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

Percentage Accurate: 68.3% → 99.8%

Time: 23.0s

Alternatives: 17

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.8%

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

   1. *-un-lft-identity 67.8%

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

   2. associate-*l* 67.8%

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

   3. times-frac 73.1%

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

   4. +-commutative 73.1%

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

   5. *-commutative 73.1%

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

   6. +-commutative 73.1%

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

   7. associate-+r+ 73.1%

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

   8. +-commutative 73.1%

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

   9. associate-+l+ 73.1%

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

4. Applied egg-rr 73.1%

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

   1. associate-/r* 75.9%

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

   2. associate-/l* 99.7%

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

   3. +-commutative 99.7%

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

6. Simplified 99.7%

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

   1. associate-*l/ 99.8%

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

   2. *-un-lft-identity 99.8%

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

   3. associate-/l* 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 95.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{-36}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + x} \cdot \frac{x}{x + 1}\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+98}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 6.6e-36)
   (* (/ (/ y (+ y x)) (+ y x)) (/ x (+ x 1.0)))
   (if (<= y 4.7e+98)
     (* x (/ y (* (* (+ y x) (+ y x)) (+ x (+ y 1.0)))))
     (/ (/ x (+ y (+ x 1.0))) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 6.6e-36) {
		tmp = ((y / (y + x)) / (y + x)) * (x / (x + 1.0));
	} else if (y <= 4.7e+98) {
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))));
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 6.6d-36) then
        tmp = ((y / (y + x)) / (y + x)) * (x / (x + 1.0d0))
    else if (y <= 4.7d+98) then
        tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0d0))))
    else
        tmp = (x / (y + (x + 1.0d0))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 6.6e-36) {
		tmp = ((y / (y + x)) / (y + x)) * (x / (x + 1.0));
	} else if (y <= 4.7e+98) {
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))));
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 6.6e-36:
		tmp = ((y / (y + x)) / (y + x)) * (x / (x + 1.0))
	elif y <= 4.7e+98:
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))))
	else:
		tmp = (x / (y + (x + 1.0))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 6.6e-36)
		tmp = Float64(Float64(Float64(y / Float64(y + x)) / Float64(y + x)) * Float64(x / Float64(x + 1.0)));
	elseif (y <= 4.7e+98)
		tmp = Float64(x * Float64(y / Float64(Float64(Float64(y + x) * Float64(y + x)) * Float64(x + Float64(y + 1.0)))));
	else
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.6e-36)
		tmp = ((y / (y + x)) / (y + x)) * (x / (x + 1.0));
	elseif (y <= 4.7e+98)
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))));
	else
		tmp = (x / (y + (x + 1.0))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 6.6e-36], N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.7e+98], N[(x * N[(y / N[(N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.59999999999999981e-36

   1. Initial program 70.0%

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

      1. *-un-lft-identity 70.0%

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

      2. associate-*l* 70.0%

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

      3. times-frac 73.5%

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

      4. +-commutative 73.5%

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

      5. *-commutative 73.5%

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

      6. +-commutative 73.5%

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

      7. associate-+r+ 73.5%

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

      8. +-commutative 73.5%

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

      9. associate-+l+ 73.5%

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

   4. Applied egg-rr 73.5%

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

      1. associate-/r* 75.8%

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

      2. associate-/l* 99.6%

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

      3. +-commutative 99.6%

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

   6. Simplified 99.6%

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

      1. frac-times 93.9%

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

      2. *-un-lft-identity 93.9%

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

      3. *-commutative 93.9%

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

      4. +-commutative 93.9%

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

      5. associate-*r/ 93.8%

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

      6. *-commutative 93.8%

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

      7. associate-/r* 99.8%

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

      8. frac-times 93.9%

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

      9. *-commutative 93.9%

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

      10. +-commutative 93.9%

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

   8. Applied egg-rr 93.9%

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

      1. times-frac 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in y around 0 80.8%

