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

Percentage Accurate: 69.3% → 99.8%

Time: 16.7s

Alternatives: 20

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

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 89.3%

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

   2. associate-+r+ 89.3%

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

   3. associate-*l/ 86.0%

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

   4. times-frac 99.8%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{\left(x + y\right) \cdot \left(y + \left(x + \left(x + 1\right)\right)\right)}\\ \mathbf{if}\;y \leq 3.4 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x + y}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.29:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+136}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* (+ x y) (+ y (+ x (+ x 1.0)))))))
   (if (<= y 3.4e-74)
     (/ (/ y (+ x (+ y 1.0))) (+ x y))
     (if (<= y 1.4e-7)
       t_0
       (if (<= y 0.29)
         (/ y (* x x))
         (if (<= y 2.25e+136) t_0 (* (/ x (+ x y)) (/ 1.0 (+ x y)))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / ((x + y) * (y + (x + (x + 1.0))));
	double tmp;
	if (y <= 3.4e-74) {
		tmp = (y / (x + (y + 1.0))) / (x + y);
	} else if (y <= 1.4e-7) {
		tmp = t_0;
	} else if (y <= 0.29) {
		tmp = y / (x * x);
	} else if (y <= 2.25e+136) {
		tmp = t_0;
	} else {
		tmp = (x / (x + y)) * (1.0 / (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) :: t_0
    real(8) :: tmp
    t_0 = x / ((x + y) * (y + (x + (x + 1.0d0))))
    if (y <= 3.4d-74) then
        tmp = (y / (x + (y + 1.0d0))) / (x + y)
    else if (y <= 1.4d-7) then
        tmp = t_0
    else if (y <= 0.29d0) then
        tmp = y / (x * x)
    else if (y <= 2.25d+136) then
        tmp = t_0
    else
        tmp = (x / (x + y)) * (1.0d0 / (x + y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / ((x + y) * (y + (x + (x + 1.0))));
	double tmp;
	if (y <= 3.4e-74) {
		tmp = (y / (x + (y + 1.0))) / (x + y);
	} else if (y <= 1.4e-7) {
		tmp = t_0;
	} else if (y <= 0.29) {
		tmp = y / (x * x);
	} else if (y <= 2.25e+136) {
		tmp = t_0;
	} else {
		tmp = (x / (x + y)) * (1.0 / (x + y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / ((x + y) * (y + (x + (x + 1.0))))
	tmp = 0
	if y <= 3.4e-74:
		tmp = (y / (x + (y + 1.0))) / (x + y)
	elif y <= 1.4e-7:
		tmp = t_0
	elif y <= 0.29:
		tmp = y / (x * x)
	elif y <= 2.25e+136:
		tmp = t_0
	else:
		tmp = (x / (x + y)) * (1.0 / (x + y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(Float64(x + y) * Float64(y + Float64(x + Float64(x + 1.0)))))
	tmp = 0.0
	if (y <= 3.4e-74)
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) / Float64(x + y));
	elseif (y <= 1.4e-7)
		tmp = t_0;
	elseif (y <= 0.29)
		tmp = Float64(y / Float64(x * x));
	elseif (y <= 2.25e+136)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(1.0 / Float64(x + y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / ((x + y) * (y + (x + (x + 1.0))));
	tmp = 0.0;
	if (y <= 3.4e-74)
		tmp = (y / (x + (y + 1.0))) / (x + y);
	elseif (y <= 1.4e-7)
		tmp = t_0;
	elseif (y <= 0.29)
		tmp = y / (x * x);
	elseif (y <= 2.25e+136)
		tmp = t_0;
	else
		tmp = (x / (x + y)) * (1.0 / (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_] := Block[{t$95$0 = N[(x / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 3.4e-74], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.4e-7], t$95$0, If[LessEqual[y, 0.29], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.25e+136], t$95$0, N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 3.4000000000000001e-74

   1. Initial program 75.6%

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

      1. times-frac 90.0%

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

      2. /-rgt-identity 90.0%

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

      3. associate-/l/ 90.0%

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

      4. *-lft-identity 90.0%

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

      5. associate-+l+ 90.0%

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

   3. Simplified 90.0%

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

      1. associate-/r* 99.8%

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

      2. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*r/ 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in y around 0 69.0%

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

      1. *-commutative 69.0%

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

      2. div-inv 69.0%

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

      3. +-commutative 69.0%

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

   10. Applied egg-rr 69.0%

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

   if 3.4000000000000001e-74 < y < 1.4000000000000001e-7 or 0.28999999999999998 < y < 2.25e136

   1. Initial program 84.3%

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

      1. times-frac 96.7%

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

      2. associate-+r+ 96.7%

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

      3. associate-*l/ 96.7%

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

      4. times-frac 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. clear-num 99.4%

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

      2. frac-times 92.0%

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

      3. *-rgt-identity 92.0%

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

      4. +-commutative 92.0%

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

      5. div-inv 91.9%

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

      6. clear-num 92.0%

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

      7. +-commutative 92.0%

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

      8. +-commutative 92.0%

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

      9. associate-+l+ 92.0%

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

      10. +-commutative 92.0%

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

   5. Applied egg-rr 92.0%

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

   6. Taylor expanded in y around -inf 84.5%

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

      1. mul-1-neg 84.5%

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

      2. unsub-neg 84.5%

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

      3. neg-mul-1 84.5%

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

      4. unsub-neg 84.5%

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

      5. distribute-lft-in 84.5%

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

      6. metadata-eval 84.5%

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

      7. neg-mul-1 84.5%

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

      8. unsub-neg 84.5%

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

   8. Simplified 84.5%

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

   if 1.4000000000000001e-7 < y < 0.28999999999999998

   1. Initial program 37.1%

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

      1. associate-*r/ 37.1%

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

      2. *-commutative 37.1%

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

      3. distribute-rgt1-in 37.1%

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

      4. fma-def 37.1%

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

      5. cube-unmult 37.6%

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

   3. Simplified 37.6%

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

   4. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 100.0%

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

   6. Simplified 100.0%

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

   if 2.25e136 < y

   1. Initial program 61.7%

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

      1. times-frac 79.3%

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

      2. associate-+r+ 79.3%

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

      3. associate-*l/ 79.3%

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

      4. times-frac 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in y around inf 86.9%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x + y}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + \left(x + \left(x + 1\right)\right)\right)}\\ \mathbf{elif}\;y \leq 0.29:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+136}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + \left(x + \left(x + 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{x + y}\\ \end{array} \]

