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

Percentage Accurate: 69.7% → 99.8%

Time: 14.8s

Alternatives: 12

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.2%

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

   1. *-commutative 71.2%

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

   2. associate-*l* 71.3%

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

   3. times-frac 95.2%

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

   4. +-commutative 95.2%

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

   5. +-commutative 95.2%

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

   6. associate-+r+ 95.2%

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

   7. +-commutative 95.2%

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

   8. associate-+l+ 95.2%

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

4. Applied egg-rr 95.2%

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

   1. div-inv 95.1%

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

   2. distribute-rgt-in 92.4%

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

   3. +-commutative 92.4%

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

   4. distribute-rgt-in 95.1%

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

   5. +-commutative 95.1%

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

6. Applied egg-rr 95.1%

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

   1. associate-*r/ 95.2%

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

   2. *-rgt-identity 95.2%

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

   3. associate-/r* 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 94.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{y + x}\\ \mathbf{if}\;x \leq -1.16 \cdot 10^{+85}:\\ \;\;\;\;\frac{t\_0}{y + x}\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(y + x\right) \cdot \frac{y + 1}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y x))))
   (if (<= x -1.16e+85)
     (/ t_0 (+ y x))
     (if (<= x -2.05e-13)
       (* x (/ y (* (* (+ y x) (+ y x)) (+ x (+ y 1.0)))))
       (/ t_0 (* (+ y x) (/ (+ y 1.0) x)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (x <= -1.16e+85) {
		tmp = t_0 / (y + x);
	} else if (x <= -2.05e-13) {
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))));
	} else {
		tmp = t_0 / ((y + x) * ((y + 1.0) / x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (y + x)
    if (x <= (-1.16d+85)) then
        tmp = t_0 / (y + x)
    else if (x <= (-2.05d-13)) then
        tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0d0))))
    else
        tmp = t_0 / ((y + x) * ((y + 1.0d0) / x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (x <= -1.16e+85) {
		tmp = t_0 / (y + x);
	} else if (x <= -2.05e-13) {
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))));
	} else {
		tmp = t_0 / ((y + x) * ((y + 1.0) / x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (y + x)
	tmp = 0
	if x <= -1.16e+85:
		tmp = t_0 / (y + x)
	elif x <= -2.05e-13:
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))))
	else:
		tmp = t_0 / ((y + x) * ((y + 1.0) / x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(y + x))
	tmp = 0.0
	if (x <= -1.16e+85)
		tmp = Float64(t_0 / Float64(y + x));
	elseif (x <= -2.05e-13)
		tmp = Float64(x * Float64(y / Float64(Float64(Float64(y + x) * Float64(y + x)) * Float64(x + Float64(y + 1.0)))));
	else
		tmp = Float64(t_0 / Float64(Float64(y + x) * Float64(Float64(y + 1.0) / x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (y + x);
	tmp = 0.0;
	if (x <= -1.16e+85)
		tmp = t_0 / (y + x);
	elseif (x <= -2.05e-13)
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))));
	else
		tmp = t_0 / ((y + x) * ((y + 1.0) / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.16e+85], N[(t$95$0 / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.05e-13], 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[(t$95$0 / N[(N[(y + x), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.15999999999999995e85