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

   if 6.59999999999999981e-36 < y < 4.6999999999999997e98

   1. Initial program 89.5%

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

      1. associate-/l* 93.2%

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

      2. associate-+l+ 93.2%

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

   3. Simplified 93.2%

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

   if 4.6999999999999997e98 < y

   1. Initial program 47.3%

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

      1. *-un-lft-identity 47.3%

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

      2. associate-*l* 47.3%

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

      3. times-frac 62.2%

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

      4. +-commutative 62.2%

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

      5. *-commutative 62.2%

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

      6. +-commutative 62.2%

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

      7. associate-+r+ 62.2%

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

      8. +-commutative 62.2%

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

      9. associate-+l+ 62.2%

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

   4. Applied egg-rr 62.2%

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

      1. associate-/r* 66.5%

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

      2. associate-/l* 99.8%

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

      3. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in y around inf 74.1%

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

      1. associate-*l/ 74.1%

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

      2. *-un-lft-identity 74.1%

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

   9. Applied egg-rr 74.1%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{-36}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + x} \cdot \frac{x}{x + 1}\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+98}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -7 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\left(y + x\right) \cdot \frac{t\_0}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -7e+154)
     (/ 1.0 (* (+ y x) (/ t_0 y)))
     (* (/ x (+ y x)) (/ y (* (+ y x) t_0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -7e+154) {
		tmp = 1.0 / ((y + x) * (t_0 / y));
	} else {
		tmp = (x / (y + x)) * (y / ((y + x) * t_0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (x <= (-7d+154)) then
        tmp = 1.0d0 / ((y + x) * (t_0 / y))
    else
        tmp = (x / (y + x)) * (y / ((y + x) * t_0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -7e+154) {
		tmp = 1.0 / ((y + x) * (t_0 / y));
	} else {
		tmp = (x / (y + x)) * (y / ((y + x) * t_0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -7e+154:
		tmp = 1.0 / ((y + x) * (t_0 / y))
	else:
		tmp = (x / (y + x)) * (y / ((y + x) * t_0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -7e+154)
		tmp = Float64(1.0 / Float64(Float64(y + x) * Float64(t_0 / y)));
	else
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(Float64(y + x) * t_0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -7e+154)
		tmp = 1.0 / ((y + x) * (t_0 / y));
	else
		tmp = (x / (y + x)) * (y / ((y + x) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7e+154], N[(1.0 / N[(N[(y + x), $MachinePrecision] * N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.00000000000000041e154

   1. Initial program 44.1%

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

      1. *-un-lft-identity 44.1%

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

      2. associate-*l* 44.1%

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

      3. times-frac 44.1%

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

      4. +-commutative 44.1%

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

      5. *-commutative 44.1%

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

      6. +-commutative 44.1%

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

      7. associate-+r+ 44.1%

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

      8. +-commutative 44.1%

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

      9. associate-+l+ 44.1%

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

   4. Applied egg-rr 44.1%

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

      1. associate-/r* 50.8%

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

      2. associate-/l* 99.8%

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

      3. +-commutative 99.8%

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

   6. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. clear-num 99.6%

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

      3. frac-times 99.7%

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

      4. metadata-eval 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in y around 0 84.4%