Alternative 3: 86.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}\\ \mathbf{if}\;y \leq 6.6 \cdot 10^{-73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + \left(x + \left(x + 1\right)\right)\right)}\\ \mathbf{elif}\;y \leq 0.142:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ y (+ x (+ y 1.0))) (+ x y))))
   (if (<= y 6.6e-73)
     t_0
     (if (<= y 2.9e-7)
       (/ x (* (+ x y) (+ y (+ x (+ x 1.0)))))
       (if (<= y 0.142) (/ y (* x x)) (* t_0 (/ x y)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y / (x + (y + 1.0))) / (x + y);
	double tmp;
	if (y <= 6.6e-73) {
		tmp = t_0;
	} else if (y <= 2.9e-7) {
		tmp = x / ((x + y) * (y + (x + (x + 1.0))));
	} else if (y <= 0.142) {
		tmp = y / (x * x);
	} else {
		tmp = t_0 * (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) :: t_0
    real(8) :: tmp
    t_0 = (y / (x + (y + 1.0d0))) / (x + y)
    if (y <= 6.6d-73) then
        tmp = t_0
    else if (y <= 2.9d-7) then
        tmp = x / ((x + y) * (y + (x + (x + 1.0d0))))
    else if (y <= 0.142d0) then
        tmp = y / (x * x)
    else
        tmp = t_0 * (x / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y / (x + (y + 1.0))) / (x + y);
	double tmp;
	if (y <= 6.6e-73) {
		tmp = t_0;
	} else if (y <= 2.9e-7) {
		tmp = x / ((x + y) * (y + (x + (x + 1.0))));
	} else if (y <= 0.142) {
		tmp = y / (x * x);
	} else {
		tmp = t_0 * (x / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y / (x + (y + 1.0))) / (x + y)
	tmp = 0
	if y <= 6.6e-73:
		tmp = t_0
	elif y <= 2.9e-7:
		tmp = x / ((x + y) * (y + (x + (x + 1.0))))
	elif y <= 0.142:
		tmp = y / (x * x)
	else:
		tmp = t_0 * (x / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y / Float64(x + Float64(y + 1.0))) / Float64(x + y))
	tmp = 0.0
	if (y <= 6.6e-73)
		tmp = t_0;
	elseif (y <= 2.9e-7)
		tmp = Float64(x / Float64(Float64(x + y) * Float64(y + Float64(x + Float64(x + 1.0)))));
	elseif (y <= 0.142)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(t_0 * Float64(x / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (y / (x + (y + 1.0))) / (x + y);
	tmp = 0.0;
	if (y <= 6.6e-73)
		tmp = t_0;
	elseif (y <= 2.9e-7)
		tmp = x / ((x + y) * (y + (x + (x + 1.0))));
	elseif (y <= 0.142)
		tmp = y / (x * x);
	else
		tmp = t_0 * (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_] := Block[{t$95$0 = N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 6.6e-73], t$95$0, If[LessEqual[y, 2.9e-7], N[(x / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.142], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(x / y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 6.60000000000000007e-73

   1. Initial program 75.6%

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

      1. times-frac 90.0%

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

      2. /-rgt-identity 90.0%

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

      3. associate-/l/ 90.0%

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

      4. *-lft-identity 90.0%

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

      5. associate-+l+ 90.0%

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

   3. Simplified 90.0%

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

      1. associate-/r* 99.8%

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

      2. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*r/ 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in y around 0 69.0%

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

      1. *-commutative 69.0%

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

      2. div-inv 69.0%

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

      3. +-commutative 69.0%

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

   10. Applied egg-rr 69.0%

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

   if 6.60000000000000007e-73 < y < 2.8999999999999998e-7

   1. Initial program 99.5%

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

      1. times-frac 99.3%

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

      2. associate-+r+ 99.3%

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

      3. associate-*l/ 99.5%

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

      4. times-frac 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. clear-num 99.7%

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

      2. frac-times 99.7%

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

      3. *-rgt-identity 99.7%

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

      4. +-commutative 99.7%

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

      5. div-inv 99.6%

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

      6. clear-num 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. associate-+l+ 99.8%