   1. Initial program 66.5%

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

      1. *-commutative 66.5%

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

      2. associate-*l* 66.5%

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

      3. times-frac 90.2%

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

      4. +-commutative 90.2%

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

      5. +-commutative 90.2%

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

      6. associate-+r+ 90.2%

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

      7. +-commutative 90.2%

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

      8. associate-+l+ 90.2%

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

   4. Applied egg-rr 90.2%

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

      1. div-inv 90.2%

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

      2. distribute-rgt-in 82.3%

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

      3. +-commutative 82.3%

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

      4. distribute-rgt-in 90.2%

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

      5. +-commutative 90.2%

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

   6. Applied egg-rr 90.2%

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

      1. associate-*r/ 90.2%

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

      2. *-rgt-identity 90.2%

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

      3. associate-/r* 99.9%

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

   8. Simplified 99.9%

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

      1. associate-/l/ 90.2%

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

      2. *-commutative 90.2%

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

      3. clear-num 90.2%

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

      4. un-div-inv 90.2%

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

      5. associate-/l* 100.0%

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

      6. +-commutative 100.0%

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

   10. Applied egg-rr 100.0%

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

   11. Taylor expanded in x around inf 93.3%

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

   if -1.15999999999999995e85 < x < -2.0500000000000001e-13

   1. Initial program 78.6%

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

      1. associate-/l* 94.6%

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

      2. associate-+l+ 94.6%

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

   3. Simplified 94.6%

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

   if -2.0500000000000001e-13 < x

   1. Initial program 71.5%

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

      1. *-commutative 71.5%

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

      2. associate-*l* 71.6%

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

      3. times-frac 95.9%

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

      4. +-commutative 95.9%

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

      5. +-commutative 95.9%

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

      6. associate-+r+ 95.9%

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

      7. +-commutative 95.9%

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

      8. associate-+l+ 95.9%

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

   4. Applied egg-rr 95.9%

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

      1. div-inv 95.7%

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

      2. distribute-rgt-in 94.4%

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

      3. +-commutative 94.4%

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

      4. distribute-rgt-in 95.7%

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

      5. +-commutative 95.7%

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

   6. Applied egg-rr 95.7%

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

      1. associate-*r/ 95.9%

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

      2. *-rgt-identity 95.9%

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

      3. associate-/r* 99.8%

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

   8. Simplified 99.8%

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

      1. associate-/l/ 95.9%

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

      2. *-commutative 95.9%

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

      3. clear-num 95.8%

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

      4. un-div-inv 95.8%

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

      5. associate-/l* 99.6%

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

      6. +-commutative 99.6%

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

   10. Applied egg-rr 99.6%

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

   11. Taylor expanded in x around 0 78.4%

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

       1. +-commutative 78.4%

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

   13. Simplified 78.4%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{+85}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + x}\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{\left(y + x\right) \cdot \frac{y + 1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;y \leq 7.8 \cdot 10^{+174}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{t\_0}\\ \end{array} \end{array} \]

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 7.8e+174)
     (* (/ y (+ y x)) (/ x (* (+ y x) t_0)))
     (/ (/ x (+ y x)) t_0))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= 7.8e+174) {
		tmp = (y / (y + x)) * (x / ((y + x) * t_0));
	} else {
		tmp = (x / (y + x)) / t_0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (y <= 7.8d+174) then
        tmp = (y / (y + x)) * (x / ((y + x) * t_0))
    else
        tmp = (x / (y + x)) / t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= 7.8e+174) {
		tmp = (y / (y + x)) * (x / ((y + x) * t_0));
	} else {
		tmp = (x / (y + x)) / t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if y <= 7.8e+174:
		tmp = (y / (y + x)) * (x / ((y + x) * t_0))
	else:
		tmp = (x / (y + x)) / t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (y <= 7.8e+174)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(Float64(y + x) * t_0)));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / t_0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (y <= 7.8e+174)
		tmp = (y / (y + x)) * (x / ((y + x) * t_0));
	else
		tmp = (x / (y + x)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 7.8e+174], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 7.79999999999999962e174