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

   if -7.00000000000000041e154 < x

   1. Initial program 71.2%

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

      1. associate-*l* 71.2%

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

      2. times-frac 94.7%

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

      3. +-commutative 94.7%

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

      4. +-commutative 94.7%

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

      5. associate-+r+ 94.7%

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

      6. +-commutative 94.7%

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

      7. associate-+l+ 94.7%

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

   4. Applied egg-rr 94.7%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\left(y + x\right) \cdot \frac{y + \left(x + 1\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + x}\\ \mathbf{elif}\;x \leq -7.4 \cdot 10^{-96}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -6.5e+154)
   (/ (/ y (+ y x)) (+ y x))
   (if (<= x -7.4e-96)
     (/ y (* (+ y x) (+ y (+ x 1.0))))
     (/ (/ x (+ y 1.0)) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -6.5e+154) {
		tmp = (y / (y + x)) / (y + x);
	} else if (x <= -7.4e-96) {
		tmp = y / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-6.5d+154)) then
        tmp = (y / (y + x)) / (y + x)
    else if (x <= (-7.4d-96)) then
        tmp = y / ((y + x) * (y + (x + 1.0d0)))
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -6.5e+154) {
		tmp = (y / (y + x)) / (y + x);
	} else if (x <= -7.4e-96) {
		tmp = y / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -6.5e+154:
		tmp = (y / (y + x)) / (y + x)
	elif x <= -7.4e-96:
		tmp = y / ((y + x) * (y + (x + 1.0)))
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -6.5e+154)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(y + x));
	elseif (x <= -7.4e-96)
		tmp = Float64(y / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.5e+154)
		tmp = (y / (y + x)) / (y + x);
	elseif (x <= -7.4e-96)
		tmp = y / ((y + x) * (y + (x + 1.0)));
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -6.5e+154], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7.4e-96], N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.5000000000000005e154

   1. Initial program 44.1%

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

      1. *-un-lft-identity 44.1%

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

      2. associate-*l* 44.1%

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

      3. times-frac 44.1%

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

      4. +-commutative 44.1%

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

      5. *-commutative 44.1%

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

      6. +-commutative 44.1%

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

      7. associate-+r+ 44.1%

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

      8. +-commutative 44.1%

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

      9. associate-+l+ 44.1%

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

   4. Applied egg-rr 44.1%

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

      1. associate-/r* 50.8%

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

      2. associate-/l* 99.8%

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

      3. +-commutative 99.8%

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

   6. Simplified 99.8%

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

      1. frac-times 73.9%

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

      2. *-un-lft-identity 73.9%

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

      3. *-commutative 73.9%

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

      4. +-commutative 73.9%

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

      5. associate-*r/ 73.9%

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

      6. *-commutative 73.9%

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

      7. associate-/r* 99.7%

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

      8. frac-times 73.9%

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

      9. *-commutative 73.9%

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

      10. +-commutative 73.9%

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

   8. Applied egg-rr 73.9%

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

      1. times-frac 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in x around inf 84.4%

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

   if -6.5000000000000005e154 < x < -7.39999999999999972e-96

   1. Initial program 73.1%

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

      1. associate-*l* 73.1%

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

      2. times-frac 96.3%

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

      3. +-commutative 96.3%

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

      4. +-commutative 96.3%

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

      5. associate-+r+ 96.3%

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

      6. +-commutative 96.3%

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

      7. associate-+l+ 96.3%

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

   4. Applied egg-rr 96.3%

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

   5. Taylor expanded in x around inf 74.1%

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

   if -7.39999999999999972e-96 < x

   1. Initial program 70.5%

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

      1. associate-/l* 80.7%

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

      2. associate-+l+ 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in x around 0 57.2%

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

      1. +-commutative 57.2%

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

   7. Simplified 57.2%

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

      1. *-un-lft-identity 57.2%

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

      2. times-frac 58.4%

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

   9. Applied egg-rr 58.4%

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

       1. associate-*l/ 58.4%

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

       2. *-un-lft-identity 58.4%

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

   11. Applied egg-rr 58.4%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + x}\\ \mathbf{elif}\;x \leq -7.4 \cdot 10^{-96}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{t\_0}\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -1.35e+154)
     (* (/ 1.0 (+ y x)) (/ y t_0))
     (if (<= x -2.9e-95) (/ y (* (+ y x) t_0)) (/ (/ x (+ y 1.0)) y)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.35e+154) {
		tmp = (1.0 / (y + x)) * (y / t_0);
	} else if (x <= -2.9e-95) {
		tmp = y / ((y + x) * 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 + 1.0d0)
    if (x <= (-1.35d+154)) then
        tmp = (1.0d0 / (y + x)) * (y / t_0)
    else if (x <= (-2.9d-95)) then
        tmp = y / ((y + x) * 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 + 1.0);
	double tmp;
	if (x <= -1.35e+154) {
		tmp = (1.0 / (y + x)) * (y / t_0);
	} else if (x <= -2.9e-95) {
		tmp = y / ((y + x) * 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 + 1.0)
	tmp = 0
	if x <= -1.35e+154:
		tmp = (1.0 / (y + x)) * (y / t_0)
	elif x <= -2.9e-95:
		tmp = y / ((y + x) * 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 + 1.0))
	tmp = 0.0
	if (x <= -1.35e+154)
		tmp = Float64(Float64(1.0 / Float64(y + x)) * Float64(y / t_0));
	elseif (x <= -2.9e-95)
		tmp = Float64(y / Float64(Float64(y + x) * t_0));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -1.35e+154)
		tmp = (1.0 / (y + x)) * (y / t_0);
	elseif (x <= -2.9e-95)
		tmp = y / ((y + x) * 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 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e+154], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.9e-95], N[(y / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35000000000000003e154

