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

Percentage Accurate: 69.3% → 99.8%

Time: 12.9s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 73.8%

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

   1. associate-*l* 73.8%

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

   2. times-frac 95.1%

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

   3. +-commutative 95.1%

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

   4. +-commutative 95.1%

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

   5. associate-+r+ 95.1%

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

   6. +-commutative 95.1%

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

   7. associate-+l+ 95.1%

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

4. Applied egg-rr 95.1%

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

   1. *-commutative 95.1%

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

   2. associate-/r* 99.8%

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

   3. clear-num 99.7%

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

   4. frac-times 99.8%

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

   5. +-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

   1. *-rgt-identity 99.8%

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

   2. associate-/l* 99.7%

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

   3. div-inv 99.8%

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

   4. times-frac 99.7%

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

   5. clear-num 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 82.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;y \leq 1.15 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{-25} \lor \neg \left(y \leq 0.0066\right):\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= y 1.15e-86)
     (/ (/ y x) t_0)
     (if (or (<= y 4.1e-25) (not (<= y 0.0066)))
       (* (/ x (+ y x)) (/ 1.0 t_0))
       (/ y (* x (+ x 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= 1.15e-86) {
		tmp = (y / x) / t_0;
	} else if ((y <= 4.1e-25) || !(y <= 0.0066)) {
		tmp = (x / (y + x)) * (1.0 / t_0);
	} else {
		tmp = y / (x * (x + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (y <= 1.15d-86) then
        tmp = (y / x) / t_0
    else if ((y <= 4.1d-25) .or. (.not. (y <= 0.0066d0))) then
        tmp = (x / (y + x)) * (1.0d0 / t_0)
    else
        tmp = y / (x * (x + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= 1.15e-86) {
		tmp = (y / x) / t_0;
	} else if ((y <= 4.1e-25) || !(y <= 0.0066)) {
		tmp = (x / (y + x)) * (1.0 / t_0);
	} else {
		tmp = y / (x * (x + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if y <= 1.15e-86:
		tmp = (y / x) / t_0
	elif (y <= 4.1e-25) or not (y <= 0.0066):
		tmp = (x / (y + x)) * (1.0 / t_0)
	else:
		tmp = y / (x * (x + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (y <= 1.15e-86)
		tmp = Float64(Float64(y / x) / t_0);
	elseif ((y <= 4.1e-25) || !(y <= 0.0066))
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(1.0 / t_0));
	else
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (y <= 1.15e-86)
		tmp = (y / x) / t_0;
	elseif ((y <= 4.1e-25) || ~((y <= 0.0066)))
		tmp = (x / (y + x)) * (1.0 / t_0);
	else
		tmp = y / (x * (x + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.15e-86], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[Or[LessEqual[y, 4.1e-25], N[Not[LessEqual[y, 0.0066]], $MachinePrecision]], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.14999999999999998e-86

   1. Initial program 74.8%

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

      1. associate-/l* 85.6%

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

      2. associate-+l+ 85.6%

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

   3. Simplified 85.6%

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

      1. associate-*r/ 74.8%

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

      2. associate-+r+ 74.8%

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

      3. associate-/r* 77.5%

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

      4. clear-num 77.2%

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

      5. associate-+r+ 77.2%

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

      6. +-commutative 77.2%

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

      7. associate-+l+ 77.2%

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

      8. *-commutative 77.2%

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

      9. associate-/l* 89.7%

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

      10. pow2 89.7%

          \[\leadsto \frac{1}{\frac{y + \left(1 + x\right)}{y \cdot \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}}}} \]

      11. +-commutative 89.7%

          \[\leadsto \frac{1}{\frac{y + \left(1 + x\right)}{y \cdot \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}}}} \]

   6. Applied egg-rr 89.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{y + \left(1 + x\right)}{y \cdot \frac{x}{{\left(y + x\right)}^{2}}}}} \]

   7. Taylor expanded in x around inf 57.1%

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

      1. *-un-lft-identity 57.1%

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

      2. clear-num 57.4%

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

      3. un-div-inv 57.5%

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

      4. +-commutative 57.5%

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

   9. Applied egg-rr 57.5%

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

       1. *-lft-identity 57.5%

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

   11. Simplified 57.5%

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

   if 1.14999999999999998e-86 < y < 4.09999999999999987e-25 or 0.0066 < y

   1. Initial program 72.2%

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

      1. associate-*l* 72.2%

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

      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%

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

      1. *-commutative 94.7%

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

      2. associate-/r* 99.8%

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

      3. clear-num 99.7%

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

      4. frac-times 99.7%

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

      5. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. *-rgt-identity 99.7%

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

      2. associate-/l* 99.6%

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

      3. div-inv 99.7%

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

      4. times-frac 99.7%

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

      5. clear-num 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in y around inf 72.9%