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

      10. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around -inf 90.4%

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

      1. mul-1-neg 90.4%

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

      2. unsub-neg 90.4%

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

      3. neg-mul-1 90.4%

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

      4. unsub-neg 90.4%

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

      5. distribute-lft-in 90.4%

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

      6. metadata-eval 90.4%

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

      7. neg-mul-1 90.4%

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

      8. unsub-neg 90.4%

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

   8. Simplified 90.4%

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

   if 2.8999999999999998e-7 < y < 0.141999999999999987

   1. Initial program 37.1%

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

      1. associate-*r/ 37.1%

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

      2. *-commutative 37.1%

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

      3. distribute-rgt1-in 37.1%

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

      4. fma-def 37.1%

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

      5. cube-unmult 37.6%

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

   3. Simplified 37.6%

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

   4. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 100.0%

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

   6. Simplified 100.0%

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

   if 0.141999999999999987 < y

   1. Initial program 63.4%

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

      1. times-frac 83.6%

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

      2. associate-+r+ 83.6%

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

      3. associate-*l/ 83.5%

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

      4. times-frac 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 83.9%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x + y}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + \left(x + \left(x + 1\right)\right)\right)}\\ \mathbf{elif}\;y \leq 0.142:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 4: 97.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x + \left(y + 1\right)}\\ \mathbf{if}\;x \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;t_0 \cdot \frac{1}{x + y}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-16}:\\ \;\;\;\;t_0 \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{\frac{y}{y + 1}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ x (+ y 1.0)))))
   (if (<= x -1.32e+154)
     (* t_0 (/ 1.0 (+ x y)))
     (if (<= x -1e-16)
       (* t_0 (/ x (* (+ x y) (+ x y))))
       (* (/ x (+ x y)) (/ (/ y (+ y 1.0)) (+ x y)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x + (y + 1.0));
	double tmp;
	if (x <= -1.32e+154) {
		tmp = t_0 * (1.0 / (x + y));
	} else if (x <= -1e-16) {
		tmp = t_0 * (x / ((x + y) * (x + y)));
	} else {
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (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) :: t_0
    real(8) :: tmp
    t_0 = y / (x + (y + 1.0d0))
    if (x <= (-1.32d+154)) then
        tmp = t_0 * (1.0d0 / (x + y))
    else if (x <= (-1d-16)) then
        tmp = t_0 * (x / ((x + y) * (x + y)))
    else
        tmp = (x / (x + y)) * ((y / (y + 1.0d0)) / (x + y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (x + (y + 1.0));
	double tmp;
	if (x <= -1.32e+154) {
		tmp = t_0 * (1.0 / (x + y));
	} else if (x <= -1e-16) {
		tmp = t_0 * (x / ((x + y) * (x + y)));
	} else {
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (x + y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (x + (y + 1.0))
	tmp = 0
	if x <= -1.32e+154:
		tmp = t_0 * (1.0 / (x + y))
	elif x <= -1e-16:
		tmp = t_0 * (x / ((x + y) * (x + y)))
	else:
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (x + y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(x + Float64(y + 1.0)))
	tmp = 0.0
	if (x <= -1.32e+154)
		tmp = Float64(t_0 * Float64(1.0 / Float64(x + y)));
	elseif (x <= -1e-16)
		tmp = Float64(t_0 * Float64(x / Float64(Float64(x + y) * Float64(x + y))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(Float64(y / Float64(y + 1.0)) / Float64(x + y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (x + (y + 1.0));
	tmp = 0.0;
	if (x <= -1.32e+154)
		tmp = t_0 * (1.0 / (x + y));
	elseif (x <= -1e-16)
		tmp = t_0 * (x / ((x + y) * (x + y)));
	else
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (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_] := Block[{t$95$0 = N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.32e+154], N[(t$95$0 * N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1e-16], N[(t$95$0 * N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.31999999999999998e154

   1. Initial program 68.3%

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

      1. times-frac 86.3%

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

      2. /-rgt-identity 86.3%

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

      3. associate-/l/ 86.3%

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

      4. *-lft-identity 86.3%

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

      5. associate-+l+ 86.3%

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

   3. Simplified 86.3%

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

      1. associate-/r* 99.9%

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

      2. div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r/ 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in y around 0 91.7%

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

   if -1.31999999999999998e154 < x < -9.9999999999999998e-17

   1. Initial program 78.0%

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

      1. times-frac 92.2%

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

      2. /-rgt-identity 92.2%

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

      3. associate-/l/ 92.2%

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

      4. *-lft-identity 92.2%

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

      5. associate-+l+ 92.2%

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

   3. Simplified 92.2%

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

   if -9.9999999999999998e-17 < x

   1. Initial program 74.7%

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

      1. times-frac 89.4%

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

      2. associate-+r+ 89.4%

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

      3. associate-*l/ 84.4%

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

      4. times-frac 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 80.4%

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

      1. +-commutative 77.2%

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

   6. Simplified 80.4%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{1}{x + y}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-16}:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{\frac{y}{y + 1}}{x + y}\\ \end{array} \]

Alternative 5: 88.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -30000000000:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-168}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -30000000000.0)
   (* (/ y (+ x (+ y 1.0))) (/ 1.0 x))
   (if (<= x -3.3e-168)
     (* (/ x (* (+ x y) (+ x y))) (/ y (+ y 1.0)))
     (/ (/ x y) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -30000000000.0) {
		tmp = (y / (x + (y + 1.0))) * (1.0 / x);
	} else if (x <= -3.3e-168) {
		tmp = (x / ((x + y) * (x + y))) * (y / (y + 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 <= (-30000000000.0d0)) then
        tmp = (y / (x + (y + 1.0d0))) * (1.0d0 / x)
    else if (x <= (-3.3d-168)) then
        tmp = (x / ((x + y) * (x + y))) * (y / (y + 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 <= -30000000000.0) {
		tmp = (y / (x + (y + 1.0))) * (1.0 / x);
	} else if (x <= -3.3e-168) {
		tmp = (x / ((x + y) * (x + y))) * (y / (y + 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 <= -30000000000.0:
		tmp = (y / (x + (y + 1.0))) * (1.0 / x)
	elif x <= -3.3e-168:
		tmp = (x / ((x + y) * (x + y))) * (y / (y + 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 <= -30000000000.0)
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) * Float64(1.0 / x));
	elseif (x <= -3.3e-168)
		tmp = Float64(Float64(x / Float64(Float64(x + y) * Float64(x + y))) * Float64(y / Float64(y + 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 <= -30000000000.0)
		tmp = (y / (x + (y + 1.0))) * (1.0 / x);
	elseif (x <= -3.3e-168)
		tmp = (x / ((x + y) * (x + y))) * (y / (y + 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, -30000000000.0], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.3e-168], N[(N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y + 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 < -3e10

   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 89.5%

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

      2. /-rgt-identity 89.5%

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

      3. associate-/l/ 89.5%

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

      4. *-lft-identity 89.5%

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

      5. associate-+l+ 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in x around inf 83.6%

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

   if -3e10 < x < -3.3000000000000001e-168

   1. Initial program 86.7%

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

      1. times-frac 97.0%

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

      2. /-rgt-identity 97.0%

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

      3. associate-/l/ 97.0%

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

      4. *-lft-identity 97.0%

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

      5. associate-+l+ 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in x around 0 96.2%

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

      1. +-commutative 96.2%

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

   6. Simplified 96.2%

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

   if -3.3000000000000001e-168 < x

   1. Initial program 72.0%

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

      1. times-frac 87.2%

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

      2. associate-+r+ 87.2%

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

      3. associate-*l/ 83.9%

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

      4. times-frac 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. clear-num 99.6%