   1. Initial program 72.1%

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

      1. *-commutative 72.1%

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

      2. associate-*l* 72.1%

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

      3. times-frac 95.3%

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

      4. +-commutative 95.3%

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

      5. +-commutative 95.3%

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

      6. associate-+r+ 95.3%

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

      7. +-commutative 95.3%

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

      8. associate-+l+ 95.3%

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

   4. Applied egg-rr 95.3%

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

   if 7.79999999999999962e174 < y

   1. Initial program 61.9%

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

      1. *-commutative 61.9%

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

      2. associate-*l* 61.9%

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

      3. times-frac 94.7%

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

      4. +-commutative 94.7%

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

      5. +-commutative 94.7%

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

      6. associate-+r+ 94.7%

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

      7. +-commutative 94.7%

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

      8. associate-+l+ 94.7%

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

   4. Applied egg-rr 94.7%

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

      1. div-inv 94.7%

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

      2. distribute-rgt-in 76.5%

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

      3. +-commutative 76.5%

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

      4. distribute-rgt-in 94.7%

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

      5. +-commutative 94.7%

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

   6. Applied egg-rr 94.7%

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

      1. associate-*r/ 94.7%

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

      2. *-rgt-identity 94.7%

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

      3. associate-/r* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in y around inf 88.9%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.8 \cdot 10^{+174}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{y + x}\\ \mathbf{if}\;x \leq -1.8:\\ \;\;\;\;t\_0 \cdot \frac{1}{y + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(y + x\right) \cdot \frac{y + 1}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y x))))
   (if (<= x -1.8)
     (* t_0 (/ 1.0 (+ y (+ x 1.0))))
     (/ t_0 (* (+ y x) (/ (+ y 1.0) x))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (x <= -1.8) {
		tmp = t_0 * (1.0 / (y + (x + 1.0)));
	} else {
		tmp = t_0 / ((y + x) * ((y + 1.0) / x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (y + x)
    if (x <= (-1.8d0)) then
        tmp = t_0 * (1.0d0 / (y + (x + 1.0d0)))
    else
        tmp = t_0 / ((y + x) * ((y + 1.0d0) / x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (x <= -1.8) {
		tmp = t_0 * (1.0 / (y + (x + 1.0)));
	} else {
		tmp = t_0 / ((y + x) * ((y + 1.0) / x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (y + x)
	tmp = 0
	if x <= -1.8:
		tmp = t_0 * (1.0 / (y + (x + 1.0)))
	else:
		tmp = t_0 / ((y + x) * ((y + 1.0) / x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(y + x))
	tmp = 0.0
	if (x <= -1.8)
		tmp = Float64(t_0 * Float64(1.0 / Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(t_0 / Float64(Float64(y + x) * Float64(Float64(y + 1.0) / x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (y + x);
	tmp = 0.0;
	if (x <= -1.8)
		tmp = t_0 * (1.0 / (y + (x + 1.0)));
	else
		tmp = t_0 / ((y + x) * ((y + 1.0) / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.8], N[(t$95$0 * N[(1.0 / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(y + x), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.80000000000000004

   1. Initial program 68.8%

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

      1. *-commutative 68.8%

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

      2. associate-*l* 68.8%

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

      3. times-frac 92.8%

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

      4. +-commutative 92.8%

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

      5. +-commutative 92.8%

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

      6. associate-+r+ 92.8%

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

      7. +-commutative 92.8%

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

      8. associate-+l+ 92.8%

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

   4. Applied egg-rr 92.8%

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

      1. div-inv 92.8%

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

      2. distribute-rgt-in 85.4%

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

      3. +-commutative 85.4%

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

      4. distribute-rgt-in 92.8%

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

      5. +-commutative 92.8%

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

   6. Applied egg-rr 92.8%

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

      1. associate-*r/ 92.8%

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

      2. *-rgt-identity 92.8%

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

      3. associate-/r* 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around inf 83.7%

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

   if -1.80000000000000004 < x

   1. Initial program 72.0%

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

      1. *-commutative 72.0%

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

      2. associate-*l* 72.0%

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

      3. times-frac 95.9%

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

      4. +-commutative 95.9%

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

      5. +-commutative 95.9%

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

      6. associate-+r+ 95.9%

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

      7. +-commutative 95.9%

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

      8. associate-+l+ 95.9%

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

   4. Applied egg-rr 95.9%

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

      1. div-inv 95.8%

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

      2. distribute-rgt-in 94.4%

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

      3. +-commutative 94.4%

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

      4. distribute-rgt-in 95.8%

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

      5. +-commutative 95.8%

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

   6. Applied egg-rr 95.8%

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

      1. associate-*r/ 95.9%

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

      2. *-rgt-identity 95.9%

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

      3. associate-/r* 99.8%

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

   8. Simplified 99.8%

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

      1. associate-/l/ 95.9%

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

      2. *-commutative 95.9%

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

      3. clear-num 95.8%

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

      4. un-div-inv 95.8%

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

      5. associate-/l* 99.6%

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

      6. +-commutative 99.6%

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

   10. Applied egg-rr 99.6%

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

   11. Taylor expanded in x around 0 78.1%

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

       1. +-commutative 78.1%

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

   13. Simplified 78.1%

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

4. Final simplification 79.4%

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

Alternative 5: 82.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-88}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.1e-88) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ y x)) (+ y (+ x 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.1e-88) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + x)) / (y + (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-88) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (y + x)) / (y + (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-88) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + x)) / (y + (x + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.1e-88:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (y + x)) / (y + (x + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.1e-88)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(y + 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-88)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (y + x)) / (y + (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-88], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.10000000000000002e-88

