((x + y) * ((y + 1.0) / x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -8.8e+146:
		tmp = 1.0 / (x * (t_0 / y))
	elif x <= -3.4e-17:
		tmp = (x / t_0) * (y / ((x + y) * (x + y)))
	else:
		tmp = (y / (x + y)) / ((x + y) * ((y + 1.0) / x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -8.8e+146)
		tmp = Float64(1.0 / Float64(x * Float64(t_0 / y)));
	elseif (x <= -3.4e-17)
		tmp = Float64(Float64(x / t_0) * Float64(y / Float64(Float64(x + y) * Float64(x + y))));
	else
		tmp = Float64(Float64(y / Float64(x + y)) / Float64(Float64(x + y) * Float64(Float64(y + 1.0) / x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -8.8e+146)
		tmp = 1.0 / (x * (t_0 / y));
	elseif (x <= -3.4e-17)
		tmp = (x / t_0) * (y / ((x + y) * (x + y)));
	else
		tmp = (y / (x + y)) / ((x + y) * ((y + 1.0) / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.8e+146], N[(1.0 / N[(x * N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.4e-17], N[(N[(x / t$95$0), $MachinePrecision] * N[(y / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.7999999999999992e146

   1. Initial program 59.8%

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

      1. times-frac 79.9%

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

      2. +-commutative 79.9%

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

      3. associate-+l+ 79.9%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in x around inf 91.3%

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

      1. *-commutative 91.3%

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

      2. clear-num 91.3%

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

      3. associate-+r+ 91.3%

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

      4. +-commutative 91.3%

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

      5. associate-+r+ 91.3%

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

      6. frac-times 91.3%

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

      7. metadata-eval 91.3%

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

      8. associate-+r+ 91.3%

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

      9. +-commutative 91.3%

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

      10. associate-+r+ 91.3%

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

   6. Applied egg-rr 91.3%

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

   if -8.7999999999999992e146 < x < -3.3999999999999998e-17

   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. associate-*l* 68.2%

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

      2. +-commutative 68.2%

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

      3. +-commutative 68.2%

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

      4. +-commutative 68.2%

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

      5. *-commutative 68.2%

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

      6. associate-*l* 68.2%

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

      7. times-frac 93.5%

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

      8. +-commutative 93.5%

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

      9. +-commutative 93.5%

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

      10. associate-+l+ 93.5%

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

   3. Simplified 93.5%

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

   if -3.3999999999999998e-17 < x

   1. Initial program 74.8%

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

      1. associate-*l* 74.8%

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

      2. +-commutative 74.8%

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

      3. +-commutative 74.8%

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

      4. +-commutative 74.8%

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

      5. *-commutative 74.8%

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

      6. associate-*l* 74.8%

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

      7. times-frac 88.8%

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

      8. +-commutative 88.8%

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

      9. +-commutative 88.8%

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

      10. associate-+l+ 88.8%

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

   3. Simplified 88.8%

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

   4. Taylor expanded in x around 0 83.3%

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

      1. +-commutative 83.3%

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

   6. Simplified 83.3%

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

      1. *-commutative 83.3%

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

      2. clear-num 83.2%

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

      3. associate-/r* 91.4%

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

      4. frac-times 85.8%

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

      5. *-un-lft-identity 85.8%

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

      6. +-commutative 85.8%

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

      7. +-commutative 85.8%

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

   8. Applied egg-rr 85.8%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{+146}:\\ \;\;\;\;\frac{1}{x \cdot \frac{y + \left(x + 1\right)}{y}}\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{y + \left(x + 1\right)} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{\left(x + y\right) \cdot \frac{y + 1}{x}}\\ \end{array} \]

Alternative 4: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.1%

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

   1. associate-*l* 72.1%

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

   2. +-commutative 72.1%

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

   3. +-commutative 72.1%

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

   4. +-commutative 72.1%

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

   5. *-commutative 72.1%

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

   6. associate-*l* 72.1%

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

   7. times-frac 88.3%

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

   8. +-commutative 88.3%

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

   9. +-commutative 88.3%

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

   10. associate-+l+ 88.3%

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

3. Simplified 88.3%

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

   1. associate-/r* 64.5%

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

   2. div-inv 64.5%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

Alternative 5: 88.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{1}{x + y}\\ \mathbf{if}\;y \leq -2.1 \cdot 10^{-293}:\\ \;\;\;\;\frac{1}{\frac{x}{y} \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \left(\frac{y}{x + y} \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t_0}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ x y))))
   (if (<= y -2.1e-293)
     (/ 1.0 (* (/ x y) (+ x 1.0)))
     (if (<= y 5.2e-15) (* x (* (/ y (+ x y)) t_0)) (/ (* x t_0) (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = 1.0 / (x + y);
	double tmp;
	if (y <= -2.1e-293) {
		tmp = 1.0 / ((x / y) * (x + 1.0));
	} else if (y <= 5.2e-15) {
		tmp = x * ((y / (x + y)) * t_0);
	} else {
		tmp = (x * 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 = 1.0d0 / (x + y)
    if (y <= (-2.1d-293)) then
        tmp = 1.0d0 / ((x / y) * (x + 1.0d0))
    else if (y <= 5.2d-15) then
        tmp = x * ((y / (x + y)) * t_0)
    else
        tmp = (x * 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 = 1.0 / (x + y);
	double tmp;
	if (y <= -2.1e-293) {
		tmp = 1.0 / ((x / y) * (x + 1.0));
	} else if (y <= 5.2e-15) {
		tmp = x * ((y / (x + y)) * t_0);
	} else {
		tmp = (x * t_0) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = 1.0 / (x + y)
	tmp = 0
	if y <= -2.1e-293:
		tmp = 1.0 / ((x / y) * (x + 1.0))
	elif y <= 5.2e-15:
		tmp = x * ((y / (x + y)) * t_0)
	else:
		tmp = (x * t_0) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(1.0 / Float64(x + y))
	tmp = 0.0
	if (y <= -2.1e-293)
		tmp = Float64(1.0 / Float64(Float64(x / y) * Float64(x + 1.0)));
	elseif (y <= 5.2e-15)
		tmp = Float64(x * Float64(Float64(y / Float64(x + y)) * t_0));
	else
		tmp = Float64(Float64(x * 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 = 1.0 / (x + y);
	tmp = 0.0;
	if (y <= -2.1e-293)
		tmp = 1.0 / ((x / y) * (x + 1.0));
	elseif (y <= 5.2e-15)
		tmp = x * ((y / (x + y)) * t_0);
	else
		tmp = (x * 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[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.1e-293], N[(1.0 / N[(N[(x / y), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.2e-15], N[(x * N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x * t$95$0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.10000000000000005e-293

   1. Initial program 70.6%

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

      1. times-frac 88.1%

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

      2. +-commutative 88.1%

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

      3. associate-+l+ 88.1%

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

   3. Simplified 88.1%

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

   4. Taylor expanded in y around 0 43.8%

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

      1. associate-/r* 49.3%

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

      2. +-commutative 49.3%

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

   6. Simplified 49.3%

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

      1. clear-num 49.3%

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

      2. inv-pow 49.3%

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

      3. div-inv 49.3%

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

      4. clear-num 49.3%

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

   8. Applied egg-rr 49.3%

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

      1. unpow-1 49.3%

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

      2. *-commutative 49.3%

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

   10. Simplified 49.3%

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

   if -2.10000000000000005e-293 < y < 5.20000000000000009e-15

   1. Initial program 73.4%

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

      1. associate-*l* 73.4%

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

      2. +-commutative 73.4%

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

      3. +-commutative 73.4%

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

      4. +-commutative 73.4%

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

      5. *-commutative 73.4%

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

      6. associate-*l* 73.4%

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

      7. times-frac 81.8%

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

      8. +-commutative 81.8%

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

      9. +-commutative 81.8%

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

      10. associate-+l+ 81.8%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in x around 0 65.2%