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

      2. frac-times 90.4%

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

      3. *-rgt-identity 90.4%

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

      4. +-commutative 90.4%

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

      5. div-inv 90.3%

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

      6. clear-num 90.4%

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

      7. +-commutative 90.4%

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

      8. +-commutative 90.4%

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

      9. associate-+l+ 90.4%

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

      10. +-commutative 90.4%

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

   5. Applied egg-rr 90.4%

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

   6. Taylor expanded in x around 0 54.1%

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

      1. associate-/r* 56.1%

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

      2. +-commutative 56.1%

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

   8. Simplified 56.1%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -30000000000:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-168}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 6: 73.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -3.25 \cdot 10^{-202}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{-179}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{elif}\;y \leq 22500000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* x x))))
   (if (<= y -3.25e-202)
     t_0
     (if (<= y 8.6e-179)
       (/ y x)
       (if (<= y 1.75e-84)
         t_0
         (if (<= y 5.5e-10)
           (- (/ x y) x)
           (if (<= y 22500000000.0) t_0 (* (/ x y) (/ 1.0 y)))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x * x);
	double tmp;
	if (y <= -3.25e-202) {
		tmp = t_0;
	} else if (y <= 8.6e-179) {
		tmp = y / x;
	} else if (y <= 1.75e-84) {
		tmp = t_0;
	} else if (y <= 5.5e-10) {
		tmp = (x / y) - x;
	} else if (y <= 22500000000.0) {
		tmp = t_0;
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (x * x)
    if (y <= (-3.25d-202)) then
        tmp = t_0
    else if (y <= 8.6d-179) then
        tmp = y / x
    else if (y <= 1.75d-84) then
        tmp = t_0
    else if (y <= 5.5d-10) then
        tmp = (x / y) - x
    else if (y <= 22500000000.0d0) then
        tmp = t_0
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (x * x);
	double tmp;
	if (y <= -3.25e-202) {
		tmp = t_0;
	} else if (y <= 8.6e-179) {
		tmp = y / x;
	} else if (y <= 1.75e-84) {
		tmp = t_0;
	} else if (y <= 5.5e-10) {
		tmp = (x / y) - x;
	} else if (y <= 22500000000.0) {
		tmp = t_0;
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (x * x)
	tmp = 0
	if y <= -3.25e-202:
		tmp = t_0
	elif y <= 8.6e-179:
		tmp = y / x
	elif y <= 1.75e-84:
		tmp = t_0
	elif y <= 5.5e-10:
		tmp = (x / y) - x
	elif y <= 22500000000.0:
		tmp = t_0
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(x * x))
	tmp = 0.0
	if (y <= -3.25e-202)
		tmp = t_0;
	elseif (y <= 8.6e-179)
		tmp = Float64(y / x);
	elseif (y <= 1.75e-84)
		tmp = t_0;
	elseif (y <= 5.5e-10)
		tmp = Float64(Float64(x / y) - x);
	elseif (y <= 22500000000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (x * x);
	tmp = 0.0;
	if (y <= -3.25e-202)
		tmp = t_0;
	elseif (y <= 8.6e-179)
		tmp = y / x;
	elseif (y <= 1.75e-84)
		tmp = t_0;
	elseif (y <= 5.5e-10)
		tmp = (x / y) - x;
	elseif (y <= 22500000000.0)
		tmp = t_0;
	else
		tmp = (x / y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.25e-202], t$95$0, If[LessEqual[y, 8.6e-179], N[(y / x), $MachinePrecision], If[LessEqual[y, 1.75e-84], t$95$0, If[LessEqual[y, 5.5e-10], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[y, 22500000000.0], t$95$0, N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.24999999999999978e-202 or 8.60000000000000052e-179 < y < 1.7500000000000001e-84 or 5.4999999999999996e-10 < y < 2.25e10

   1. Initial program 76.9%

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

      1. associate-*r/ 82.5%

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

      2. *-commutative 82.5%

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

      3. distribute-rgt1-in 52.3%

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

      4. fma-def 82.5%

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

      5. cube-unmult 82.6%

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

   3. Simplified 82.6%

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

   4. Taylor expanded in x around inf 51.1%

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 51.1%

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

   6. Simplified 51.1%

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

   if -3.24999999999999978e-202 < y < 8.60000000000000052e-179

   1. Initial program 71.7%

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

      1. times-frac 84.1%

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

      2. /-rgt-identity 84.1%

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

      3. associate-/l/ 84.1%

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

      4. *-lft-identity 84.1%

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

      5. associate-+l+ 84.1%

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

   3. Simplified 84.1%

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

   4. Taylor expanded in y around 0 89.7%

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

   5. Taylor expanded in x around 0 69.9%

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

   if 1.7500000000000001e-84 < y < 5.4999999999999996e-10

   1. Initial program 99.5%

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

      1. times-frac 99.2%

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

      2. /-rgt-identity 99.2%

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

      3. associate-/l/ 99.2%

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

      4. *-lft-identity 99.2%

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

      5. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 49.2%

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

      1. distribute-rgt-in 49.2%

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

      2. *-lft-identity 49.2%

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

   6. Simplified 49.2%

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

   7. Taylor expanded in y around 0 49.2%

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

      1. neg-mul-1 49.2%

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

      2. unsub-neg 49.2%

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

   9. Simplified 49.2%

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

   if 2.25e10 < y

   1. Initial program 62.2%

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

      1. associate-*r/ 78.3%

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

      2. *-commutative 78.3%

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

      3. distribute-rgt1-in 72.8%

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

      4. fma-def 78.2%

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

      5. cube-unmult 78.3%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in y around inf 73.7%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 73.7%

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

   6. Simplified 73.7%

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

      1. associate-/r* 79.1%

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

      2. div-inv 79.1%

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

   8. Applied egg-rr 79.1%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.25 \cdot 10^{-202}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{-179}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-84}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{elif}\;y \leq 22500000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]