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

   if 4.09999999999999987e-25 < y < 0.0066

   1. Initial program 56.7%

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

      1. associate-/l* 56.7%

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

      2. associate-+l+ 56.7%

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

   3. Simplified 56.7%

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

   5. Taylor expanded in y around 0 100.0%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{-25} \lor \neg \left(y \leq 0.0066\right):\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{1}{y + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;y \leq 3.8 \cdot 10^{-156}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+112}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{1}{t\_0}\\ \end{array} \end{array} \]

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 (<= y 3.8e-156)
     (/ (/ y x) t_0)
     (if (<= y 3.9e+112)
       (* x (/ y (* (* (+ y x) (+ y x)) (+ x (+ y 1.0)))))
       (* (/ x (+ y x)) (/ 1.0 t_0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= 3.8e-156) {
		tmp = (y / x) / t_0;
	} else if (y <= 3.9e+112) {
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))));
	} else {
		tmp = (x / (y + x)) * (1.0 / 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 (y <= 3.8d-156) then
        tmp = (y / x) / t_0
    else if (y <= 3.9d+112) then
        tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0d0))))
    else
        tmp = (x / (y + x)) * (1.0d0 / 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 (y <= 3.8e-156) {
		tmp = (y / x) / t_0;
	} else if (y <= 3.9e+112) {
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))));
	} else {
		tmp = (x / (y + x)) * (1.0 / t_0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if y <= 3.8e-156:
		tmp = (y / x) / t_0
	elif y <= 3.9e+112:
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))))
	else:
		tmp = (x / (y + x)) * (1.0 / 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 (y <= 3.8e-156)
		tmp = Float64(Float64(y / x) / t_0);
	elseif (y <= 3.9e+112)
		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 + x)) * Float64(1.0 / 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 (y <= 3.8e-156)
		tmp = (y / x) / t_0;
	elseif (y <= 3.9e+112)
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))));
	else
		tmp = (x / (y + x)) * (1.0 / 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[y, 3.8e-156], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 3.9e+112], 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 + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.80000000000000008e-156

   1. Initial program 74.0%

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

      1. associate-/l* 85.0%

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

      2. associate-+l+ 85.0%

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

   3. Simplified 85.0%

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

      1. associate-*r/ 74.0%

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

      2. associate-+r+ 74.0%

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

      3. associate-/r* 76.7%

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

      4. clear-num 76.4%

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

      5. associate-+r+ 76.4%

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

      6. +-commutative 76.4%

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

      7. associate-+l+ 76.4%

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

      8. *-commutative 76.4%

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

      9. associate-/l* 89.2%

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

      10. pow2 89.2%

          \[\leadsto \frac{1}{\frac{y + \left(1 + x\right)}{y \cdot \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}}}} \]

      11. +-commutative 89.2%

          \[\leadsto \frac{1}{\frac{y + \left(1 + x\right)}{y \cdot \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}}}} \]

   6. Applied egg-rr 89.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{y + \left(1 + x\right)}{y \cdot \frac{x}{{\left(y + x\right)}^{2}}}}} \]