   1. Initial program 44.1%

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

      1. *-un-lft-identity 44.1%

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

      2. associate-*l* 44.1%

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

      3. times-frac 44.1%

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

      4. +-commutative 44.1%

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

      5. *-commutative 44.1%

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

      6. +-commutative 44.1%

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

      7. associate-+r+ 44.1%

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

      8. +-commutative 44.1%

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

      9. associate-+l+ 44.1%

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

   4. Applied egg-rr 44.1%

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

      1. associate-/r* 50.8%

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

      2. associate-/l* 99.8%

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

      3. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in y around 0 84.4%

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

   if -1.35000000000000003e154 < x < -2.90000000000000002e-95

   1. Initial program 73.1%

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

      1. associate-*l* 73.1%

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

      2. times-frac 96.3%

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

      3. +-commutative 96.3%

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

      4. +-commutative 96.3%

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

      5. associate-+r+ 96.3%

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

      6. +-commutative 96.3%

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

      7. associate-+l+ 96.3%

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

   4. Applied egg-rr 96.3%

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

   5. Taylor expanded in x around inf 74.1%

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

   if -2.90000000000000002e-95 < x

   1. Initial program 70.5%

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

      1. associate-/l* 80.7%

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

      2. associate-+l+ 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in x around 0 57.2%

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

      1. +-commutative 57.2%

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

   7. Simplified 57.2%

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

      1. *-un-lft-identity 57.2%

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

      2. times-frac 58.4%

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

   9. Applied egg-rr 58.4%

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

       1. associate-*l/ 58.4%

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

       2. *-un-lft-identity 58.4%

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

   11. Applied egg-rr 58.4%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{y + \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{+155}:\\ \;\;\;\;\frac{1}{\left(y + x\right) \cdot \frac{t\_0}{y}}\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -1.4e+155)
     (/ 1.0 (* (+ y x) (/ t_0 y)))
     (if (<= x -2.05e-95) (/ y (* (+ y x) t_0)) (/ (/ x (+ y 1.0)) y)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.4e+155) {
		tmp = 1.0 / ((y + x) * (t_0 / y));
	} else if (x <= -2.05e-95) {
		tmp = y / ((y + x) * 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 + 1.0d0)
    if (x <= (-1.4d+155)) then
        tmp = 1.0d0 / ((y + x) * (t_0 / y))
    else if (x <= (-2.05d-95)) then
        tmp = y / ((y + x) * 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 + 1.0);
	double tmp;
	if (x <= -1.4e+155) {
		tmp = 1.0 / ((y + x) * (t_0 / y));
	} else if (x <= -2.05e-95) {
		tmp = y / ((y + x) * 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 + 1.0)
	tmp = 0
	if x <= -1.4e+155:
		tmp = 1.0 / ((y + x) * (t_0 / y))
	elif x <= -2.05e-95:
		tmp = y / ((y + x) * 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 + 1.0))
	tmp = 0.0
	if (x <= -1.4e+155)
		tmp = Float64(1.0 / Float64(Float64(y + x) * Float64(t_0 / y)));
	elseif (x <= -2.05e-95)
		tmp = Float64(y / Float64(Float64(y + x) * t_0));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -1.4e+155)
		tmp = 1.0 / ((y + x) * (t_0 / y));
	elseif (x <= -2.05e-95)
		tmp = y / ((y + x) * 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 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.4e+155], N[(1.0 / N[(N[(y + x), $MachinePrecision] * N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.05e-95], N[(y / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.40000000000000008e155