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

      1. +-commutative 65.2%

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

   6. Simplified 65.2%

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

   7. Taylor expanded in y around 0 65.2%

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

      1. associate-/r* 83.2%

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

      2. div-inv 83.2%

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

   9. Applied egg-rr 83.2%

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

   if 5.20000000000000009e-15 < y

   1. Initial program 73.7%

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

      1. 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. associate-*l* 73.7%

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

      6. *-commutative 73.7%

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

      7. associate-*r/ 82.3%

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

      8. *-commutative 82.3%

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

      9. distribute-rgt1-in 79.6%

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

      10. fma-def 82.3%

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

      11. +-commutative 82.3%

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

      12. +-commutative 82.3%

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

      13. cube-unmult 82.4%

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

      14. +-commutative 82.4%

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

   3. Simplified 82.4%

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

      1. associate-*r/ 73.7%

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

      2. *-commutative 73.7%

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

      3. fma-udef 73.6%

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

      4. cube-mult 73.6%

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

      5. distribute-rgt1-in 73.7%

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

      6. *-commutative 73.7%

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

      7. +-commutative 73.7%

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

      8. associate-+r+ 73.7%

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

      9. frac-times 93.6%

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

      10. *-commutative 93.6%

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

      11. clear-num 93.6%

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

      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 99.8%

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

      14. *-un-lft-identity 99.8%

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

      15. associate-+r+ 99.8%

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

      16. +-commutative 99.8%

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

      17. associate-+l+ 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around 0 78.9%

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

      1. +-commutative 78.9%

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

   8. Simplified 78.9%

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

      1. div-inv 78.9%

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

   10. Applied egg-rr 78.9%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-293}:\\ \;\;\;\;\frac{1}{\frac{x}{y} \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \left(\frac{y}{x + y} \cdot \frac{1}{x + y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{x + y}}{y + 1}\\ \end{array} \]

Alternative 6: 88.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-294}:\\ \;\;\;\;\frac{1}{\frac{x}{y} \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \left(\frac{y}{x + y} \cdot \frac{1}{x + y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -1e-294)
   (/ 1.0 (* (/ x y) (+ x 1.0)))
   (if (<= y 2.8e-13)
     (* x (* (/ y (+ x y)) (/ 1.0 (+ x y))))
     (/ (/ x (+ x y)) (+ y (+ x (+ x 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -1e-294) {
		tmp = 1.0 / ((x / y) * (x + 1.0));
	} else if (y <= 2.8e-13) {
		tmp = x * ((y / (x + y)) * (1.0 / (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 <= (-1d-294)) then
        tmp = 1.0d0 / ((x / y) * (x + 1.0d0))
    else if (y <= 2.8d-13) then
        tmp = x * ((y / (x + y)) * (1.0d0 / (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 <= -1e-294) {
		tmp = 1.0 / ((x / y) * (x + 1.0));
	} else if (y <= 2.8e-13) {
		tmp = x * ((y / (x + y)) * (1.0 / (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 <= -1e-294:
		tmp = 1.0 / ((x / y) * (x + 1.0))
	elif y <= 2.8e-13:
		tmp = x * ((y / (x + y)) * (1.0 / (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 <= -1e-294)
		tmp = Float64(1.0 / Float64(Float64(x / y) * Float64(x + 1.0)));
	elseif (y <= 2.8e-13)
		tmp = Float64(x * Float64(Float64(y / Float64(x + y)) * Float64(1.0 / 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 <= -1e-294)
		tmp = 1.0 / ((x / y) * (x + 1.0));
	elseif (y <= 2.8e-13)
		tmp = x * ((y / (x + y)) * (1.0 / (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, -1e-294], N[(1.0 / N[(N[(x / y), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.8e-13], N[(x * N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x + y), $MachinePrecision]), $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 3 regimes

2. if y < -1.00000000000000002e-294

   1. Initial program 70.8%

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

      1. times-frac 88.2%

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

      2. +-commutative 88.2%

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

      3. associate-+l+ 88.2%

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

   3. Simplified 88.2%

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

   4. Taylor expanded in y around 0 44.2%

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

      1. associate-/r* 49.7%

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

      2. +-commutative 49.7%

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

   6. Simplified 49.7%

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

      1. clear-num 49.7%

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

      2. inv-pow 49.7%

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

      3. div-inv 49.7%

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

      4. clear-num 49.7%

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

   8. Applied egg-rr 49.7%

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

      1. unpow-1 49.7%

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

      2. *-commutative 49.7%

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

   10. Simplified 49.7%

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

   if -1.00000000000000002e-294 < y < 2.8000000000000002e-13

   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. associate-*l* 72.9%

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

      2. +-commutative 72.9%

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

      3. +-commutative 72.9%

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

      4. +-commutative 72.9%

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

      5. *-commutative 72.9%

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

      6. associate-*l* 72.9%

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

      7. times-frac 81.5%

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

      8. +-commutative 81.5%

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

      9. +-commutative 81.5%

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

      10. associate-+l+ 81.5%

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

   3. Simplified 81.5%

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

   4. Taylor expanded in x around 0 64.6%

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

      1. +-commutative 64.6%

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

   6. Simplified 64.6%

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

   7. Taylor expanded in y around 0 64.6%

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

      1. associate-/r* 82.9%

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

      2. div-inv 82.9%

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

   9. Applied egg-rr 82.9%

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

   if 2.8000000000000002e-13 < y

   1. Initial program 73.7%

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

      1. 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. associate-*l* 73.7%

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

      6. *-commutative 73.7%

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

      7. associate-*r/ 82.3%

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

      8. *-commutative 82.3%

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

      9. distribute-rgt1-in 79.6%

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

      10. fma-def 82.3%

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

      11. +-commutative 82.3%

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

      12. +-commutative 82.3%

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

      13. cube-unmult 82.4%

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

      14. +-commutative 82.4%

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

   3. Simplified 82.4%

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

      1. associate-*r/ 73.7%

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

      2. *-commutative 73.7%

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

      3. fma-udef 73.6%

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

      4. cube-mult 73.6%

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

      5. distribute-rgt1-in 73.7%

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

      6. *-commutative 73.7%

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

      7. +-commutative 73.7%

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

      8. associate-+r+ 73.7%

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

      9. frac-times 93.6%

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

      10. *-commutative 93.6%

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

      11. clear-num 93.6%

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

      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 99.8%

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

      14. *-un-lft-identity 99.8%

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

      15. associate-+r+ 99.8%

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

      16. +-commutative 99.8%

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

      17. associate-+l+ 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around -inf 80.1%

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

      1. mul-1-neg 80.1%

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

      2. unsub-neg 80.1%

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

      3. neg-mul-1 80.1%

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

      4. distribute-lft-in 80.1%

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

      5. metadata-eval 80.1%

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

      6. neg-mul-1 80.1%

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

      7. +-commutative 80.1%

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

      8. +-commutative 80.1%

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

      9. unsub-neg 80.1%

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

      10. +-commutative 80.1%

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

      11. unsub-neg 80.1%

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

   8. Simplified 80.1%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-294}:\\ \;\;\;\;\frac{1}{\frac{x}{y} \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \left(\frac{y}{x + y} \cdot \frac{1}{x + y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \]