   7. Taylor expanded in x around inf 55.7%

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

      1. *-un-lft-identity 55.7%

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

      2. clear-num 56.0%

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

      3. un-div-inv 56.1%

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

      4. +-commutative 56.1%

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

   9. Applied egg-rr 56.1%

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

       1. *-lft-identity 56.1%

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

   11. Simplified 56.1%

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

   if 3.80000000000000008e-156 < y < 3.89999999999999968e112

   1. Initial program 75.9%

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

      1. associate-/l* 83.7%

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

      2. associate-+l+ 83.7%

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

   3. Simplified 83.7%

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

   if 3.89999999999999968e112 < y

   1. Initial program 69.9%

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

      1. associate-*l* 69.9%

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

      2. times-frac 94.6%

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

      3. +-commutative 94.6%

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

      4. +-commutative 94.6%

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

      5. associate-+r+ 94.6%

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

      6. +-commutative 94.6%

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

      7. associate-+l+ 94.6%

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

   4. Applied egg-rr 94.6%

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

      1. *-commutative 94.6%

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

      2. associate-/r* 99.9%

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

      3. clear-num 99.9%

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

      4. frac-times 99.9%

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

      5. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. *-rgt-identity 99.9%

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

      2. associate-/l* 99.9%

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

      3. div-inv 99.9%

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

      4. times-frac 99.9%

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

      5. clear-num 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in y around inf 89.6%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-156}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+112}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{1}{y + \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-25} \lor \neg \left(y \leq 0.00052\right):\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.1e-86)
   (/ (/ y x) (+ y (+ x 1.0)))
   (if (or (<= y 5e-25) (not (<= y 0.00052)))
     (* (/ x (+ y x)) (/ 1.0 (+ y 1.0)))
     (/ y (* x (+ x 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.1e-86) {
		tmp = (y / x) / (y + (x + 1.0));
	} else if ((y <= 5e-25) || !(y <= 0.00052)) {
		tmp = (x / (y + x)) * (1.0 / (y + 1.0));
	} else {
		tmp = y / (x * (x + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.1d-86) then
        tmp = (y / x) / (y + (x + 1.0d0))
    else if ((y <= 5d-25) .or. (.not. (y <= 0.00052d0))) then
        tmp = (x / (y + x)) * (1.0d0 / (y + 1.0d0))
    else
        tmp = y / (x * (x + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.1e-86) {
		tmp = (y / x) / (y + (x + 1.0));
	} else if ((y <= 5e-25) || !(y <= 0.00052)) {
		tmp = (x / (y + x)) * (1.0 / (y + 1.0));
	} else {
		tmp = y / (x * (x + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.1e-86:
		tmp = (y / x) / (y + (x + 1.0))
	elif (y <= 5e-25) or not (y <= 0.00052):
		tmp = (x / (y + x)) * (1.0 / (y + 1.0))
	else:
		tmp = y / (x * (x + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.1e-86)
		tmp = Float64(Float64(y / x) / Float64(y + Float64(x + 1.0)));
	elseif ((y <= 5e-25) || !(y <= 0.00052))
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(1.0 / Float64(y + 1.0)));
	else
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.1e-86)
		tmp = (y / x) / (y + (x + 1.0));
	elseif ((y <= 5e-25) || ~((y <= 0.00052)))
		tmp = (x / (y + x)) * (1.0 / (y + 1.0));
	else
		tmp = y / (x * (x + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.1e-86], N[(N[(y / x), $MachinePrecision] / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 5e-25], N[Not[LessEqual[y, 0.00052]], $MachinePrecision]], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.1000000000000001e-86

   1. Initial program 74.8%

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

      1. associate-/l* 85.6%

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

      2. associate-+l+ 85.6%

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

   3. Simplified 85.6%

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

      1. associate-*r/ 74.8%

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

      2. associate-+r+ 74.8%

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

      3. associate-/r* 77.5%

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

      4. clear-num 77.2%

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

      5. associate-+r+ 77.2%

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

      6. +-commutative 77.2%

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

      7. associate-+l+ 77.2%

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

      8. *-commutative 77.2%

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

      9. associate-/l* 89.7%

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

      10. pow2 89.7%

          \[\leadsto \frac{1}{\frac{y + \left(1 + x\right)}{y \cdot \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}}}} \]

      11. +-commutative 89.7%

          \[\leadsto \frac{1}{\frac{y + \left(1 + x\right)}{y \cdot \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}}}} \]

   6. Applied egg-rr 89.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{y + \left(1 + x\right)}{y \cdot \frac{x}{{\left(y + x\right)}^{2}}}}} \]

   7. Taylor expanded in x around inf 57.1%

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

      1. *-un-lft-identity 57.1%

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

      2. clear-num 57.4%

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

      3. un-div-inv 57.5%

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

      4. +-commutative 57.5%

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

   9. Applied egg-rr 57.5%

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

       1. *-lft-identity 57.5%

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

   11. Simplified 57.5%

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

   if 1.1000000000000001e-86 < y < 4.99999999999999962e-25 or 5.19999999999999954e-4 < y

   1. Initial program 71.4%

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

      1. associate-*l* 71.4%

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

      2. times-frac 94.8%

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

      3. +-commutative 94.8%

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

      4. +-commutative 94.8%

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

      5. associate-+r+ 94.8%

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

      6. +-commutative 94.8%

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

      7. associate-+l+ 94.8%

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

   4. Applied egg-rr 94.8%

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

   5. Taylor expanded in x around 0 71.6%

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

      1. +-commutative 71.6%

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

   7. Simplified 71.6%

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

   if 4.99999999999999962e-25 < y < 5.19999999999999954e-4

   1. Initial program 71.0%

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

      1. associate-/l* 71.0%

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

      2. associate-+l+ 71.0%

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

   3. Simplified 71.0%

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

   5. Taylor expanded in y around 0 100.0%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-25} \lor \neg \left(y \leq 0.00052\right):\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{1}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{if}\;y \leq 4.8 \cdot 10^{-88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 0.00062:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* x (+ x 1.0)))))
   (if (<= y 4.8e-88)
     t_0
     (if (<= y 4.5e-25)
       (/ x y)
       (if (<= y 0.00062) t_0 (/ x (* y (+ y 1.0))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x * (x + 1.0));
	double tmp;
	if (y <= 4.8e-88) {
		tmp = t_0;
	} else if (y <= 4.5e-25) {
		tmp = x / y;
	} else if (y <= 0.00062) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = y / (x * (x + 1.0d0))
    if (y <= 4.8d-88) then
        tmp = t_0
    else if (y <= 4.5d-25) then
        tmp = x / y
    else if (y <= 0.00062d0) then
        tmp = t_0
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (x * (x + 1.0));
	double tmp;
	if (y <= 4.8e-88) {
		tmp = t_0;
	} else if (y <= 4.5e-25) {
		tmp = x / y;
	} else if (y <= 0.00062) {
		tmp = t_0;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (x * (x + 1.0))
	tmp = 0
	if y <= 4.8e-88:
		tmp = t_0
	elif y <= 4.5e-25:
		tmp = x / y
	elif y <= 0.00062:
		tmp = t_0
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(x * Float64(x + 1.0)))
	tmp = 0.0
	if (y <= 4.8e-88)
		tmp = t_0;
	elseif (y <= 4.5e-25)
		tmp = Float64(x / y);
	elseif (y <= 0.00062)
		tmp = t_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)
	t_0 = y / (x * (x + 1.0));
	tmp = 0.0;
	if (y <= 4.8e-88)
		tmp = t_0;
	elseif (y <= 4.5e-25)
		tmp = x / y;
	elseif (y <= 0.00062)
		tmp = t_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_] := Block[{t$95$0 = N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 4.8e-88], t$95$0, If[LessEqual[y, 4.5e-25], N[(x / y), $MachinePrecision], If[LessEqual[y, 0.00062], t$95$0, N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.7999999999999999e-88 or 4.5000000000000001e-25 < y < 6.2e-4