   1. Initial program 72.3%

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

      1. associate-/l* 85.9%

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

      2. associate-+l+ 85.9%

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

   3. Simplified 85.9%

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

   5. Taylor expanded in y around 0 57.3%

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

      1. associate-/r* 60.0%

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

      2. +-commutative 60.0%

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

   7. Simplified 60.0%

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

   if 1.10000000000000002e-88 < y

   1. Initial program 68.8%

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

      1. *-commutative 68.8%

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

      2. associate-*l* 68.8%

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

      3. times-frac 93.5%

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

      4. +-commutative 93.5%

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

      5. +-commutative 93.5%

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

      6. associate-+r+ 93.5%

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

      7. +-commutative 93.5%

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

      8. associate-+l+ 93.5%

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

   4. Applied egg-rr 93.5%

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

      1. div-inv 93.3%

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

      2. distribute-rgt-in 87.8%

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

      3. +-commutative 87.8%

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

      4. distribute-rgt-in 93.3%

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

      5. +-commutative 93.3%

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

   6. Applied egg-rr 93.3%

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

      1. associate-*r/ 93.5%

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

      2. *-rgt-identity 93.5%

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

      3. associate-/r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in y around inf 65.5%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-88}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-88}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.1e-88) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ y 1.0)) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.1e-88) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.1d-88) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.1e-88) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.1e-88:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.1e-88)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.1e-88)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.1e-88], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.10000000000000002e-88

   1. Initial program 72.3%

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

      1. associate-/l* 85.9%

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

      2. associate-+l+ 85.9%

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

   3. Simplified 85.9%

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

   5. Taylor expanded in y around 0 57.3%

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

      1. associate-/r* 60.0%

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

      2. +-commutative 60.0%

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

   7. Simplified 60.0%

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

   if 1.10000000000000002e-88 < y

   1. Initial program 68.8%

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

      1. *-commutative 68.8%

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

      2. associate-*l* 68.8%

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

      3. times-frac 93.5%

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

      4. +-commutative 93.5%

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

      5. +-commutative 93.5%

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

      6. associate-+r+ 93.5%

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

      7. +-commutative 93.5%

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

      8. associate-+l+ 93.5%

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

   4. Applied egg-rr 93.5%

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

      1. add-sqr-sqrt 93.2%

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

      2. associate-*l/ 93.2%

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

      3. *-commutative 93.2%

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

      4. clear-num 93.0%

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

      5. associate-*l/ 93.0%

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

      6. add-sqr-sqrt 93.3%

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

      7. frac-times 88.6%

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

      8. *-un-lft-identity 88.6%

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

      9. distribute-rgt-in 83.2%

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

      10. +-commutative 83.2%

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

      11. distribute-rgt-in 88.6%

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

      12. +-commutative 88.6%

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

      13. +-commutative 88.6%

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

   6. Applied egg-rr 88.6%

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

   7. Taylor expanded in x around 0 63.2%

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

      1. associate-/l* 63.1%

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

      2. +-commutative 63.1%

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

   9. Simplified 63.1%

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

       1. *-un-lft-identity 63.1%

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

       2. +-commutative 63.1%

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

       3. *-commutative 63.1%

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

       4. times-frac 64.3%

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

       5. *-un-lft-identity 64.3%

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

       6. *-commutative 64.3%

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

       7. times-frac 64.6%

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

       8. clear-num 64.5%

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

   11. Applied egg-rr 64.5%

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

       1. associate-*l/ 64.6%

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

       2. *-inverses 64.6%

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

       3. *-rgt-identity 64.6%

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

       4. *-lft-identity 64.6%

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

   13. Simplified 64.6%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-88}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4.5e-95) (/ y x) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.5e-95) {
		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 <= 4.5d-95) 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 <= 4.5e-95) {
		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 <= 4.5e-95:
		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 <= 4.5e-95)
		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 <= 4.5e-95)
		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, 4.5e-95], 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 < 4.5e-95

   1. Initial program 71.9%

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

      1. associate-/l* 86.2%

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

      2. associate-+l+ 86.2%

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

   3. Simplified 86.2%

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

   5. Taylor expanded in y around 0 56.6%

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

      1. associate-/r* 59.4%

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

      2. +-commutative 59.4%

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

   7. Simplified 59.4%

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

   8. Taylor expanded in x around 0 34.1%

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

   if 4.5e-95 < 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* 81.4%

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

      2. associate-+l+ 81.4%

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

   3. Simplified 81.4%

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

   5. Taylor expanded in x around 0 65.4%

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

      1. +-commutative 65.4%

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

   7. Simplified 65.4%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.9% accurate, 1.4× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1e-88) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1e-88) {
		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 <= 1d-88) 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 <= 1e-88) {
		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 <= 1e-88:
		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 <= 1e-88)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1e-88)
		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, 1e-88], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.99999999999999934e-89