Alternative 7: 91.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{y + \left(x + 1\right)} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{y}{x + y}}{\left(x + y\right) \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.4e-6)
   (* (/ y (+ y (+ x 1.0))) (/ 1.0 x))
   (/ (* x (/ y (+ x y))) (* (+ x y) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.4e-6) {
		tmp = (y / (y + (x + 1.0))) * (1.0 / x);
	} else {
		tmp = (x * (y / (x + y))) / ((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 <= (-2.4d-6)) then
        tmp = (y / (y + (x + 1.0d0))) * (1.0d0 / x)
    else
        tmp = (x * (y / (x + y))) / ((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 <= -2.4e-6) {
		tmp = (y / (y + (x + 1.0))) * (1.0 / x);
	} else {
		tmp = (x * (y / (x + y))) / ((x + y) * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.4e-6:
		tmp = (y / (y + (x + 1.0))) * (1.0 / x)
	else:
		tmp = (x * (y / (x + y))) / ((x + y) * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.4e-6)
		tmp = Float64(Float64(y / Float64(y + Float64(x + 1.0))) * Float64(1.0 / x));
	else
		tmp = Float64(Float64(x * Float64(y / Float64(x + y))) / Float64(Float64(x + y) * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.4e-6)
		tmp = (y / (y + (x + 1.0))) * (1.0 / x);
	else
		tmp = (x * (y / (x + y))) / ((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, -2.4e-6], N[(N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.3999999999999999e-6

   1. Initial program 63.9%

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

      1. times-frac 86.5%

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

      2. +-commutative 86.5%

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

      3. associate-+l+ 86.5%

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

   3. Simplified 86.5%

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

   4. Taylor expanded in x around inf 80.7%

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

   if -2.3999999999999999e-6 < x

   1. Initial program 74.8%

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

      1. associate-*l* 74.8%

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

      2. +-commutative 74.8%

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

      3. +-commutative 74.8%

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

      4. +-commutative 74.8%

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

      5. *-commutative 74.8%

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

      6. associate-*l* 74.8%

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

      7. times-frac 88.8%

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

      8. +-commutative 88.8%

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

      9. +-commutative 88.8%

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

      10. associate-+l+ 88.8%

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

   3. Simplified 88.8%

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

   4. Taylor expanded in x around 0 83.3%

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

      1. +-commutative 83.3%

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

   6. Simplified 83.3%

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

      1. associate-/r* 91.4%

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

      2. frac-times 87.6%

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

      3. +-commutative 87.6%

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

      4. +-commutative 87.6%

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

   8. Applied egg-rr 87.6%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{y + \left(x + 1\right)} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{y}{x + y}}{\left(x + y\right) \cdot \left(y + 1\right)}\\ \end{array} \]

Alternative 8: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.56 \cdot 10^{-24}:\\ \;\;\;\;\frac{y}{y + \left(x + 1\right)} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(x + y\right) \cdot \frac{y + 1}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.56e-24)
   (* (/ y (+ y (+ x 1.0))) (/ 1.0 x))
   (/ (/ x (+ x y)) (* (+ x y) (/ (+ y 1.0) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.56e-24) {
		tmp = (y / (y + (x + 1.0))) * (1.0 / x);
	} 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 <= (-1.56d-24)) then
        tmp = (y / (y + (x + 1.0d0))) * (1.0d0 / x)
    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 <= -1.56e-24) {
		tmp = (y / (y + (x + 1.0))) * (1.0 / x);
	} 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 <= -1.56e-24:
		tmp = (y / (y + (x + 1.0))) * (1.0 / x)
	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 <= -1.56e-24)
		tmp = Float64(Float64(y / Float64(y + Float64(x + 1.0))) * Float64(1.0 / x));
	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 <= -1.56e-24)
		tmp = (y / (y + (x + 1.0))) * (1.0 / x);
	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, -1.56e-24], N[(N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $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 < -1.56e-24

   1. Initial program 65.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 79.8%

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

   if -1.56e-24 < x

   1. Initial program 74.5%

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

      1. associate-*l* 74.5%

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

      2. +-commutative 74.5%

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

      3. +-commutative 74.5%

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

      4. +-commutative 74.5%

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

      5. associate-*l* 74.5%

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

      6. *-commutative 74.5%

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

      7. associate-*r/ 83.4%

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

      8. *-commutative 83.4%

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

      9. distribute-rgt1-in 71.4%

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

      10. fma-def 83.4%

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

      11. +-commutative 83.4%

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

      12. +-commutative 83.4%

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

      13. cube-unmult 83.5%

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

      14. +-commutative 83.5%

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

   3. Simplified 83.5%

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

      1. associate-*r/ 74.5%

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

      2. *-commutative 74.5%

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

      3. fma-udef 62.5%

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

      4. cube-mult 62.5%

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

      5. distribute-rgt1-in 74.5%

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

      6. *-commutative 74.5%

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

      7. +-commutative 74.5%

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

      8. associate-+r+ 74.5%

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

      9. frac-times 88.7%

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

      10. *-commutative 88.7%

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

      11. clear-num 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 99.7%

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

      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 85.7%

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

      1. +-commutative 85.7%

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

   8. Simplified 85.7%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.56 \cdot 10^{-24}:\\ \;\;\;\;\frac{y}{y + \left(x + 1\right)} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(x + y\right) \cdot \frac{y + 1}{y}}\\ \end{array} \]

Alternative 9: 91.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{x \cdot \frac{y + \left(x + 1\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{\left(x + y\right) \cdot \frac{y + 1}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.4e-18)
   (/ 1.0 (* x (/ (+ y (+ x 1.0)) y)))
   (/ (/ y (+ x y)) (* (+ x y) (/ (+ y 1.0) x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.4e-18) {
		tmp = 1.0 / (x * ((y + (x + 1.0)) / y));
	} else {
		tmp = (y / (x + y)) / ((x + y) * ((y + 1.0) / x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.4d-18)) then
        tmp = 1.0d0 / (x * ((y + (x + 1.0d0)) / y))
    else
        tmp = (y / (x + y)) / ((x + y) * ((y + 1.0d0) / x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.4e-18) {
		tmp = 1.0 / (x * ((y + (x + 1.0)) / y));
	} else {
		tmp = (y / (x + y)) / ((x + y) * ((y + 1.0) / x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.4e-18:
		tmp = 1.0 / (x * ((y + (x + 1.0)) / y))
	else:
		tmp = (y / (x + y)) / ((x + y) * ((y + 1.0) / x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.4e-18)
		tmp = Float64(1.0 / Float64(x * Float64(Float64(y + Float64(x + 1.0)) / y)));
	else
		tmp = Float64(Float64(y / Float64(x + y)) / Float64(Float64(x + y) * Float64(Float64(y + 1.0) / x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.4e-18)
		tmp = 1.0 / (x * ((y + (x + 1.0)) / y));
	else
		tmp = (y / (x + y)) / ((x + y) * ((y + 1.0) / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.4e-18], N[(1.0 / N[(x * N[(N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.39999999999999994e-18