Alternative 7: 83.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x + \left(y + 1\right)}\\ \mathbf{if}\;x \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;t_0 \cdot \frac{1}{x + y}\\ \mathbf{elif}\;x \leq -1.9:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -2.45 \cdot 10^{-55}:\\ \;\;\;\;\frac{t_0}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ x (+ y 1.0)))))
   (if (<= x -1.32e+154)
     (* t_0 (/ 1.0 (+ x y)))
     (if (<= x -1.9)
       (* (/ x (* (+ x y) (+ x y))) (/ y x))
       (if (<= x -2.45e-55) (/ t_0 (+ x y)) (/ (/ x y) (+ y 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x + (y + 1.0));
	double tmp;
	if (x <= -1.32e+154) {
		tmp = t_0 * (1.0 / (x + y));
	} else if (x <= -1.9) {
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	} else if (x <= -2.45e-55) {
		tmp = t_0 / (x + y);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (x + (y + 1.0d0))
    if (x <= (-1.32d+154)) then
        tmp = t_0 * (1.0d0 / (x + y))
    else if (x <= (-1.9d0)) then
        tmp = (x / ((x + y) * (x + y))) * (y / x)
    else if (x <= (-2.45d-55)) then
        tmp = t_0 / (x + y)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (x + (y + 1.0));
	double tmp;
	if (x <= -1.32e+154) {
		tmp = t_0 * (1.0 / (x + y));
	} else if (x <= -1.9) {
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	} else if (x <= -2.45e-55) {
		tmp = t_0 / (x + y);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (x + (y + 1.0))
	tmp = 0
	if x <= -1.32e+154:
		tmp = t_0 * (1.0 / (x + y))
	elif x <= -1.9:
		tmp = (x / ((x + y) * (x + y))) * (y / x)
	elif x <= -2.45e-55:
		tmp = t_0 / (x + y)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(x + Float64(y + 1.0)))
	tmp = 0.0
	if (x <= -1.32e+154)
		tmp = Float64(t_0 * Float64(1.0 / Float64(x + y)));
	elseif (x <= -1.9)
		tmp = Float64(Float64(x / Float64(Float64(x + y) * Float64(x + y))) * Float64(y / x));
	elseif (x <= -2.45e-55)
		tmp = Float64(t_0 / Float64(x + y));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (x + (y + 1.0));
	tmp = 0.0;
	if (x <= -1.32e+154)
		tmp = t_0 * (1.0 / (x + y));
	elseif (x <= -1.9)
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	elseif (x <= -2.45e-55)
		tmp = t_0 / (x + y);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.32e+154], N[(t$95$0 * N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.9], N[(N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.45e-55], N[(t$95$0 / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.31999999999999998e154

   1. Initial program 68.3%

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

      1. times-frac 86.3%

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

      2. /-rgt-identity 86.3%

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

      3. associate-/l/ 86.3%

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

      4. *-lft-identity 86.3%

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

      5. associate-+l+ 86.3%

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

   3. Simplified 86.3%

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

      1. associate-/r* 99.9%

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

      2. div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r/ 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in y around 0 91.7%

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

   if -1.31999999999999998e154 < x < -1.8999999999999999

   1. Initial program 76.3%

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

      1. times-frac 91.7%

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

      2. /-rgt-identity 91.7%

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

      3. associate-/l/ 91.7%

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

      4. *-lft-identity 91.7%

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

      5. associate-+l+ 91.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in x around inf 82.1%

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

   if -1.8999999999999999 < x < -2.45000000000000018e-55

   1. Initial program 91.2%

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

      1. times-frac 99.5%

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

      2. /-rgt-identity 99.5%

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

      3. associate-/l/ 99.5%

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

      4. *-lft-identity 99.5%

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

      5. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

      1. associate-/r* 99.3%

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

      2. div-inv 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*r/ 99.5%

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

      3. +-commutative 99.5%

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

      4. +-commutative 99.5%

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

   7. Applied egg-rr 99.5%

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

   8. Taylor expanded in y around 0 65.2%

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

      1. *-commutative 65.2%

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

      2. div-inv 65.3%

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

      3. +-commutative 65.3%

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

   10. Applied egg-rr 65.3%

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

   if -2.45000000000000018e-55 < x

   1. Initial program 74.0%

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

      1. times-frac 88.8%

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

      2. associate-+r+ 88.8%

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

      3. associate-*l/ 84.2%

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

      4. times-frac 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. clear-num 99.6%

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

      2. frac-times 91.7%

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

      3. *-rgt-identity 91.7%

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

      4. +-commutative 91.7%

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

      5. div-inv 91.6%

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

      6. clear-num 91.6%

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

      7. +-commutative 91.6%

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

      8. +-commutative 91.6%

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

      9. associate-+l+ 91.6%

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

      10. +-commutative 91.6%

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

   5. Applied egg-rr 91.6%

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

   6. Taylor expanded in x around 0 56.2%

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

      1. associate-/r* 57.9%

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

      2. +-commutative 57.9%

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

   8. Simplified 57.9%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{1}{x + y}\\ \mathbf{elif}\;x \leq -1.9:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -2.45 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 8: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -90000000000000:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{\frac{y}{y + 1}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -90000000000000.0)
   (* (/ y (+ x (+ y 1.0))) (/ 1.0 x))
   (* (/ x (+ x y)) (/ (/ y (+ y 1.0)) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -90000000000000.0) {
		tmp = (y / (x + (y + 1.0))) * (1.0 / x);
	} else {
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (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 <= (-90000000000000.0d0)) then
        tmp = (y / (x + (y + 1.0d0))) * (1.0d0 / x)
    else
        tmp = (x / (x + y)) * ((y / (y + 1.0d0)) / (x + y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -90000000000000.0) {
		tmp = (y / (x + (y + 1.0))) * (1.0 / x);
	} else {
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (x + y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -90000000000000.0:
		tmp = (y / (x + (y + 1.0))) * (1.0 / x)
	else:
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (x + y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -90000000000000.0)
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) * Float64(1.0 / x));
	else
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(Float64(y / Float64(y + 1.0)) / 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 <= -90000000000000.0)
		tmp = (y / (x + (y + 1.0))) * (1.0 / x);
	else
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (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, -90000000000000.0], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9e13

   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 89.5%

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

      2. /-rgt-identity 89.5%

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

      3. associate-/l/ 89.5%

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

      4. *-lft-identity 89.5%

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

      5. associate-+l+ 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in x around inf 83.6%

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

   if -9e13 < 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 89.2%

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

      2. associate-+r+ 89.2%

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

      3. associate-*l/ 84.4%

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

      4. times-frac 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 80.9%

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

      1. +-commutative 77.3%

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

   6. Simplified 80.9%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -90000000000000:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{\frac{y}{y + 1}}{x + y}\\ \end{array} \]