   1. Initial program 74.8%

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

      1. associate-/l* 85.4%

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

      2. associate-+l+ 85.4%

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

   3. Simplified 85.4%

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

   5. Taylor expanded in y around 0 56.8%

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

   if 4.7999999999999999e-88 < y < 4.5000000000000001e-25

   1. Initial program 85.7%

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

      1. associate-/l* 91.1%

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

      2. associate-+l+ 91.1%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in x around 0 62.5%

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

      1. +-commutative 62.5%

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

   7. Simplified 62.5%

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

   8. Taylor expanded in y around 0 62.5%

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

   if 6.2e-4 < y

   1. Initial program 69.1%

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

      1. associate-/l* 82.9%

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

      2. associate-+l+ 82.9%

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

   3. Simplified 82.9%

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

   5. Taylor expanded in x around 0 72.6%

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

      1. +-commutative 72.6%

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

   7. Simplified 72.6%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-88}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 0.00062:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{if}\;y \leq 1.15 \cdot 10^{-86}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 0.000125:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* x (+ x 1.0)))))
   (if (<= y 1.15e-86)
     t_0
     (if (<= y 3.8e-25)
       (/ x y)
       (if (<= y 0.000125) t_0 (/ (/ x y) (+ y 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x * (x + 1.0));
	double tmp;
	if (y <= 1.15e-86) {
		tmp = t_0;
	} else if (y <= 3.8e-25) {
		tmp = x / y;
	} else if (y <= 0.000125) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = y / (x * (x + 1.0d0))
    if (y <= 1.15d-86) then
        tmp = t_0
    else if (y <= 3.8d-25) then
        tmp = x / y
    else if (y <= 0.000125d0) then
        tmp = t_0
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (x * (x + 1.0));
	double tmp;
	if (y <= 1.15e-86) {
		tmp = t_0;
	} else if (y <= 3.8e-25) {
		tmp = x / y;
	} else if (y <= 0.000125) {
		tmp = t_0;
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (x * (x + 1.0))
	tmp = 0
	if y <= 1.15e-86:
		tmp = t_0
	elif y <= 3.8e-25:
		tmp = x / y
	elif y <= 0.000125:
		tmp = t_0
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(x * Float64(x + 1.0)))
	tmp = 0.0
	if (y <= 1.15e-86)
		tmp = t_0;
	elseif (y <= 3.8e-25)
		tmp = Float64(x / y);
	elseif (y <= 0.000125)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (x * (x + 1.0));
	tmp = 0.0;
	if (y <= 1.15e-86)
		tmp = t_0;
	elseif (y <= 3.8e-25)
		tmp = x / y;
	elseif (y <= 0.000125)
		tmp = t_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_] := Block[{t$95$0 = N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.15e-86], t$95$0, If[LessEqual[y, 3.8e-25], N[(x / y), $MachinePrecision], If[LessEqual[y, 0.000125], t$95$0, N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.14999999999999998e-86 or 3.7999999999999998e-25 < y < 1.25e-4

   1. Initial program 74.8%

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

      1. associate-/l* 85.4%

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

      2. associate-+l+ 85.4%

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

   3. Simplified 85.4%

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

   5. Taylor expanded in y around 0 56.8%

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

   if 1.14999999999999998e-86 < y < 3.7999999999999998e-25

   1. Initial program 85.7%

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

      1. associate-/l* 91.1%

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

      2. associate-+l+ 91.1%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in x around 0 62.5%