   1. Initial program 44.1%

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

      1. *-un-lft-identity 44.1%

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

      2. associate-*l* 44.1%

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

      3. times-frac 44.1%

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

      4. +-commutative 44.1%

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

      5. *-commutative 44.1%

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

      6. +-commutative 44.1%

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

      7. associate-+r+ 44.1%

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

      8. +-commutative 44.1%

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

      9. associate-+l+ 44.1%

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

   4. Applied egg-rr 44.1%

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

      1. associate-/r* 50.8%

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

      2. associate-/l* 99.8%

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

      3. +-commutative 99.8%

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

   6. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. clear-num 99.6%

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

      3. frac-times 99.7%

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

      4. metadata-eval 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in y around 0 84.4%

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

   if -1.40000000000000008e155 < x < -2.0499999999999999e-95

   1. Initial program 73.1%

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

      1. associate-*l* 73.1%

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

      2. times-frac 96.3%

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

      3. +-commutative 96.3%

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

      4. +-commutative 96.3%

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

      5. associate-+r+ 96.3%

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

      6. +-commutative 96.3%

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

      7. associate-+l+ 96.3%

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

   4. Applied egg-rr 96.3%

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

   5. Taylor expanded in x around inf 74.1%

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

   if -2.0499999999999999e-95 < x

   1. Initial program 70.5%

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

      1. associate-/l* 80.7%

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

      2. associate-+l+ 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in x around 0 57.2%

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

      1. +-commutative 57.2%

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

   7. Simplified 57.2%

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

      1. *-un-lft-identity 57.2%

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

      2. times-frac 58.4%

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

   9. Applied egg-rr 58.4%

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

       1. associate-*l/ 58.4%

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

       2. *-un-lft-identity 58.4%

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

   11. Applied egg-rr 58.4%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+155}:\\ \;\;\;\;\frac{1}{\left(y + x\right) \cdot \frac{y + \left(x + 1\right)}{y}}\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.35 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + x} \cdot \frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.35e+36)
   (* (/ (/ y (+ y x)) (+ y x)) (/ x (+ x 1.0)))
   (/ (/ x (+ y (+ x 1.0))) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.35e+36) {
		tmp = ((y / (y + x)) / (y + x)) * (x / (x + 1.0));
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.35d+36) then
        tmp = ((y / (y + x)) / (y + x)) * (x / (x + 1.0d0))
    else
        tmp = (x / (y + (x + 1.0d0))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.35e+36) {
		tmp = ((y / (y + x)) / (y + x)) * (x / (x + 1.0));
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.35e+36:
		tmp = ((y / (y + x)) / (y + x)) * (x / (x + 1.0))
	else:
		tmp = (x / (y + (x + 1.0))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.35e+36)
		tmp = Float64(Float64(Float64(y / Float64(y + x)) / Float64(y + x)) * Float64(x / Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.35e+36)
		tmp = ((y / (y + x)) / (y + x)) * (x / (x + 1.0));
	else
		tmp = (x / (y + (x + 1.0))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 2.35e+36], N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.34999999999999994e36

   1. Initial program 72.3%

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

      1. *-un-lft-identity 72.3%

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

      2. associate-*l* 72.3%

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

      3. times-frac 75.5%

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

      4. +-commutative 75.5%

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

      5. *-commutative 75.5%

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

      6. +-commutative 75.5%

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

      7. associate-+r+ 75.5%

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

      8. +-commutative 75.5%

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

      9. associate-+l+ 75.5%

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

   4. Applied egg-rr 75.5%

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

      1. associate-/r* 77.6%

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

      2. associate-/l* 99.6%

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

      3. +-commutative 99.6%

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

   6. Simplified 99.6%

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

      1. frac-times 94.3%

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

      2. *-un-lft-identity 94.3%

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

      3. *-commutative 94.3%

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

      4. +-commutative 94.3%

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

      5. associate-*r/ 94.3%

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

      6. *-commutative 94.3%

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

      7. associate-/r* 99.8%

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

      8. frac-times 94.3%

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

      9. *-commutative 94.3%

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

      10. +-commutative 94.3%

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

   8. Applied egg-rr 94.3%

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

      1. times-frac 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in y around 0 81.3%