Alternative 9: 82.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-56}:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{1}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -9.5e-56)
   (* (/ y (+ x (+ y 1.0))) (/ 1.0 (+ x y)))
   (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -9.5e-56) {
		tmp = (y / (x + (y + 1.0))) * (1.0 / (x + y));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-9.5d-56)) then
        tmp = (y / (x + (y + 1.0d0))) * (1.0d0 / (x + y))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -9.5e-56) {
		tmp = (y / (x + (y + 1.0))) * (1.0 / (x + y));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -9.5e-56:
		tmp = (y / (x + (y + 1.0))) * (1.0 / (x + y))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -9.5e-56)
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) * Float64(1.0 / Float64(x + y)));
	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 <= -9.5e-56)
		tmp = (y / (x + (y + 1.0))) * (1.0 / (x + y));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -9.5e-56], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.4999999999999991e-56

   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. 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. /-rgt-identity 90.2%

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

      3. associate-/l/ 90.2%

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

      4. *-lft-identity 90.2%

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

      5. associate-+l+ 90.2%

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

   3. Simplified 90.2%

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

      1. associate-/r* 99.8%

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

      2. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*r/ 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in y around 0 78.2%

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

   if -9.4999999999999991e-56 < x

   1. Initial program 73.8%

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

      1. times-frac 88.8%

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

      2. associate-+r+ 88.8%

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

      3. associate-*l/ 84.1%

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

      4. times-frac 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. clear-num 99.6%

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

      2. frac-times 91.6%

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

      3. *-rgt-identity 91.6%

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

      4. +-commutative 91.6%

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

      5. div-inv 91.6%

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

      6. clear-num 91.6%

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

      7. +-commutative 91.6%

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

      8. +-commutative 91.6%

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

      9. associate-+l+ 91.6%

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

      10. +-commutative 91.6%

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

   5. Applied egg-rr 91.6%

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

   6. Taylor expanded in x around 0 55.9%

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

      1. associate-/r* 57.7%

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

      2. +-commutative 57.7%

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

   8. Simplified 57.7%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-56}:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{1}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 10: 81.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.8e-55)
   (* (/ y (+ x (+ y 1.0))) (/ 1.0 x))
   (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.8e-55) {
		tmp = (y / (x + (y + 1.0))) * (1.0 / x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.8d-55)) then
        tmp = (y / (x + (y + 1.0d0))) * (1.0d0 / x)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.8e-55) {
		tmp = (y / (x + (y + 1.0))) * (1.0 / x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.8e-55:
		tmp = (y / (x + (y + 1.0))) * (1.0 / x)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.8e-55)
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) * Float64(1.0 / x));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.8e-55)
		tmp = (y / (x + (y + 1.0))) * (1.0 / x);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.8e-55], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8e-55

   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. 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. /-rgt-identity 90.2%

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

      3. associate-/l/ 90.2%

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

      4. *-lft-identity 90.2%

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

      5. associate-+l+ 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in x around inf 77.9%

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

   if -1.8e-55 < x

   1. Initial program 73.8%

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

      1. times-frac 88.8%

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

      2. associate-+r+ 88.8%

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

      3. associate-*l/ 84.1%

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

      4. times-frac 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. clear-num 99.6%

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

      2. frac-times 91.6%

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

      3. *-rgt-identity 91.6%

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

      4. +-commutative 91.6%

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

      5. div-inv 91.6%

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

      6. clear-num 91.6%

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

      7. +-commutative 91.6%

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

      8. +-commutative 91.6%

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

      9. associate-+l+ 91.6%

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

      10. +-commutative 91.6%

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

   5. Applied egg-rr 91.6%

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

   6. Taylor expanded in x around 0 55.9%

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

      1. associate-/r* 57.7%

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

      2. +-commutative 57.7%

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

   8. Simplified 57.7%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 11: 77.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq 10500000000000:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.2e+14)
   (/ y (* x x))
   (if (<= x 10500000000000.0) (/ x (* y (+ y 1.0))) (* (/ x y) (/ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.2e+14) {
		tmp = y / (x * x);
	} else if (x <= 10500000000000.0) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.2d+14)) then
        tmp = y / (x * x)
    else if (x <= 10500000000000.0d0) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.2e+14) {
		tmp = y / (x * x);
	} else if (x <= 10500000000000.0) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.2e+14:
		tmp = y / (x * x)
	elif x <= 10500000000000.0:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.2e+14)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= 10500000000000.0)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.2e+14)
		tmp = y / (x * x);
	elseif (x <= 10500000000000.0)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.2e+14], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 10500000000000.0], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2e14

   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-*r/ 80.2%

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

      2. *-commutative 80.2%

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

      3. distribute-rgt1-in 32.7%

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

      4. fma-def 80.2%

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

      5. cube-unmult 80.2%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in x around inf 80.4%

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 80.4%

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

   6. Simplified 80.4%

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

   if -2.2e14 < x < 1.05e13

   1. Initial program 81.4%

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

      1. times-frac 90.6%

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

      2. /-rgt-identity 90.6%

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

      3. associate-/l/ 90.6%

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

      4. *-lft-identity 90.6%

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

      5. associate-+l+ 90.6%

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

   3. Simplified 90.6%

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

   4. Taylor expanded in x around 0 73.2%

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

      1. +-commutative 73.2%

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

   6. Simplified 73.2%

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

   if 1.05e13 < x

   1. Initial program 64.1%

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

      1. associate-*r/ 76.5%

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

      2. *-commutative 76.5%

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

      3. distribute-rgt1-in 73.2%

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

      4. fma-def 76.5%

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

      5. cube-unmult 76.5%

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

   3. Simplified 76.5%

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

   4. Taylor expanded in y around inf 24.9%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 24.9%

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

   6. Simplified 24.9%

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

      1. associate-/r* 29.3%

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

      2. div-inv 29.2%

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

   8. Applied egg-rr 29.2%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq 10500000000000:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]

Alternative 12: 80.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.95 \cdot 10^{-55}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.95e-55)
   (/ y (* x (+ x 1.0)))
   (if (<= x 2e+16) (/ x (* y (+ y 1.0))) (* (/ x y) (/ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.95e-55) {
		tmp = y / (x * (x + 1.0));
	} else if (x <= 2e+16) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.95d-55)) then
        tmp = y / (x * (x + 1.0d0))
    else if (x <= 2d+16) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.95e-55) {
		tmp = y / (x * (x + 1.0));
	} else if (x <= 2e+16) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.95e-55:
		tmp = y / (x * (x + 1.0))
	elif x <= 2e+16:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.95e-55)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (x <= 2e+16)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.95e-55)
		tmp = y / (x * (x + 1.0));
	elseif (x <= 2e+16)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.95e-55], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e+16], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.9499999999999999e-55