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

      1. +-commutative 62.5%

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

   7. Simplified 62.5%

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

   8. Taylor expanded in y around 0 62.5%

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

   if 1.25e-4 < y

   1. Initial program 69.1%

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

      1. associate-/l* 82.9%

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

      2. associate-+l+ 82.9%

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

   3. Simplified 82.9%

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

   5. Taylor expanded in x around 0 72.5%

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

      1. associate-/r* 72.5%

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

      2. +-commutative 72.5%

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

   7. Simplified 72.5%

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

      1. associate-*r/ 72.6%

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

      2. un-div-inv 72.7%

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

   9. Applied egg-rr 72.7%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 0.000125:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 0.00078:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.15e-86)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 4e-25)
     (/ x y)
     (if (<= y 0.00078) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.15e-86) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 4e-25) {
		tmp = x / y;
	} else if (y <= 0.00078) {
		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 <= 1.15d-86) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 4d-25) then
        tmp = x / y
    else if (y <= 0.00078d0) 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 <= 1.15e-86) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 4e-25) {
		tmp = x / y;
	} else if (y <= 0.00078) {
		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 <= 1.15e-86:
		tmp = (y / x) / (x + 1.0)
	elif y <= 4e-25:
		tmp = x / y
	elif y <= 0.00078:
		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 <= 1.15e-86)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 4e-25)
		tmp = Float64(x / y);
	elseif (y <= 0.00078)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.15e-86)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 4e-25)
		tmp = x / y;
	elseif (y <= 0.00078)
		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, 1.15e-86], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4e-25], N[(x / y), $MachinePrecision], If[LessEqual[y, 0.00078], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 1.14999999999999998e-86

   1. Initial program 74.8%

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

      1. associate-/l* 85.6%

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

      2. associate-+l+ 85.6%

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

   3. Simplified 85.6%

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

   5. Taylor expanded in y around 0 56.1%

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

      1. associate-/r* 57.3%

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

      2. +-commutative 57.3%

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

   7. Simplified 57.3%

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

   if 1.14999999999999998e-86 < y < 4.00000000000000015e-25

   1. Initial program 85.7%

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

      1. associate-/l* 91.1%

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

      2. associate-+l+ 91.1%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in x around 0 62.5%

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

      1. +-commutative 62.5%

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

   7. Simplified 62.5%

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

   8. Taylor expanded in y around 0 62.5%

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

   if 4.00000000000000015e-25 < y < 7.79999999999999986e-4

   1. Initial program 71.0%

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

      1. associate-/l* 71.0%

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

      2. associate-+l+ 71.0%

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

   3. Simplified 71.0%

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

   5. Taylor expanded in y around 0 100.0%

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

   if 7.79999999999999986e-4 < y

   1. Initial program 69.1%

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

      1. associate-/l* 82.9%

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

      2. associate-+l+ 82.9%

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

   3. Simplified 82.9%

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

   5. Taylor expanded in x around 0 72.5%

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

      1. associate-/r* 72.5%

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

      2. +-commutative 72.5%

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

   7. Simplified 72.5%

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

      1. associate-*r/ 72.6%

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

      2. un-div-inv 72.7%

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

   9. Applied egg-rr 72.7%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 0.00078:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 0.0038:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 8e-87)
   (/ (/ y x) (+ y (+ x 1.0)))
   (if (<= y 3.8e-25)
     (/ x y)
     (if (<= y 0.0038) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 8e-87) {
		tmp = (y / x) / (y + (x + 1.0));
	} else if (y <= 3.8e-25) {
		tmp = x / y;
	} else if (y <= 0.0038) {
		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 <= 8d-87) then
        tmp = (y / x) / (y + (x + 1.0d0))
    else if (y <= 3.8d-25) then
        tmp = x / y
    else if (y <= 0.0038d0) 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 <= 8e-87) {
		tmp = (y / x) / (y + (x + 1.0));
	} else if (y <= 3.8e-25) {
		tmp = x / y;
	} else if (y <= 0.0038) {
		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 <= 8e-87:
		tmp = (y / x) / (y + (x + 1.0))
	elif y <= 3.8e-25:
		tmp = x / y
	elif y <= 0.0038:
		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 <= 8e-87)
		tmp = Float64(Float64(y / x) / Float64(y + Float64(x + 1.0)));
	elseif (y <= 3.8e-25)
		tmp = Float64(x / y);
	elseif (y <= 0.0038)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8e-87)
		tmp = (y / x) / (y + (x + 1.0));
	elseif (y <= 3.8e-25)
		tmp = x / y;
	elseif (y <= 0.0038)
		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, 8e-87], N[(N[(y / x), $MachinePrecision] / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e-25], N[(x / y), $MachinePrecision], If[LessEqual[y, 0.0038], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 8.00000000000000014e-87

   1. Initial program 74.8%

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

      1. associate-/l* 85.6%

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

      2. associate-+l+ 85.6%

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

   3. Simplified 85.6%

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

      1. associate-*r/ 74.8%

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

      2. associate-+r+ 74.8%

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

      3. associate-/r* 77.5%

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

      4. clear-num 77.2%

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

      5. associate-+r+ 77.2%

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

      6. +-commutative 77.2%

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

      7. associate-+l+ 77.2%

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

      8. *-commutative 77.2%

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

      9. associate-/l* 89.7%

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

      10. pow2 89.7%

          \[\leadsto \frac{1}{\frac{y + \left(1 + x\right)}{y \cdot \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}}}} \]

      11. +-commutative 89.7%

          \[\leadsto \frac{1}{\frac{y + \left(1 + x\right)}{y \cdot \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}}}} \]

   6. Applied egg-rr 89.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{y + \left(1 + x\right)}{y \cdot \frac{x}{{\left(y + x\right)}^{2}}}}} \]