   1. Initial program 64.5%

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

      1. times-frac 86.7%

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

      2. +-commutative 86.7%

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

      3. associate-+l+ 86.7%

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

   3. Simplified 86.7%

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

   4. Taylor expanded in x around inf 79.5%

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

      1. *-commutative 79.5%

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

      2. clear-num 79.4%

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

      3. associate-+r+ 79.4%

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

      4. +-commutative 79.4%

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

      5. associate-+r+ 79.4%

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

      6. frac-times 79.3%

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

      7. metadata-eval 79.3%

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

      8. associate-+r+ 79.3%

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

      9. +-commutative 79.3%

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

      10. associate-+r+ 79.3%

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

   6. Applied egg-rr 79.3%

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

   if -2.39999999999999994e-18 < x

   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. *-commutative 74.6%

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

      6. associate-*l* 74.6%

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

      7. times-frac 88.8%

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

      8. +-commutative 88.8%

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

      9. +-commutative 88.8%

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

      10. associate-+l+ 88.8%

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

   3. Simplified 88.8%

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

   4. Taylor expanded in x around 0 83.2%

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

      1. +-commutative 83.2%

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

   6. Simplified 83.2%

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

      1. *-commutative 83.2%

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

      2. clear-num 83.1%

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

      3. associate-/r* 91.3%

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

      4. frac-times 85.7%

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

      5. *-un-lft-identity 85.7%

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

      6. +-commutative 85.7%

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

      7. +-commutative 85.7%

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

   8. Applied egg-rr 85.7%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{x \cdot \frac{y + \left(x + 1\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{\left(x + y\right) \cdot \frac{y + 1}{x}}\\ \end{array} \]

Alternative 10: 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 72.1%

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

   1. associate-*l* 72.1%

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

   2. +-commutative 72.1%

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

   3. +-commutative 72.1%

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

   4. +-commutative 72.1%

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

   5. associate-*l* 72.1%

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

   6. *-commutative 72.1%

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

   7. associate-*r/ 81.4%

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

   8. *-commutative 81.4%

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

   9. distribute-rgt1-in 62.4%

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

   10. fma-def 81.4%

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

   11. +-commutative 81.4%

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

   12. +-commutative 81.4%

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

   13. cube-unmult 81.4%

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

   14. +-commutative 81.4%

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

3. Simplified 81.4%

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

   1. associate-*r/ 72.2%

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

   2. *-commutative 72.2%

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

   3. fma-udef 55.3%

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

   4. cube-mult 55.3%

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

   5. distribute-rgt1-in 72.1%

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

   6. *-commutative 72.1%

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

   7. +-commutative 72.1%

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

   8. associate-+r+ 72.1%

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

   9. frac-times 88.2%

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

   10. *-commutative 88.2%

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

   11. clear-num 88.2%

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

   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 99.7%

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

   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. Final simplification 99.7%

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

Alternative 11: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return (y / (x + y)) / ((x + y) * ((x + (y + 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 = (y / (x + y)) / ((x + y) * ((x + (y + 1.0d0)) / x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.1%

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

   1. associate-*l* 72.1%

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

   2. +-commutative 72.1%

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

   3. +-commutative 72.1%

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

   4. +-commutative 72.1%

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

   5. associate-*l* 72.1%

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

   6. *-commutative 72.1%

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

   7. associate-*r/ 81.4%

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

   8. *-commutative 81.4%

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

   9. distribute-rgt1-in 62.4%

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

   10. fma-def 81.4%

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

   11. +-commutative 81.4%

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

   12. +-commutative 81.4%

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

   13. cube-unmult 81.4%

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

   14. +-commutative 81.4%

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

3. Simplified 81.4%

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

   1. associate-*r/ 72.2%

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

   2. fma-udef 55.3%

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

   3. cube-mult 55.3%

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

   4. distribute-rgt1-in 72.1%

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

   5. *-commutative 72.1%

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

   6. +-commutative 72.1%

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

   7. associate-+r+ 72.1%

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

   8. frac-times 88.3%

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

   9. *-commutative 88.3%

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

   10. associate-/r* 99.7%

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

   11. clear-num 99.7%

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

   12. frac-times 99.7%

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

   13. *-un-lft-identity 99.7%

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

   14. associate-+r+ 99.7%

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

   15. +-commutative 99.7%

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

   16. associate-+l+ 99.7%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

Alternative 12: 84.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.05e-106)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 5.2e-15)
     (* x (/ y (* (+ x y) (+ x y))))
     (/ (/ x (+ x y)) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.05e-106) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 5.2e-15) {
		tmp = x * (y / ((x + y) * (x + y)));
	} 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 <= 1.05d-106) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 5.2d-15) then
        tmp = x * (y / ((x + y) * (x + y)))
    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 <= 1.05e-106) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 5.2e-15) {
		tmp = x * (y / ((x + y) * (x + y)));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.05e-106:
		tmp = (y / x) / (x + 1.0)
	elif y <= 5.2e-15:
		tmp = x * (y / ((x + y) * (x + y)))
	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 <= 1.05e-106)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 5.2e-15)
		tmp = Float64(x * Float64(y / Float64(Float64(x + y) * Float64(x + y))));
	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 <= 1.05e-106)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 5.2e-15)
		tmp = x * (y / ((x + y) * (x + y)));
	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, 1.05e-106], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.2e-15], N[(x * N[(y / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.05000000000000002e-106

   1. Initial program 69.2%

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

      1. times-frac 84.7%

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

      2. +-commutative 84.7%

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

      3. associate-+l+ 84.7%

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

   3. Simplified 84.7%

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

   4. Taylor expanded in y around 0 53.9%

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

      1. associate-/r* 58.0%

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

      2. +-commutative 58.0%

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

   6. Simplified 58.0%

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

   if 1.05000000000000002e-106 < y < 5.20000000000000009e-15

   1. Initial program 94.1%

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

      1. associate-*l* 94.1%

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

      2. +-commutative 94.1%

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

      3. +-commutative 94.1%

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

      4. +-commutative 94.1%

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

      5. *-commutative 94.1%

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

      6. associate-*l* 94.1%

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

      7. times-frac 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 79.3%

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

      1. +-commutative 79.3%

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

   6. Simplified 79.3%

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

   7. Taylor expanded in y around 0 79.3%

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

   if 5.20000000000000009e-15 < y

   1. Initial program 73.7%

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

      1. 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. associate-*l* 73.7%

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

      6. *-commutative 73.7%

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

      7. associate-*r/ 82.3%

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

      8. *-commutative 82.3%

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

      9. distribute-rgt1-in 79.6%

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

      10. fma-def 82.3%

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

      11. +-commutative 82.3%

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

      12. +-commutative 82.3%

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

      13. cube-unmult 82.4%

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

      14. +-commutative 82.4%

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

   3. Simplified 82.4%

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

      1. associate-*r/ 73.7%

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

      2. *-commutative 73.7%

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

      3. fma-udef 73.6%

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

      4. cube-mult 73.6%

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

      5. distribute-rgt1-in 73.7%

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

      6. *-commutative 73.7%

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

      7. +-commutative 73.7%

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

      8. associate-+r+ 73.7%

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

      9. frac-times 93.6%

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

      10. *-commutative 93.6%

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

      11. clear-num 93.6%

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

      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 99.8%

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

      14. *-un-lft-identity 99.8%

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

      15. associate-+r+ 99.8%

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

      16. +-commutative 99.8%

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

      17. associate-+l+ 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around 0 78.9%