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

   if 2.34999999999999994e36 < y

   1. Initial program 54.1%

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

      1. *-un-lft-identity 54.1%

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

      2. associate-*l* 54.1%

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

      3. times-frac 65.8%

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

      4. +-commutative 65.8%

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

      5. *-commutative 65.8%

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

      6. +-commutative 65.8%

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

      7. associate-+r+ 65.8%

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

      8. +-commutative 65.8%

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

      9. associate-+l+ 65.8%

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

   4. Applied egg-rr 65.8%

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

      1. associate-/r* 70.7%

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

      2. associate-/l* 99.7%

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

      3. +-commutative 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in y around inf 72.7%

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

      1. associate-*l/ 72.7%

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

      2. *-un-lft-identity 72.7%

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

   9. Applied egg-rr 72.7%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.35 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + x} \cdot \frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.8%

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

   1. *-un-lft-identity 67.8%

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

   2. associate-*l* 67.8%

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

   3. times-frac 73.1%

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

   4. +-commutative 73.1%

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

   5. *-commutative 73.1%

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

   6. +-commutative 73.1%

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

   7. associate-+r+ 73.1%

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

   8. +-commutative 73.1%

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

   9. associate-+l+ 73.1%

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

4. Applied egg-rr 73.1%

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

   1. associate-/r* 75.9%

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

   2. associate-/l* 99.7%

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

   3. +-commutative 99.7%

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

6. Simplified 99.7%

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

   1. frac-times 92.1%

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

   2. *-un-lft-identity 92.1%

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

   3. *-commutative 92.1%

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

   4. +-commutative 92.1%

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

   5. associate-*r/ 92.1%

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

   6. *-commutative 92.1%

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

   7. associate-/r* 99.8%

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

   8. frac-times 92.1%

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

   9. *-commutative 92.1%

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

   10. +-commutative 92.1%

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

8. Applied egg-rr 92.1%

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

   1. times-frac 99.7%

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

10. Simplified 99.7%

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

11. Final simplification 99.7%

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

Alternative 9: 82.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.85e-106) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ y (+ x 1.0))) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.85e-106) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.85d-106) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (y + (x + 1.0d0))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.85e-106) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.85e-106:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (y + (x + 1.0))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.85e-106)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.85e-106)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (y + (x + 1.0))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.85e-106], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.8499999999999999e-106

   1. Initial program 69.4%

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

      1. associate-/l* 81.1%

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

      2. associate-+l+ 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in y around 0 57.5%

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

      1. associate-/r* 59.3%

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

      2. +-commutative 59.3%

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

   7. Simplified 59.3%

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

   if 1.8499999999999999e-106 < y

   1. Initial program 64.6%

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

      1. *-un-lft-identity 64.6%

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

      2. associate-*l* 64.6%

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

      3. times-frac 73.5%

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

      4. +-commutative 73.5%

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

      5. *-commutative 73.5%

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

      6. +-commutative 73.5%

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

      7. associate-+r+ 73.5%

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

      8. +-commutative 73.5%

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

      9. associate-+l+ 73.5%

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

   4. Applied egg-rr 73.5%

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

      1. associate-/r* 77.0%

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

      2. associate-/l* 99.7%

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

      3. +-commutative 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in y around inf 66.9%

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

      1. associate-*l/ 66.9%

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

      2. *-un-lft-identity 66.9%

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

   9. Applied egg-rr 66.9%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 71.1%

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

      1. associate-/l* 81.9%

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

      2. associate-+l+ 81.9%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in x around 0 42.4%