   7. Taylor expanded in x around inf 57.1%

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

      1. *-un-lft-identity 57.1%

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

      2. clear-num 57.4%

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

      3. un-div-inv 57.5%

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

      4. +-commutative 57.5%

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

   9. Applied egg-rr 57.5%

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

       1. *-lft-identity 57.5%

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

   11. Simplified 57.5%

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

   if 8.00000000000000014e-87 < y < 3.7999999999999998e-25

   1. Initial program 85.7%

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

      1. associate-/l* 91.1%

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

      2. associate-+l+ 91.1%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in x around 0 62.5%

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

      1. +-commutative 62.5%

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

   7. Simplified 62.5%

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

   8. Taylor expanded in y around 0 62.5%

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

   if 3.7999999999999998e-25 < y < 0.00379999999999999999

   1. Initial program 56.7%

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

      1. associate-/l* 56.7%

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

      2. associate-+l+ 56.7%

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

   3. Simplified 56.7%

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

   5. Taylor expanded in y around 0 100.0%

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

   if 0.00379999999999999999 < y

   1. Initial program 69.9%

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

      1. associate-/l* 83.9%

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

      2. associate-+l+ 83.9%

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

   3. Simplified 83.9%

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

   5. Taylor expanded in x around 0 73.6%

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

      1. associate-/r* 73.6%

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

      2. +-commutative 73.6%

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

   7. Simplified 73.6%

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

      1. associate-*r/ 73.7%

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

      2. un-div-inv 73.7%

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

   9. Applied egg-rr 73.7%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 0.0038:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ t_1 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -9 \cdot 10^{+157}:\\ \;\;\;\;t\_0 \cdot \frac{\frac{y}{x}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{y}{\left(y + x\right) \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))) (t_1 (+ y (+ x 1.0))))
   (if (<= x -9e+157) (* t_0 (/ (/ y x) t_1)) (* t_0 (/ y (* (+ y x) t_1))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y + x);
	double t_1 = y + (x + 1.0);
	double tmp;
	if (x <= -9e+157) {
		tmp = t_0 * ((y / x) / t_1);
	} else {
		tmp = t_0 * (y / ((y + x) * t_1));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = x / (y + x)
    t_1 = y + (x + 1.0d0)
    if (x <= (-9d+157)) then
        tmp = t_0 * ((y / x) / t_1)
    else
        tmp = t_0 * (y / ((y + x) * t_1))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double t_1 = y + (x + 1.0);
	double tmp;
	if (x <= -9e+157) {
		tmp = t_0 * ((y / x) / t_1);
	} else {
		tmp = t_0 * (y / ((y + x) * t_1));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y + x)
	t_1 = y + (x + 1.0)
	tmp = 0
	if x <= -9e+157:
		tmp = t_0 * ((y / x) / t_1)
	else:
		tmp = t_0 * (y / ((y + x) * t_1))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	t_1 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -9e+157)
		tmp = Float64(t_0 * Float64(Float64(y / x) / t_1));
	else
		tmp = Float64(t_0 * Float64(y / Float64(Float64(y + x) * t_1)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	t_1 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -9e+157)
		tmp = t_0 * ((y / x) / t_1);
	else
		tmp = t_0 * (y / ((y + x) * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9e+157], N[(t$95$0 * N[(N[(y / x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(y / N[(N[(y + x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.9999999999999997e157

   1. Initial program 55.3%

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

      1. associate-*l* 55.3%

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

      2. times-frac 82.1%

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

      3. +-commutative 82.1%

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

      4. +-commutative 82.1%

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

      5. associate-+r+ 82.1%

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

      6. +-commutative 82.1%

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

      7. associate-+l+ 82.1%

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

   4. Applied egg-rr 82.1%

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

      1. *-commutative 82.1%

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

      2. associate-/r* 99.9%

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

      3. clear-num 99.9%

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

      4. frac-times 99.9%

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

      5. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. *-rgt-identity 99.9%

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

      2. associate-/l* 99.8%

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

      3. div-inv 99.9%

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

      4. times-frac 99.9%

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

      5. clear-num 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in y around 0 89.4%

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

   if -8.9999999999999997e157 < x

   1. Initial program 76.3%

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

      1. associate-*l* 76.3%

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

      2. times-frac 96.9%

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

      3. +-commutative 96.9%

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

      4. +-commutative 96.9%

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

      5. associate-+r+ 96.9%

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

      6. +-commutative 96.9%

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

      7. associate-+l+ 96.9%

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

   4. Applied egg-rr 96.9%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+157}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{x}}{y + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;y \leq 5.2 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot t\_0} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= y 5.2e-87) (/ (/ y x) t_0) (* (/ y (* (+ y x) t_0)) (/ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= 5.2e-87) {
		tmp = (y / x) / t_0;
	} else {
		tmp = (y / ((y + x) * t_0)) * (x / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (y <= 5.2d-87) then
        tmp = (y / x) / t_0
    else
        tmp = (y / ((y + x) * t_0)) * (x / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= 5.2e-87) {
		tmp = (y / x) / t_0;
	} else {
		tmp = (y / ((y + x) * t_0)) * (x / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if y <= 5.2e-87:
		tmp = (y / x) / t_0
	else:
		tmp = (y / ((y + x) * t_0)) * (x / 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 (y <= 5.2e-87)
		tmp = Float64(Float64(y / x) / t_0);
	else
		tmp = Float64(Float64(y / Float64(Float64(y + x) * t_0)) * Float64(x / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (y <= 5.2e-87)
		tmp = (y / x) / t_0;
	else
		tmp = (y / ((y + x) * t_0)) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 5.2e-87], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(y / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 5.20000000000000005e-87