   1. Initial program 72.3%

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

      1. associate-/l* 85.9%

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

      2. associate-+l+ 85.9%

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

   3. Simplified 85.9%

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

   5. Taylor expanded in y around 0 57.3%

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

   if 9.99999999999999934e-89 < y

   1. Initial program 68.8%

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

      1. associate-/l* 81.8%

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

      2. associate-+l+ 81.8%

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

   3. Simplified 81.8%

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

   5. Taylor expanded in x around 0 67.8%

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

      1. +-commutative 67.8%

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

   7. Simplified 67.8%

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

4. Final simplification 60.5%

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

Alternative 9: 80.9% accurate, 1.4× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1e-88) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1e-88) {
		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 <= 1d-88) 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 <= 1e-88) {
		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 <= 1e-88:
		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 <= 1e-88)
		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 <= 1e-88)
		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, 1e-88], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.99999999999999934e-89

   1. Initial program 72.3%

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

      1. associate-/l* 85.9%

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

      2. associate-+l+ 85.9%

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

   3. Simplified 85.9%

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

   5. Taylor expanded in y around 0 57.3%

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

   if 9.99999999999999934e-89 < y

   1. Initial program 68.8%

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

      1. associate-/l* 81.8%

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

      2. associate-+l+ 81.8%

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

   3. Simplified 81.8%

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

   5. Taylor expanded in x around 0 67.6%

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

      1. +-commutative 67.6%

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

      2. div-inv 67.8%

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

      3. associate-/r* 64.2%

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

   7. Applied egg-rr 64.2%

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

4. Final simplification 59.5%

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

Alternative 10: 82.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-88}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.1e-88) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.1e-88) {
		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 <= 1.1d-88) 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 <= 1.1e-88) {
		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 <= 1.1e-88:
		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 <= 1.1e-88)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.1e-88)
		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, 1.1e-88], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.10000000000000002e-88

   1. Initial program 72.3%

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

      1. associate-/l* 85.9%

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

      2. associate-+l+ 85.9%

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

   3. Simplified 85.9%

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

   5. Taylor expanded in y around 0 57.3%

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

      1. associate-/r* 60.0%

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

      2. +-commutative 60.0%

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

   7. Simplified 60.0%

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

   if 1.10000000000000002e-88 < y

   1. Initial program 68.8%

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

      1. associate-/l* 81.8%

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

      2. associate-+l+ 81.8%

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

   3. Simplified 81.8%

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

   5. Taylor expanded in x around 0 67.6%

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

      1. +-commutative 67.6%

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

      2. div-inv 67.8%

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

      3. associate-/r* 64.2%

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

   7. Applied egg-rr 64.2%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-88}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4.305 \cdot 10^{-172}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -4.305e-172) (/ y x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4.305e-172) {
		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 (x <= (-4.305d-172)) 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 (x <= -4.305e-172) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -4.305e-172:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -4.305e-172)
		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 (x <= -4.305e-172)
		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[x, -4.305e-172], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.3049999999999998e-172

   1. Initial program 72.1%

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

      1. associate-/l* 88.9%

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

      2. associate-+l+ 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in y around 0 66.2%

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

      1. associate-/r* 69.8%

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

      2. +-commutative 69.8%

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

   7. Simplified 69.8%

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

   8. Taylor expanded in x around 0 36.4%

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

   if -4.3049999999999998e-172 < x

   1. Initial program 70.8%

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

      1. associate-/l* 82.3%

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

      2. associate-+l+ 82.3%

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

   3. Simplified 82.3%

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

   5. Taylor expanded in x around 0 56.5%

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

      1. +-commutative 56.5%

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

   7. Simplified 56.5%

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

   8. Taylor expanded in y around 0 33.6%

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

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.305 \cdot 10^{-172}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 26.1% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.2%

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

   1. associate-/l* 84.6%

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

   2. associate-+l+ 84.6%

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

3. Simplified 84.6%

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

5. Taylor expanded in x around 0 48.5%

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

   1. +-commutative 48.5%

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

7. Simplified 48.5%

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

8. Taylor expanded in y around 0 24.7%

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

9. Final simplification 24.7%

   \[\leadsto \frac{x}{y} \]
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 2024077 
(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))))