   1. Initial program 74.3%

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

      1. times-frac 90.1%

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

      2. /-rgt-identity 90.1%

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

      3. associate-/l/ 90.1%

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

      4. *-lft-identity 90.1%

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

      5. associate-+l+ 90.1%

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

   3. Simplified 90.1%

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

   4. Taylor expanded in y around 0 75.9%

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

   if -2.9499999999999999e-55 < x < 2e16

   1. Initial program 80.8%

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

      1. times-frac 90.3%

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

      2. /-rgt-identity 90.3%

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

      3. associate-/l/ 90.3%

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

      4. *-lft-identity 90.3%

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

      5. associate-+l+ 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in x around 0 76.4%

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

      1. +-commutative 76.4%

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

   6. Simplified 76.4%

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

   if 2e16 < x

   1. Initial program 63.5%

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

      1. associate-*r/ 76.1%

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

      2. *-commutative 76.1%

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

      3. distribute-rgt1-in 72.8%

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

      4. fma-def 76.1%

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

      5. cube-unmult 76.2%

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

   3. Simplified 76.2%

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

   4. Taylor expanded in y around inf 25.2%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 25.2%

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

   6. Simplified 25.2%

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

      1. associate-/r* 29.6%

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

      2. div-inv 29.6%

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

   8. Applied egg-rr 29.6%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.95 \cdot 10^{-55}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]

Alternative 13: 81.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-55}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.9e-55) (* (/ 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 <= -1.9e-55) {
		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 <= (-1.9d-55)) 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 <= -1.9e-55) {
		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 <= -1.9e-55:
		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 <= -1.9e-55)
		tmp = Float64(Float64(1.0 / x) * Float64(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 <= -1.9e-55)
		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, -1.9e-55], N[(N[(1.0 / x), $MachinePrecision] * N[(y / 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 < -1.8999999999999998e-55

   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. 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. /-rgt-identity 90.2%

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

      3. associate-/l/ 90.2%

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

      4. *-lft-identity 90.2%

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

      5. associate-+l+ 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in y around 0 75.1%

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

      1. associate-/r* 77.4%

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

      2. div-inv 77.4%

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

      3. +-commutative 77.4%

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

   6. Applied egg-rr 77.4%

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

   if -1.8999999999999998e-55 < x

   1. Initial program 73.8%

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

      1. times-frac 88.8%

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

      2. associate-+r+ 88.8%

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

      3. associate-*l/ 84.1%

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

      4. times-frac 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. clear-num 99.6%

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

      2. frac-times 91.6%

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

      3. *-rgt-identity 91.6%

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

      4. +-commutative 91.6%

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

      5. div-inv 91.6%

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

      6. clear-num 91.6%

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

      7. +-commutative 91.6%

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

      8. +-commutative 91.6%

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

      9. associate-+l+ 91.6%

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

      10. +-commutative 91.6%

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

   5. Applied egg-rr 91.6%

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

   6. Taylor expanded in x around 0 55.9%

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

      1. associate-/r* 57.7%

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

      2. +-commutative 57.7%

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

   8. Simplified 57.7%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-55}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 14: 63.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.28 \cdot 10^{-79}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.28e-79) (/ y x) (if (<= y 0.75) (- (/ x y) x) (/ x (* y y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.28e-79) {
		tmp = y / x;
	} else if (y <= 0.75) {
		tmp = (x / y) - x;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.28d-79) then
        tmp = y / x
    else if (y <= 0.75d0) then
        tmp = (x / y) - x
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.28e-79) {
		tmp = y / x;
	} else if (y <= 0.75) {
		tmp = (x / y) - x;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.28e-79:
		tmp = y / x
	elif y <= 0.75:
		tmp = (x / y) - x
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.28e-79)
		tmp = Float64(y / x);
	elseif (y <= 0.75)
		tmp = Float64(Float64(x / y) - x);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.28e-79)
		tmp = y / x;
	elseif (y <= 0.75)
		tmp = (x / y) - x;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.28e-79], N[(y / x), $MachinePrecision], If[LessEqual[y, 0.75], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.28e-79

   1. Initial program 75.6%

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

      1. times-frac 90.0%

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

      2. /-rgt-identity 90.0%

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

      3. associate-/l/ 90.0%

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

      4. *-lft-identity 90.0%

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

      5. associate-+l+ 90.0%

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

   3. Simplified 90.0%

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

   4. Taylor expanded in y around 0 67.6%

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

   5. Taylor expanded in x around 0 35.7%

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

   if 1.28e-79 < y < 0.75

   1. Initial program 91.3%

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

      1. times-frac 99.4%

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

      2. /-rgt-identity 99.4%

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

      3. associate-/l/ 99.4%

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

      4. *-lft-identity 99.4%

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

      5. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 44.9%

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

      1. distribute-rgt-in 44.9%

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

      2. *-lft-identity 44.9%

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

   6. Simplified 44.9%

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

   7. Taylor expanded in y around 0 44.6%

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

      1. neg-mul-1 44.6%

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

      2. unsub-neg 44.6%

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

   9. Simplified 44.6%

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

   if 0.75 < y

   1. Initial program 63.4%

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

      1. associate-*r/ 78.9%

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

      2. *-commutative 78.9%

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

      3. distribute-rgt1-in 72.0%

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

      4. fma-def 78.9%

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

      5. cube-unmult 78.9%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in y around inf 71.7%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 71.7%

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

   6. Simplified 71.7%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.28 \cdot 10^{-79}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 15: 80.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-56}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -4.7e-56) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4.7e-56) {
		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 <= (-4.7d-56)) 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 <= -4.7e-56) {
		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 <= -4.7e-56:
		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 <= -4.7e-56)
		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 <= -4.7e-56)
		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, -4.7e-56], 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 2 regimes

2. if x < -4.7e-56

   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. 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. /-rgt-identity 90.2%

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

      3. associate-/l/ 90.2%

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

      4. *-lft-identity 90.2%

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

      5. associate-+l+ 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in y around 0 75.1%