   1. Initial program 74.8%

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

      1. associate-/l* 85.6%

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

      2. associate-+l+ 85.6%

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

   3. Simplified 85.6%

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

      1. associate-*r/ 74.8%

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

      2. associate-+r+ 74.8%

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

      3. associate-/r* 77.5%

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

      4. clear-num 77.2%

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

      5. associate-+r+ 77.2%

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

      6. +-commutative 77.2%

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

      7. associate-+l+ 77.2%

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

      8. *-commutative 77.2%

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

      9. associate-/l* 89.7%

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

      10. pow2 89.7%

          \[\leadsto \frac{1}{\frac{y + \left(1 + x\right)}{y \cdot \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}}}} \]

      11. +-commutative 89.7%

          \[\leadsto \frac{1}{\frac{y + \left(1 + x\right)}{y \cdot \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}}}} \]

   6. Applied egg-rr 89.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{y + \left(1 + x\right)}{y \cdot \frac{x}{{\left(y + x\right)}^{2}}}}} \]

   7. Taylor expanded in x around inf 57.1%

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

      1. *-un-lft-identity 57.1%

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

      2. clear-num 57.4%

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

      3. un-div-inv 57.5%

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

      4. +-commutative 57.5%

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

   9. Applied egg-rr 57.5%

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

       1. *-lft-identity 57.5%

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

   11. Simplified 57.5%

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

   if 5.20000000000000005e-87 < y

   1. Initial program 71.4%

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

      1. associate-*l* 71.4%

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

      2. times-frac 95.0%

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

      3. +-commutative 95.0%

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

      4. +-commutative 95.0%

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

      5. associate-+r+ 95.0%

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

      6. +-commutative 95.0%

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

      7. associate-+l+ 95.0%

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

   4. Applied egg-rr 95.0%

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

   5. Taylor expanded in x around 0 78.7%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{-130}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 6.2e-130) (/ y x) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 6.2e-130) {
		tmp = y / x;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 6.2d-130) then
        tmp = y / x
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 6.2e-130) {
		tmp = y / x;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 6.2e-130:
		tmp = y / x
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 6.2e-130)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.2e-130)
		tmp = y / x;
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 6.2e-130], N[(y / x), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.20000000000000021e-130

   1. Initial program 74.6%

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

      1. associate-/l* 85.3%

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

      2. associate-+l+ 85.3%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in y around 0 55.6%

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

      1. associate-/r* 56.8%

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

      2. +-commutative 56.8%

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

   7. Simplified 56.8%

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

   8. Taylor expanded in x around 0 35.8%

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

   if 6.20000000000000021e-130 < y

   1. Initial program 72.2%

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

      1. associate-/l* 84.3%

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

      2. associate-+l+ 84.3%

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

   3. Simplified 84.3%

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

   5. Taylor expanded in x around 0 66.6%

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

      1. +-commutative 66.6%

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

   7. Simplified 66.6%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{-130}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 44.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-129}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 2.1e-129) (/ y x) (/ 1.0 (/ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.1e-129) {
		tmp = y / x;
	} else {
		tmp = 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 <= 2.1d-129) then
        tmp = y / x
    else
        tmp = 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 <= 2.1e-129) {
		tmp = y / x;
	} else {
		tmp = 1.0 / (y / x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.1e-129:
		tmp = y / x
	else:
		tmp = 1.0 / (y / x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.1e-129)
		tmp = Float64(y / x);
	else
		tmp = Float64(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 <= 2.1e-129)
		tmp = y / x;
	else
		tmp = 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, 2.1e-129], N[(y / x), $MachinePrecision], N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.1e-129

   1. Initial program 74.6%

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

      1. associate-/l* 85.3%

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

      2. associate-+l+ 85.3%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in y around 0 55.6%

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

      1. associate-/r* 56.8%

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

      2. +-commutative 56.8%

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

   7. Simplified 56.8%

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

   8. Taylor expanded in x around 0 35.8%

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

   if 2.1e-129 < y

   1. Initial program 72.2%

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

      1. associate-/l* 84.3%

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

      2. associate-+l+ 84.3%

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

   3. Simplified 84.3%

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

      1. associate-*r/ 72.2%

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

      2. associate-+r+ 72.2%

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

      3. associate-/r* 82.0%

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

      4. clear-num 81.9%

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

      5. associate-+r+ 81.9%

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

      6. +-commutative 81.9%

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

      7. associate-+l+ 81.9%

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

      8. *-commutative 81.9%

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

      9. associate-/l* 95.1%

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

      10. pow2 95.1%

          \[\leadsto \frac{1}{\frac{y + \left(1 + x\right)}{y \cdot \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}}}} \]