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

      1. +-commutative 42.4%

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

   7. Simplified 42.4%

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

      1. *-un-lft-identity 42.4%

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

      2. times-frac 44.2%

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

   9. Applied egg-rr 44.2%

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

   10. Taylor expanded in y around 0 25.8%

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

   if 1 < y

   1. Initial program 59.2%

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

      1. *-un-lft-identity 59.2%

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

      2. associate-*l* 59.2%

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

      3. times-frac 69.6%

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

      4. +-commutative 69.6%

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

      5. *-commutative 69.6%

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

      6. +-commutative 69.6%

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

      7. associate-+r+ 69.6%

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

      8. +-commutative 69.6%

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

      9. associate-+l+ 69.6%

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

   4. Applied egg-rr 69.6%

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

      1. associate-/r* 73.9%

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

      2. associate-/l* 99.7%

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

      3. +-commutative 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in y around inf 65.3%

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

   8. Taylor expanded in y around inf 64.6%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 49.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 3.55 \cdot 10^{+61}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x 3.55e+61) (/ x (* y (+ y 1.0))) (* (/ 1.0 y) (/ x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= 3.55e+61) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (1.0 / y) * (x / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 3.55d+61) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (1.0d0 / y) * (x / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= 3.55e+61) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (1.0 / y) * (x / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= 3.55e+61:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (1.0 / y) * (x / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= 3.55e+61)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(1.0 / y) * Float64(x / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 3.55e+61)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (1.0 / y) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, 3.55e+61], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.55e61

   1. Initial program 69.6%

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

      1. associate-/l* 82.2%

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

      2. associate-+l+ 82.2%

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

   3. Simplified 82.2%

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

   5. Taylor expanded in x around 0 58.0%

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

      1. +-commutative 58.0%

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

   7. Simplified 58.0%

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

   if 3.55e61 < x

   1. Initial program 60.0%

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

      1. *-un-lft-identity 60.0%

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

      2. associate-*l* 60.0%

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

      3. times-frac 63.8%

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

      4. +-commutative 63.8%

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

      5. *-commutative 63.8%

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

      6. +-commutative 63.8%

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

      7. associate-+r+ 63.8%

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

      8. +-commutative 63.8%

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

      9. associate-+l+ 63.8%

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

   4. Applied egg-rr 63.8%

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

      1. associate-/r* 71.7%

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

      2. associate-/l* 99.7%

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

      3. +-commutative 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in y around inf 21.3%

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

   8. Taylor expanded in y around inf 20.6%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.55 \cdot 10^{+61}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.9 \cdot 10^{-108}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 5.9e-108) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 5.9e-108) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5.9d-108) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 5.9e-108) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 5.9e-108:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 5.9e-108)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.9e-108)
		tmp = y / (x * (x + 1.0));
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 5.9e-108], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.89999999999999965e-108

   1. Initial program 69.4%

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

      1. associate-/l* 81.1%

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

      2. associate-+l+ 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in y around 0 57.5%

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

   if 5.89999999999999965e-108 < y

   1. Initial program 64.6%

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

      1. associate-/l* 79.2%

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

      2. associate-+l+ 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in x around 0 69.2%

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

      1. +-commutative 69.2%

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

   7. Simplified 69.2%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.9 \cdot 10^{-108}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 81.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-106}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.9e-106) (/ y (* x (+ x 1.0))) (/ (/ x (+ y 1.0)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.9e-106) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.9d-106) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.9e-106) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.9e-106:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.9e-106)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.9e-106)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.9e-106], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.9e-106

   1. Initial program 69.4%

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

      1. associate-/l* 81.1%

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

      2. associate-+l+ 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in y around 0 57.5%

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

   if 1.9e-106 < y

   1. Initial program 64.6%

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

      1. associate-/l* 79.2%

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

      2. associate-+l+ 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in x around 0 69.2%

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

      1. +-commutative 69.2%

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

   7. Simplified 69.2%

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

      1. *-un-lft-identity 69.2%

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

      2. times-frac 65.7%

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

   9. Applied egg-rr 65.7%

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

       1. associate-*l/ 65.7%

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

       2. *-un-lft-identity 65.7%

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

   11. Applied egg-rr 65.7%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-106}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 82.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.9e-106) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ y 1.0)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.9e-106) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.9d-106) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.9e-106) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.9e-106:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.9e-106)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.9e-106)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.9e-106], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.9e-106