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

      1. +-commutative 78.9%

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

   8. Simplified 78.9%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]

Alternative 13: 88.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-293}:\\ \;\;\;\;\frac{1}{\frac{x}{y} \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \frac{\frac{y}{x + y}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -2.2e-293)
   (/ 1.0 (* (/ x y) (+ x 1.0)))
   (if (<= y 5.2e-15)
     (* x (/ (/ y (+ x y)) (+ x y)))
     (/ (/ x (+ x y)) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -2.2e-293) {
		tmp = 1.0 / ((x / y) * (x + 1.0));
	} else if (y <= 5.2e-15) {
		tmp = x * ((y / (x + y)) / (x + y));
	} 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 <= (-2.2d-293)) then
        tmp = 1.0d0 / ((x / y) * (x + 1.0d0))
    else if (y <= 5.2d-15) then
        tmp = x * ((y / (x + y)) / (x + y))
    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 <= -2.2e-293) {
		tmp = 1.0 / ((x / y) * (x + 1.0));
	} else if (y <= 5.2e-15) {
		tmp = x * ((y / (x + y)) / (x + y));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -2.2e-293:
		tmp = 1.0 / ((x / y) * (x + 1.0))
	elif y <= 5.2e-15:
		tmp = x * ((y / (x + y)) / (x + y))
	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 <= -2.2e-293)
		tmp = Float64(1.0 / Float64(Float64(x / y) * Float64(x + 1.0)));
	elseif (y <= 5.2e-15)
		tmp = Float64(x * Float64(Float64(y / Float64(x + y)) / Float64(x + y)));
	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 <= -2.2e-293)
		tmp = 1.0 / ((x / y) * (x + 1.0));
	elseif (y <= 5.2e-15)
		tmp = x * ((y / (x + y)) / (x + y));
	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, -2.2e-293], N[(1.0 / N[(N[(x / y), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.2e-15], N[(x * N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.2e-293

   1. Initial program 70.6%

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

      1. times-frac 88.1%

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

      2. +-commutative 88.1%

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

      3. associate-+l+ 88.1%

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

   3. Simplified 88.1%

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

   4. Taylor expanded in y around 0 43.8%

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

      1. associate-/r* 49.3%

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

      2. +-commutative 49.3%

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

   6. Simplified 49.3%

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

      1. clear-num 49.3%

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

      2. inv-pow 49.3%

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

      3. div-inv 49.3%

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

      4. clear-num 49.3%

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

   8. Applied egg-rr 49.3%

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

      1. unpow-1 49.3%

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

      2. *-commutative 49.3%

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

   10. Simplified 49.3%

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

   if -2.2e-293 < y < 5.20000000000000009e-15

   1. Initial program 73.4%

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

      1. associate-*l* 73.4%

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

      2. +-commutative 73.4%

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

      3. +-commutative 73.4%

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

      4. +-commutative 73.4%

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

      5. *-commutative 73.4%

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

      6. associate-*l* 73.4%

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

      7. times-frac 81.8%

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

      8. +-commutative 81.8%

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

      9. +-commutative 81.8%

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

      10. associate-+l+ 81.8%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in x around 0 65.2%

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

      1. +-commutative 65.2%

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

   6. Simplified 65.2%

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

   7. Taylor expanded in y around 0 65.2%

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

      1. associate-/r* 83.2%

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

      2. div-inv 83.2%

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

   9. Applied egg-rr 83.2%

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

       1. associate-*r/ 83.2%

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

       2. *-rgt-identity 83.2%

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

   11. Simplified 83.2%

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

   if 5.20000000000000009e-15 < y

   1. Initial program 73.7%

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

      1. 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. associate-*l* 73.7%

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

      6. *-commutative 73.7%

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

      7. associate-*r/ 82.3%

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

      8. *-commutative 82.3%

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

      9. distribute-rgt1-in 79.6%

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

      10. fma-def 82.3%

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

      11. +-commutative 82.3%

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

      12. +-commutative 82.3%

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

      13. cube-unmult 82.4%

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

      14. +-commutative 82.4%

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

   3. Simplified 82.4%

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

      1. associate-*r/ 73.7%

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

      2. *-commutative 73.7%

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

      3. fma-udef 73.6%

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

      4. cube-mult 73.6%

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

      5. distribute-rgt1-in 73.7%

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

      6. *-commutative 73.7%

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

      7. +-commutative 73.7%

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

      8. associate-+r+ 73.7%

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

      9. frac-times 93.6%

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

      10. *-commutative 93.6%

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

      11. clear-num 93.6%

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

      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 99.8%

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

      14. *-un-lft-identity 99.8%

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

      15. associate-+r+ 99.8%

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

      16. +-commutative 99.8%

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

      17. associate-+l+ 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around 0 78.9%

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

      1. +-commutative 78.9%

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

   8. Simplified 78.9%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-293}:\\ \;\;\;\;\frac{1}{\frac{x}{y} \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \frac{\frac{y}{x + y}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]

Alternative 14: 88.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{-296}:\\ \;\;\;\;\frac{1}{\frac{x}{y} \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \frac{\frac{y}{x + y}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -9.2e-296)
   (/ 1.0 (* (/ x y) (+ x 1.0)))
   (if (<= y 5e-15)
     (* x (/ (/ y (+ x y)) (+ x y)))
     (/ (* x (/ 1.0 (+ x y))) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -9.2e-296) {
		tmp = 1.0 / ((x / y) * (x + 1.0));
	} else if (y <= 5e-15) {
		tmp = x * ((y / (x + y)) / (x + y));
	} else {
		tmp = (x * (1.0 / (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 <= (-9.2d-296)) then
        tmp = 1.0d0 / ((x / y) * (x + 1.0d0))
    else if (y <= 5d-15) then
        tmp = x * ((y / (x + y)) / (x + y))
    else
        tmp = (x * (1.0d0 / (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 <= -9.2e-296) {
		tmp = 1.0 / ((x / y) * (x + 1.0));
	} else if (y <= 5e-15) {
		tmp = x * ((y / (x + y)) / (x + y));
	} else {
		tmp = (x * (1.0 / (x + y))) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -9.2e-296:
		tmp = 1.0 / ((x / y) * (x + 1.0))
	elif y <= 5e-15:
		tmp = x * ((y / (x + y)) / (x + y))
	else:
		tmp = (x * (1.0 / (x + y))) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -9.2e-296)
		tmp = Float64(1.0 / Float64(Float64(x / y) * Float64(x + 1.0)));
	elseif (y <= 5e-15)
		tmp = Float64(x * Float64(Float64(y / Float64(x + y)) / Float64(x + y)));
	else
		tmp = Float64(Float64(x * Float64(1.0 / 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 <= -9.2e-296)
		tmp = 1.0 / ((x / y) * (x + 1.0));
	elseif (y <= 5e-15)
		tmp = x * ((y / (x + y)) / (x + y));
	else
		tmp = (x * (1.0 / (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, -9.2e-296], N[(1.0 / N[(N[(x / y), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e-15], N[(x * N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.20000000000000016e-296