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

   if -4.7e-56 < x

   1. Initial program 73.8%

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

      1. times-frac 88.8%

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

      2. associate-+r+ 88.8%

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

      3. associate-*l/ 84.1%

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

      4. times-frac 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. clear-num 99.6%

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

      2. frac-times 91.6%

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

      3. *-rgt-identity 91.6%

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

      4. +-commutative 91.6%

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

      5. div-inv 91.6%

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

      6. clear-num 91.6%

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

      7. +-commutative 91.6%

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

      8. +-commutative 91.6%

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

      9. associate-+l+ 91.6%

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

      10. +-commutative 91.6%

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

   5. Applied egg-rr 91.6%

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

   6. Taylor expanded in x around 0 55.9%

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

      1. associate-/r* 57.7%

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

      2. +-commutative 57.7%

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

   8. Simplified 57.7%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-56}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 16: 81.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.65e-55) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.65e-55) {
		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 <= (-1.65d-55)) 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 <= -1.65e-55) {
		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 <= -1.65e-55:
		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 <= -1.65e-55)
		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 <= -1.65e-55)
		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, -1.65e-55], 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 < -1.65e-55

   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. 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. /-rgt-identity 90.2%

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

      3. associate-/l/ 90.2%

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

      4. *-lft-identity 90.2%

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

      5. associate-+l+ 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in y around 0 75.1%

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

      1. *-un-lft-identity 75.1%

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

      2. times-frac 77.4%

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

      3. +-commutative 77.4%

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

   6. Applied egg-rr 77.4%

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

      1. associate-*l/ 77.4%

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

      2. *-lft-identity 77.4%

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

      3. +-commutative 77.4%

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

   8. Simplified 77.4%

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

   if -1.65e-55 < x

   1. Initial program 73.8%

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

      1. times-frac 88.8%

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

      2. associate-+r+ 88.8%

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

      3. associate-*l/ 84.1%

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

      4. times-frac 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. clear-num 99.6%

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

      2. frac-times 91.6%

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

      3. *-rgt-identity 91.6%

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

      4. +-commutative 91.6%

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

      5. div-inv 91.6%

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

      6. clear-num 91.6%

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

      7. +-commutative 91.6%

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

      8. +-commutative 91.6%

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

      9. associate-+l+ 91.6%

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

      10. +-commutative 91.6%

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

   5. Applied egg-rr 91.6%

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

   6. Taylor expanded in x around 0 55.9%

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

      1. associate-/r* 57.7%

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

      2. +-commutative 57.7%

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

   8. Simplified 57.7%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 17: 33.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-252}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} - x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -7.8e-252) (/ y x) (- (/ x y) x)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -7.8e-252) {
		tmp = y / x;
	} else {
		tmp = (x / y) - x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7.8d-252)) then
        tmp = y / x
    else
        tmp = (x / y) - x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -7.8e-252) {
		tmp = y / x;
	} else {
		tmp = (x / y) - x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -7.8e-252:
		tmp = y / x
	else:
		tmp = (x / y) - x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -7.8e-252)
		tmp = Float64(y / x);
	else
		tmp = Float64(Float64(x / y) - x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.8e-252)
		tmp = y / x;
	else
		tmp = (x / y) - x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -7.8e-252], N[(y / x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.7999999999999998e-252

   1. Initial program 77.2%

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

      1. times-frac 91.3%

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

      2. /-rgt-identity 91.3%

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

      3. associate-/l/ 91.3%

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

      4. *-lft-identity 91.3%

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

      5. associate-+l+ 91.3%

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

   3. Simplified 91.3%

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

   4. Taylor expanded in y around 0 60.5%

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

   5. Taylor expanded in x around 0 28.6%

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

   if -7.7999999999999998e-252 < x

   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 87.2%

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

      2. /-rgt-identity 87.2%

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

      3. associate-/l/ 87.2%

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

      4. *-lft-identity 87.2%

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

      5. associate-+l+ 87.2%

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

   3. Simplified 87.2%

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

   4. Taylor expanded in x around 0 49.0%

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

      1. distribute-rgt-in 49.0%

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

      2. *-lft-identity 49.0%

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

   6. Simplified 49.0%

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

   7. Taylor expanded in y around 0 13.3%

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

      1. neg-mul-1 13.3%

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

      2. unsub-neg 13.3%

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

   9. Simplified 13.3%

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

4. Final simplification 21.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-252}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} - x\\ \end{array} \]

Alternative 18: 64.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2400000000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2400000000000.0) (/ y (* x x)) (/ x (* y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2400000000000.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / (y * 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 <= (-2400000000000.0d0)) then
        tmp = y / (x * x)
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2400000000000.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2400000000000.0:
		tmp = y / (x * x)
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2400000000000.0)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2400000000000.0)
		tmp = y / (x * x);
	else
		tmp = x / (y * 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, -2400000000000.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4e12

   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-*r/ 80.2%

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

      2. *-commutative 80.2%

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

      3. distribute-rgt1-in 32.7%

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

      4. fma-def 80.2%

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

      5. cube-unmult 80.2%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in x around inf 80.4%

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 80.4%

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

   6. Simplified 80.4%

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

   if -2.4e12 < 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. associate-*r/ 84.4%

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

      2. *-commutative 84.4%

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

      3. distribute-rgt1-in 75.1%

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

      4. fma-def 84.4%

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

      5. cube-unmult 84.4%

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

   3. Simplified 84.4%

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

   4. Taylor expanded in y around inf 43.1%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 43.1%

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

   6. Simplified 43.1%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2400000000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 19: 4.3% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 89.3%

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

   2. associate-+r+ 89.3%

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

   3. associate-*l/ 86.0%

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

   4. times-frac 99.8%

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

3. Applied egg-rr 99.8%

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

4. Taylor expanded in y around inf 38.1%

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

5. Taylor expanded in x around inf 4.3%

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

6. Final simplification 4.3%

   \[\leadsto \frac{1}{x} \]

Alternative 20: 26.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{y}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 89.3%

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

   2. /-rgt-identity 89.3%

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

   3. associate-/l/ 89.3%

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

   4. *-lft-identity 89.3%

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

   5. associate-+l+ 89.3%

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

3. Simplified 89.3%

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

4. Taylor expanded in y around 0 55.0%

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

5. Taylor expanded in x around 0 26.2%

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

6. Final simplification 26.2%

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

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 2023195 
(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))))