   1. Initial program 69.4%

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

      1. associate-/l* 81.1%

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

      2. associate-+l+ 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in y around 0 57.5%

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

      1. associate-/r* 59.3%

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

      2. +-commutative 59.3%

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

   7. Simplified 59.3%

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

   if 1.9e-106 < y

   1. Initial program 64.6%

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

      1. associate-/l* 79.2%

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

      2. associate-+l+ 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in x around 0 69.2%

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

      1. +-commutative 69.2%

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

   7. Simplified 69.2%

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

      1. *-un-lft-identity 69.2%

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

      2. times-frac 65.7%

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

   9. Applied egg-rr 65.7%

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

       1. associate-*l/ 65.7%

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

       2. *-un-lft-identity 65.7%

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

   11. Applied egg-rr 65.7%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 27.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.55e-31) (/ 1.0 (+ x 1.0)) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.55e-31) {
		tmp = 1.0 / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.55d-31)) then
        tmp = 1.0d0 / (x + 1.0d0)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.55e-31) {
		tmp = 1.0 / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.55e-31:
		tmp = 1.0 / (x + 1.0)
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.55e-31)
		tmp = Float64(1.0 / Float64(x + 1.0));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.55e-31)
		tmp = 1.0 / (x + 1.0);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.55e-31], N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5499999999999999e-31

   1. Initial program 52.8%

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

      1. *-un-lft-identity 52.8%

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

      2. associate-*l* 52.8%

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

      3. times-frac 62.0%

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

      4. +-commutative 62.0%

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

      5. *-commutative 62.0%

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

      6. +-commutative 62.0%

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

      7. associate-+r+ 62.0%

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

      8. +-commutative 62.0%

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

      9. associate-+l+ 62.0%

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

   4. Applied egg-rr 62.0%

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

      1. associate-/r* 67.1%

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

      2. associate-/l* 99.7%

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

      3. +-commutative 99.7%

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

   6. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. clear-num 99.6%

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

      3. frac-times 98.8%

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

      4. metadata-eval 98.8%

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

   8. Applied egg-rr 98.8%

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

   9. Taylor expanded in y around 0 69.9%

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

   10. Taylor expanded in y around inf 6.1%

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

       1. +-commutative 6.1%

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

   12. Simplified 6.1%

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

   if -2.5499999999999999e-31 < x

   1. Initial program 72.7%

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

      1. associate-/l* 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. associate-+l+ 82.5%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in x around 0 56.1%

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

      1. +-commutative 56.1%

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

   7. Simplified 56.1%

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

      1. *-un-lft-identity 56.1%

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

      2. times-frac 57.1%

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

   9. Applied egg-rr 57.1%

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

   10. Taylor expanded in y around 0 32.3%

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

4. Final simplification 25.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 4.1% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.8%

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

   1. *-un-lft-identity 67.8%

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

   2. associate-*l* 67.8%

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

   3. times-frac 73.1%

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

   4. +-commutative 73.1%

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

   5. *-commutative 73.1%

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

   6. +-commutative 73.1%

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

   7. associate-+r+ 73.1%

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

   8. +-commutative 73.1%

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

   9. associate-+l+ 73.1%

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

4. Applied egg-rr 73.1%

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

   1. associate-/r* 75.9%

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

   2. associate-/l* 99.7%

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

   3. +-commutative 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in y around inf 38.0%

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

8. Taylor expanded in y around 0 4.2%

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

9. Final simplification 4.2%

   \[\leadsto \frac{1}{y} \]
10. Add Preprocessing

Alternative 17: 25.5% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.8%

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

   1. associate-/l* 80.4%

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

   2. associate-+l+ 80.4%

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

3. Simplified 80.4%

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

5. Taylor expanded in x around 0 50.0%

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

   1. +-commutative 50.0%

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

7. Simplified 50.0%

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

   1. *-un-lft-identity 50.0%

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

   2. times-frac 50.1%

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

9. Applied egg-rr 50.1%

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

10. Taylor expanded in y around 0 25.0%

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

11. Final simplification 25.0%

    \[\leadsto \frac{x}{y} \]
12. Add Preprocessing

Developer target: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024055 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))