   1. Initial program 70.8%

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

      1. times-frac 88.2%

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

      2. +-commutative 88.2%

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

      3. associate-+l+ 88.2%

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

   3. Simplified 88.2%

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

   4. Taylor expanded in y around 0 44.2%

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

      1. associate-/r* 49.7%

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

      2. +-commutative 49.7%

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

   6. Simplified 49.7%

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

      1. clear-num 49.7%

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

      2. inv-pow 49.7%

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

      3. div-inv 49.7%

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

      4. clear-num 49.7%

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

   8. Applied egg-rr 49.7%

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

      1. unpow-1 49.7%

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

      2. *-commutative 49.7%

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

   10. Simplified 49.7%

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

   if -9.20000000000000016e-296 < y < 4.99999999999999999e-15

   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. associate-*l* 72.9%

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

      2. +-commutative 72.9%

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

      3. +-commutative 72.9%

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

      4. +-commutative 72.9%

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

      5. *-commutative 72.9%

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

      6. associate-*l* 72.9%

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

      7. times-frac 81.5%

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

      8. +-commutative 81.5%

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

      9. +-commutative 81.5%

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

      10. associate-+l+ 81.5%

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

   3. Simplified 81.5%

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

   4. Taylor expanded in x around 0 64.6%

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

      1. +-commutative 64.6%

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

   6. Simplified 64.6%

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

   7. Taylor expanded in y around 0 64.6%

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

      1. associate-/r* 82.9%

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

      2. div-inv 82.9%

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

   9. Applied egg-rr 82.9%

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

       1. associate-*r/ 82.9%

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

       2. *-rgt-identity 82.9%

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

   11. Simplified 82.9%

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

   if 4.99999999999999999e-15 < y

   1. Initial program 73.7%

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

      1. 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. associate-*l* 73.7%

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

      6. *-commutative 73.7%

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

      7. associate-*r/ 82.3%

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

      8. *-commutative 82.3%

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

      9. distribute-rgt1-in 79.6%

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

      10. fma-def 82.3%

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

      11. +-commutative 82.3%

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

      12. +-commutative 82.3%

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

      13. cube-unmult 82.4%

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

      14. +-commutative 82.4%

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

   3. Simplified 82.4%

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

      1. associate-*r/ 73.7%

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

      2. *-commutative 73.7%

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

      3. fma-udef 73.6%

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

      4. cube-mult 73.6%

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

      5. distribute-rgt1-in 73.7%

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

      6. *-commutative 73.7%

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

      7. +-commutative 73.7%

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

      8. associate-+r+ 73.7%

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

      9. frac-times 93.6%

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

      10. *-commutative 93.6%

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

      11. clear-num 93.6%

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

      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 99.8%

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

      14. *-un-lft-identity 99.8%

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

      15. associate-+r+ 99.8%

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

      16. +-commutative 99.8%

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

      17. associate-+l+ 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around 0 78.9%

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

      1. +-commutative 78.9%

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

   8. Simplified 78.9%

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

      1. div-inv 78.9%

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

   10. Applied egg-rr 78.9%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{-296}:\\ \;\;\;\;\frac{1}{\frac{x}{y} \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \frac{\frac{y}{x + y}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{x + y}}{y + 1}\\ \end{array} \]

Alternative 15: 82.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-37}:\\ \;\;\;\;\frac{y}{y + \left(x + 1\right)} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -8.2e-37)
   (* (/ y (+ y (+ x 1.0))) (/ 1.0 x))
   (/ (/ x (+ x y)) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -8.2e-37) {
		tmp = (y / (y + (x + 1.0))) * (1.0 / x);
	} 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 (x <= (-8.2d-37)) then
        tmp = (y / (y + (x + 1.0d0))) * (1.0d0 / x)
    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 (x <= -8.2e-37) {
		tmp = (y / (y + (x + 1.0))) * (1.0 / x);
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -8.2e-37:
		tmp = (y / (y + (x + 1.0))) * (1.0 / x)
	else:
		tmp = (x / (x + y)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -8.2e-37)
		tmp = Float64(Float64(y / Float64(y + Float64(x + 1.0))) * Float64(1.0 / x));
	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 (x <= -8.2e-37)
		tmp = (y / (y + (x + 1.0))) * (1.0 / x);
	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[x, -8.2e-37], N[(N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $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 x < -8.1999999999999996e-37

   1. Initial program 65.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 79.8%

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

   if -8.1999999999999996e-37 < x

   1. Initial program 74.5%

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

      1. associate-*l* 74.5%

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

      2. +-commutative 74.5%

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

      3. +-commutative 74.5%

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

      4. +-commutative 74.5%

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

      5. associate-*l* 74.5%

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

      6. *-commutative 74.5%

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

      7. associate-*r/ 83.4%

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

      8. *-commutative 83.4%

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

      9. distribute-rgt1-in 71.4%

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

      10. fma-def 83.4%

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

      11. +-commutative 83.4%

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

      12. +-commutative 83.4%

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

      13. cube-unmult 83.5%

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

      14. +-commutative 83.5%

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

   3. Simplified 83.5%

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

      1. associate-*r/ 74.5%

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

      2. *-commutative 74.5%

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

      3. fma-udef 62.5%

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

      4. cube-mult 62.5%

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

      5. distribute-rgt1-in 74.5%

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

      6. *-commutative 74.5%

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

      7. +-commutative 74.5%

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

      8. associate-+r+ 74.5%

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

      9. frac-times 88.7%

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

      10. *-commutative 88.7%

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

      11. clear-num 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 99.7%

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

      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 64.2%

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

      1. +-commutative 64.2%

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

   8. Simplified 64.2%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-37}:\\ \;\;\;\;\frac{y}{y + \left(x + 1\right)} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]

Alternative 16: 80.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+184}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-37}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.5e+184)
   (* (/ 1.0 x) (/ y x))
   (if (<= x -8.5e-37) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.5e+184) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -8.5e-37) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.5d+184)) then
        tmp = (1.0d0 / x) * (y / x)
    else if (x <= (-8.5d-37)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.5e+184) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -8.5e-37) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.5e+184:
		tmp = (1.0 / x) * (y / x)
	elif x <= -8.5e-37:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.5e+184)
		tmp = Float64(Float64(1.0 / x) * Float64(y / x));
	elseif (x <= -8.5e-37)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.5e+184)
		tmp = (1.0 / x) * (y / x);
	elseif (x <= -8.5e-37)
		tmp = y / (x * (x + 1.0));
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.5e+184], N[(N[(1.0 / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8.5e-37], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4999999999999999e184

   1. Initial program 60.3%

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

      1. times-frac 81.7%

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

      2. +-commutative 81.7%

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

      3. associate-+l+ 81.7%

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

   3. Simplified 81.7%

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

   4. Taylor expanded in x around inf 96.4%

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

   5. Taylor expanded in x around inf 96.2%

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

   if -2.4999999999999999e184 < x < -8.5000000000000007e-37

   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. times-frac 90.2%

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

      2. +-commutative 90.2%

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

      3. associate-+l+ 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in y around 0 68.7%

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

   if -8.5000000000000007e-37 < x

   1. Initial program 74.5%

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

      1. times-frac 88.7%

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

      2. +-commutative 88.7%

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

      3. associate-+l+ 88.7%

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

   3. Simplified 88.7%

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

   4. Taylor expanded in x around 0 63.7%

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

      1. +-commutative 63.7%

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

   6. Simplified 63.7%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+184}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-37}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]

Alternative 17: 81.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+183}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-37}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.9e+183)
   (* (/ 1.0 x) (/ y x))
   (if (<= x -3.8e-37) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.9e+183) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -3.8e-37) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.9d+183)) then
        tmp = (1.0d0 / x) * (y / x)
    else if (x <= (-3.8d-37)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.9e+183) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -3.8e-37) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.9e+183:
		tmp = (1.0 / x) * (y / x)
	elif x <= -3.8e-37:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.9e+183)
		tmp = Float64(Float64(1.0 / x) * Float64(y / x));
	elseif (x <= -3.8e-37)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.9e+183)
		tmp = (1.0 / x) * (y / x);
	elseif (x <= -3.8e-37)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.9e+183], N[(N[(1.0 / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.8e-37], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.9000000000000001e183

   1. Initial program 60.3%

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

      1. times-frac 81.7%

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

      2. +-commutative 81.7%

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

      3. associate-+l+ 81.7%

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

   3. Simplified 81.7%

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

   4. Taylor expanded in x around inf 96.4%

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

   5. Taylor expanded in x around inf 96.2%

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

   if -2.9000000000000001e183 < x < -3.8000000000000004e-37

   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. times-frac 90.2%

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

      2. +-commutative 90.2%

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

      3. associate-+l+ 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in y around 0 68.7%