      11. +-commutative 95.1%

          \[\leadsto \frac{1}{\frac{y + \left(1 + x\right)}{y \cdot \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}}}} \]

   6. Applied egg-rr 95.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{y + \left(1 + x\right)}{y \cdot \frac{x}{{\left(y + x\right)}^{2}}}}} \]

   7. Taylor expanded in x around 0 66.4%

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

   8. Taylor expanded in y around 0 31.0%

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

4. Final simplification 34.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-129}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 28.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3300000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -3300000.0) (/ 1.0 x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3300000.0) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3300000.0d0)) then
        tmp = 1.0d0 / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -3300000.0) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3300000.0:
		tmp = 1.0 / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3300000.0)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3300000.0)
		tmp = 1.0 / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -3300000.0], N[(1.0 / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.3e6

   1. Initial program 64.9%

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

      1. associate-/l* 75.2%

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

      2. associate-+l+ 75.2%

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

   3. Simplified 75.2%

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

      1. associate-*r/ 64.9%

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

      2. associate-+r+ 64.9%

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

      3. associate-/r* 73.6%

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

      4. clear-num 73.5%

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

      5. associate-+r+ 73.5%

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

      6. +-commutative 73.5%

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

      7. associate-+l+ 73.5%

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

      8. *-commutative 73.5%

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

      9. associate-/l* 87.3%

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

      10. pow2 87.3%

          \[\leadsto \frac{1}{\frac{y + \left(1 + x\right)}{y \cdot \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}}}} \]

      11. +-commutative 87.3%

          \[\leadsto \frac{1}{\frac{y + \left(1 + x\right)}{y \cdot \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}}}} \]

   6. Applied egg-rr 87.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{y + \left(1 + x\right)}{y \cdot \frac{x}{{\left(y + x\right)}^{2}}}}} \]

   7. Taylor expanded in x around inf 73.9%

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

   8. Taylor expanded in y around inf 5.6%

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

   if -3.3e6 < x

   1. Initial program 77.4%

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

      1. associate-/l* 89.0%

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

      2. associate-+l+ 89.0%

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

   3. Simplified 89.0%

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

   5. Taylor expanded in x around 0 62.2%

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

      1. +-commutative 62.2%

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

   7. Simplified 62.2%

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

   8. Taylor expanded in y around 0 35.4%

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

4. Final simplification 26.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3300000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 44.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-129}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 1.3e-129) (/ y x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.3e-129) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.3d-129) then
        tmp = y / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.3e-129) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.3e-129:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.3e-129)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.3e-129)
		tmp = y / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.3e-129], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.3e-129

   1. Initial program 74.6%

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

      1. associate-/l* 85.3%

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

      2. associate-+l+ 85.3%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in y around 0 55.6%

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

      1. associate-/r* 56.8%

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

      2. +-commutative 56.8%

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

   7. Simplified 56.8%

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

   8. Taylor expanded in x around 0 35.8%

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

   if 1.3e-129 < y

   1. Initial program 72.2%

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

      1. associate-/l* 84.3%

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

      2. associate-+l+ 84.3%

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

   3. Simplified 84.3%

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

   5. Taylor expanded in x around 0 66.6%

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

      1. +-commutative 66.6%

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

   7. Simplified 66.6%

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

   8. Taylor expanded in y around 0 30.4%

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

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-129}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 4.3% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.8%

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

   1. associate-/l* 85.0%

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

   2. associate-+l+ 85.0%

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

3. Simplified 85.0%

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

   1. associate-*r/ 73.8%

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

   2. associate-+r+ 73.8%

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

   3. associate-/r* 78.8%

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

   4. clear-num 78.6%

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

   5. associate-+r+ 78.6%

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

   6. +-commutative 78.6%

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

   7. associate-+l+ 78.6%

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

   8. *-commutative 78.6%

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

   9. associate-/l* 91.3%

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

   10. pow2 91.3%

       \[\leadsto \frac{1}{\frac{y + \left(1 + x\right)}{y \cdot \frac{x}{\color{blue}{{\left(x + y\right)}^{2}}}}} \]

   11. +-commutative 91.3%

       \[\leadsto \frac{1}{\frac{y + \left(1 + x\right)}{y \cdot \frac{x}{{\color{blue}{\left(y + x\right)}}^{2}}}} \]

6. Applied egg-rr 91.3%

   \[\leadsto \color{blue}{\frac{1}{\frac{y + \left(1 + x\right)}{y \cdot \frac{x}{{\left(y + x\right)}^{2}}}}} \]

7. Taylor expanded in x around inf 49.4%

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

8. Taylor expanded in y around inf 4.1%

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

9. Final simplification 4.1%

   \[\leadsto \frac{1}{x} \]
10. 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 2024095 
(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))))