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

   if -3.8000000000000004e-37 < x

   1. Initial program 74.5%

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

      1. associate-*l* 74.5%

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

      2. +-commutative 74.5%

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

      3. +-commutative 74.5%

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

      4. +-commutative 74.5%

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

      5. associate-*l* 74.5%

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

      6. *-commutative 74.5%

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

      7. associate-*r/ 83.4%

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

      8. *-commutative 83.4%

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

      9. distribute-rgt1-in 71.4%

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

      10. fma-def 83.4%

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

      11. +-commutative 83.4%

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

      12. +-commutative 83.4%

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

      13. cube-unmult 83.5%

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

      14. +-commutative 83.5%

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

   3. Simplified 83.5%

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

      1. associate-*r/ 74.5%

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

      2. *-commutative 74.5%

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

      3. fma-udef 62.5%

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

      4. cube-mult 62.5%

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

      5. distribute-rgt1-in 74.5%

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

      6. *-commutative 74.5%

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

      7. +-commutative 74.5%

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

      8. associate-+r+ 74.5%

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

      9. frac-times 88.7%

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

      10. *-commutative 88.7%

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

      11. clear-num 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 99.7%

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

      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 64.2%

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

      1. +-commutative 64.2%

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

   8. Simplified 64.2%

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

   9. Taylor expanded in x around 0 63.8%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+183}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-37}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 18: 81.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-36}:\\ \;\;\;\;\frac{1}{\frac{x}{y} \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.05e-36) (/ 1.0 (* (/ x y) (+ x 1.0))) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.05e-36) {
		tmp = 1.0 / ((x / y) * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.05d-36)) then
        tmp = 1.0d0 / ((x / y) * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.05e-36) {
		tmp = 1.0 / ((x / y) * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.05e-36:
		tmp = 1.0 / ((x / y) * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.05e-36)
		tmp = Float64(1.0 / Float64(Float64(x / y) * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.05e-36)
		tmp = 1.0 / ((x / y) * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.05e-36], N[(1.0 / N[(N[(x / y), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.04999999999999995e-36

   1. Initial program 65.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 y around 0 73.8%

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

      1. associate-/r* 79.3%

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

      2. +-commutative 79.3%

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

   6. Simplified 79.3%

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

      1. clear-num 79.2%

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

      2. inv-pow 79.2%

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

      3. div-inv 79.2%

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

      4. clear-num 79.2%

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

   8. Applied egg-rr 79.2%

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

      1. unpow-1 79.2%

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

      2. *-commutative 79.2%

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

   10. Simplified 79.2%

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

   if -1.04999999999999995e-36 < x

   1. Initial program 74.5%

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

      1. associate-*l* 74.5%

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

      2. +-commutative 74.5%

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

      3. +-commutative 74.5%

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

      4. +-commutative 74.5%

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

      5. associate-*l* 74.5%

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

      6. *-commutative 74.5%

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

      7. associate-*r/ 83.4%

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

      8. *-commutative 83.4%

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

      9. distribute-rgt1-in 71.4%

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

      10. fma-def 83.4%

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

      11. +-commutative 83.4%

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

      12. +-commutative 83.4%

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

      13. cube-unmult 83.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 83.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 83.5%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 74.5%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. *-commutative 74.5%

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      3. fma-udef 62.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      4. cube-mult 62.5%

         \[\leadsto \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. distribute-rgt1-in 74.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      6. *-commutative 74.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. +-commutative 74.5%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      8. associate-+r+ 74.5%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      9. frac-times 88.7%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

      10. *-commutative 88.7%

          \[\leadsto \color{blue}{\frac{y}{y + \left(x + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      11. clear-num 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 99.7%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{y + \left(x + 1\right)}{y} \cdot \left(x + y\right)}} \]

      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 64.2%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{1 + y}} \]
   7. Step-by-step derivation

      1. +-commutative 64.2%

         \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{y + 1}} \]

   8. Simplified 64.2%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{y + 1}} \]

   9. Taylor expanded in x around 0 63.8%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-36}:\\ \;\;\;\;\frac{1}{\frac{x}{y} \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 19: 82.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.9e-104) (/ (/ 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 <= 1.9e-104) {
		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 <= 1.9d-104) 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 <= 1.9e-104) {
		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 <= 1.9e-104:
		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 <= 1.9e-104)
		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 <= 1.9e-104)
		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, 1.9e-104], 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 < 1.9e-104

   1. Initial program 69.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 84.7%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 84.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 84.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 84.7%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around 0 53.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 58.0%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 58.0%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   6. Simplified 58.0%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if 1.9e-104 < y

   1. Initial program 77.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 77.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 77.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 77.5%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 77.5%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 77.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 77.5%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. associate-*r/ 85.5%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 85.5%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 81.1%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 85.5%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 85.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 85.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 85.6%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 85.6%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 85.6%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 77.5%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. *-commutative 77.5%

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      3. fma-udef 75.3%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      4. cube-mult 75.3%

         \[\leadsto \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. distribute-rgt1-in 77.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      6. *-commutative 77.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. +-commutative 77.5%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      8. associate-+r+ 77.5%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      9. frac-times 94.7%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

      10. *-commutative 94.7%

          \[\leadsto \color{blue}{\frac{y}{y + \left(x + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      11. clear-num 94.7%

          \[\leadsto \color{blue}{\frac{1}{\frac{y + \left(x + 1\right)}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      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 99.8%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{y + \left(x + 1\right)}{y} \cdot \left(x + y\right)}} \]

      14. *-un-lft-identity 99.8%

          \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{\frac{y + \left(x + 1\right)}{y} \cdot \left(x + y\right)} \]

      15. associate-+r+ 99.8%

          \[\leadsto \frac{\frac{x}{x + y}}{\frac{\color{blue}{\left(y + x\right) + 1}}{y} \cdot \left(x + y\right)} \]

      16. +-commutative 99.8%

          \[\leadsto \frac{\frac{x}{x + y}}{\frac{\color{blue}{\left(x + y\right)} + 1}{y} \cdot \left(x + y\right)} \]

      17. associate-+l+ 99.8%

          \[\leadsto \frac{\frac{x}{x + y}}{\frac{\color{blue}{x + \left(y + 1\right)}}{y} \cdot \left(x + y\right)} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}} \]

   6. Taylor expanded in x around 0 72.5%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{1 + y}} \]
   7. Step-by-step derivation

      1. +-commutative 72.5%

         \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{y + 1}} \]

   8. Simplified 72.5%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{y + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]

Alternative 20: 67.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-201}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

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) (/ y x))
   (if (<= x -3.5e-201) (- (/ 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) * (y / x);
	} else if (x <= -3.5e-201) {
		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) * (y / x)
    else if (x <= (-3.5d-201)) 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) * (y / x);
	} else if (x <= -3.5e-201) {
		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) * (y / x)
	elif x <= -3.5e-201:
		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(Float64(1.0 / x) * Float64(y / x));
	elseif (x <= -3.5e-201)
		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) * (y / x);
	elseif (x <= -3.5e-201)
		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[(N[(1.0 / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.5e-201], 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 62.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 86.0%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 86.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 86.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 86.0%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around inf 81.7%

      \[\leadsto \color{blue}{\frac{1}{x}} \cdot \frac{y}{y + \left(x + 1\right)} \]

   5. Taylor expanded in x around inf 80.0%

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{y}{x}} \]

   if -1 < x < -3.50000000000000008e-201

   1. Initial program 78.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 88.3%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 88.3%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 88.3%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 88.3%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around 0 33.4%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 33.4%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 33.4%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   6. Simplified 33.4%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   7. Taylor expanded in x around 0 31.9%

      \[\leadsto \color{blue}{-1 \cdot y + \frac{y}{x}} \]
   8. Step-by-step derivation

      1. neg-mul-1 31.9%

         \[\leadsto \color{blue}{\left(-y\right)} + \frac{y}{x} \]

      2. +-commutative 31.9%

         \[\leadsto \color{blue}{\frac{y}{x} + \left(-y\right)} \]

      3. unsub-neg 31.9%

         \[\leadsto \color{blue}{\frac{y}{x} - y} \]

   9. Simplified 31.9%

      \[\leadsto \color{blue}{\frac{y}{x} - y} \]

   if -3.50000000000000008e-201 < x

   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 89.1%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 89.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 89.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 89.1%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around 0 61.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 61.8%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   6. Simplified 61.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   7. Taylor expanded in y around 0 45.4%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-201}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 21: 78.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -4.2e+18) (* (/ 1.0 x) (/ y x)) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4.2e+18) {
		tmp = (1.0 / x) * (y / x);
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.2d+18)) then
        tmp = (1.0d0 / x) * (y / x)
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -4.2e+18) {
		tmp = (1.0 / x) * (y / x);
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -4.2e+18:
		tmp = (1.0 / x) * (y / x)
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -4.2e+18)
		tmp = Float64(Float64(1.0 / x) * Float64(y / x));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.2e+18)
		tmp = (1.0 / x) * (y / x);
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -4.2e+18], N[(N[(1.0 / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.2e18

   1. Initial program 61.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 85.6%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 85.6%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 85.6%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 85.6%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around inf 82.7%

      \[\leadsto \color{blue}{\frac{1}{x}} \cdot \frac{y}{y + \left(x + 1\right)} \]

   5. Taylor expanded in x around inf 82.3%

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{y}{x}} \]

   if -4.2e18 < x

   1. Initial program 75.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 89.0%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 89.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 89.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 89.0%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around 0 63.3%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 63.3%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   6. Simplified 63.3%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]

Alternative 22: 82.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.3e-37) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.3e-37) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.3d-37)) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -3.3e-37) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.3e-37:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.3e-37)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.3e-37)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -3.3e-37], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.29999999999999982e-37

   1. Initial program 65.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 y around 0 73.8%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 79.3%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 79.3%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   6. Simplified 79.3%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if -3.29999999999999982e-37 < x

   1. Initial program 74.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 74.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 74.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 74.5%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 74.5%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 74.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 74.5%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. associate-*r/ 83.4%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 83.4%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 71.4%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 83.4%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 83.4%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 83.4%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 83.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 83.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 83.5%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 74.5%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. *-commutative 74.5%

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      3. fma-udef 62.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      4. cube-mult 62.5%

         \[\leadsto \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. distribute-rgt1-in 74.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      6. *-commutative 74.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. +-commutative 74.5%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      8. associate-+r+ 74.5%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      9. frac-times 88.7%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

      10. *-commutative 88.7%

          \[\leadsto \color{blue}{\frac{y}{y + \left(x + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      11. clear-num 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 99.7%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{y + \left(x + 1\right)}{y} \cdot \left(x + y\right)}} \]

      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 64.2%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{1 + y}} \]
   7. Step-by-step derivation

      1. +-commutative 64.2%

         \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{y + 1}} \]

   8. Simplified 64.2%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{y + 1}} \]

   9. Taylor expanded in x around 0 63.8%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 23: 27.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -0.45:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -0.45) (/ 1.0 x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -0.45) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.45d0)) then
        tmp = 1.0d0 / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -0.45) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -0.45:
		tmp = 1.0 / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -0.45)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.45)
		tmp = 1.0 / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -0.45], N[(1.0 / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.450000000000000011

   1. Initial program 62.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 86.0%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 86.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 86.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 86.0%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around inf 81.7%

      \[\leadsto \color{blue}{\frac{1}{x}} \cdot \frac{y}{y + \left(x + 1\right)} \]

   5. Taylor expanded in y around inf 6.1%

      \[\leadsto \color{blue}{\frac{1}{x}} \]

   if -0.450000000000000011 < x

   1. Initial program 75.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 88.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 88.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 88.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 88.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around 0 63.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 63.5%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   6. Simplified 63.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   7. Taylor expanded in y around 0 41.3%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.45:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 24: 42.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-212}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 3.5e-212) (/ y x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.5e-212) {
		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 <= 3.5d-212) 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 <= 3.5e-212) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.5e-212:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.5e-212)
		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 <= 3.5e-212)
		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, 3.5e-212], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.4999999999999998e-212

   1. Initial program 69.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 86.5%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 86.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 86.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 86.5%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around 0 51.7%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 56.4%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 56.4%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   6. Simplified 56.4%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   7. Taylor expanded in x around 0 32.6%

      \[\leadsto \color{blue}{\frac{y}{x}} \]

   if 3.4999999999999998e-212 < y

   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. times-frac 90.5%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 90.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 90.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 90.5%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around 0 63.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 63.8%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   6. Simplified 63.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   7. Taylor expanded in y around 0 39.4%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-212}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 25: 4.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{1}{x} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ 1.0 x))

C

assert(x < y);
double code(double x, double y) {
	return 1.0 / x;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / x
end function

Java

assert x < y;
public static double code(double x, double y) {
	return 1.0 / x;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return 1.0 / x

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(1.0 / x)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 1.0 / x;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(1.0 / x), $MachinePrecision]

Derivation

1. Initial program 72.1%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. times-frac 88.2%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. +-commutative 88.2%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   3. associate-+l+ 88.2%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

3. Simplified 88.2%

   \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

4. Taylor expanded in x around inf 47.9%

   \[\leadsto \color{blue}{\frac{1}{x}} \cdot \frac{y}{y + \left(x + 1\right)} \]

5. Taylor expanded in y around inf 4.1%

   \[\leadsto \color{blue}{\frac{1}{x}} \]

6. Final simplification 4.1%

   \[\leadsto \frac{1}{x} \]

Alternative 26: 3.5% accurate, 17.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ 1 \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 1.0)

C

assert(x < y);
double code(double x, double y) {
	return 1.0;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

assert x < y;
public static double code(double x, double y) {
	return 1.0;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return 1.0

Julia

x, y = sort([x, y])
function code(x, y)
	return 1.0
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := 1.0

Derivation

1. Initial program 72.1%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 72.1%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

   2. +-commutative 72.1%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

   3. +-commutative 72.1%

      \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

   4. +-commutative 72.1%

      \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

   5. associate-*l* 72.1%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

   6. *-commutative 72.1%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

   7. associate-*r/ 81.4%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

   8. *-commutative 81.4%

      \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   9. distribute-rgt1-in 62.4%

      \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   10. fma-def 81.4%

       \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

   11. +-commutative 81.4%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   12. +-commutative 81.4%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   13. cube-unmult 81.4%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

   14. +-commutative 81.4%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

3. Simplified 81.4%

   \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ 72.2%

      \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   2. *-commutative 72.2%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

   3. fma-udef 55.3%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   4. cube-mult 55.3%

      \[\leadsto \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   5. distribute-rgt1-in 72.1%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   6. *-commutative 72.1%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   7. +-commutative 72.1%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

   8. associate-+r+ 72.1%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

   9. frac-times 88.2%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   10. *-commutative 88.2%

       \[\leadsto \color{blue}{\frac{y}{y + \left(x + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   11. clear-num 88.2%

       \[\leadsto \color{blue}{\frac{1}{\frac{y + \left(x + 1\right)}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

   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 99.7%

       \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{y + \left(x + 1\right)}{y} \cdot \left(x + y\right)}} \]

   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 53.8%

   \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{1 + y}} \]
7. Step-by-step derivation

   1. +-commutative 53.8%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{y + 1}} \]

8. Simplified 53.8%

   \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{y + 1}} \]

9. Taylor expanded in y around 0 3.6%

   \[\leadsto \color{blue}{1} \]

10. Final simplification 3.6%

    \[\leadsto 1 \]

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 2023320 
(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))))