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

Percentage Accurate: 69.0% → 99.8%

Time: 14.6s

Alternatives: 15

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: 69.0% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

   1. associate-*l* 63.6%

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

   2. times-frac 91.5%

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

   3. +-commutative 91.5%

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

   4. +-commutative 91.5%

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

   5. associate-+r+ 91.5%

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

   6. +-commutative 91.5%

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

   7. associate-+l+ 91.5%

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

4. Applied egg-rr 91.5%

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

   1. *-commutative 91.5%

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

   2. clear-num 91.4%

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

   3. frac-times 87.4%

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

   4. *-rgt-identity 87.4%

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

   5. +-commutative 87.4%

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

6. Applied egg-rr 87.4%

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

   1. associate-/r* 91.4%

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

   2. associate-/r* 99.7%

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

   3. associate-/r* 98.9%

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

   4. *-commutative 98.9%

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

   5. associate-/r* 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 91.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{+101}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{t\_0}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -1.75e+101)
     (/ (/ y (+ y x)) t_0)
     (if (<= x -1.6e-162)
       (* x (/ y (* (* (+ y x) (+ y x)) (+ x (+ y 1.0)))))
       (/ (/ x y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.75e+101) {
		tmp = (y / (y + x)) / t_0;
	} else if (x <= -1.6e-162) {
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (x <= (-1.75d+101)) then
        tmp = (y / (y + x)) / t_0
    else if (x <= (-1.6d-162)) then
        tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0d0))))
    else
        tmp = (x / y) / t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.75e+101) {
		tmp = (y / (y + x)) / t_0;
	} else if (x <= -1.6e-162) {
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -1.75e+101:
		tmp = (y / (y + x)) / t_0
	elif x <= -1.6e-162:
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))))
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1.75e+101)
		tmp = Float64(Float64(y / Float64(y + x)) / t_0);
	elseif (x <= -1.6e-162)
		tmp = Float64(x * Float64(y / Float64(Float64(Float64(y + x) * Float64(y + x)) * Float64(x + Float64(y + 1.0)))));
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -1.75e+101)
		tmp = (y / (y + x)) / t_0;
	elseif (x <= -1.6e-162)
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))));
	else
		tmp = (x / y) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.75e+101], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, -1.6e-162], 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 / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.75000000000000012e101

   1. Initial program 43.3%

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

      1. associate-*l* 43.3%

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

      2. times-frac 79.1%

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

      3. +-commutative 79.1%

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

      4. +-commutative 79.1%

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

      5. associate-+r+ 79.1%

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

      6. +-commutative 79.1%

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

      7. associate-+l+ 79.1%

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

   4. Applied egg-rr 79.1%

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

      1. *-commutative 79.1%

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

      2. clear-num 79.0%

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

      3. frac-times 77.0%

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

      4. *-rgt-identity 77.0%

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

      5. +-commutative 77.0%

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

   6. Applied egg-rr 77.0%

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

      1. associate-/r* 79.0%

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

      2. associate-/r* 99.8%

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

      3. associate-/r* 99.0%

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

      4. *-commutative 99.0%

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

      5. associate-/r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in y around 0 79.0%

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

   if -1.75000000000000012e101 < x < -1.59999999999999988e-162

   1. Initial program 82.9%

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

      1. associate-/l* 87.7%

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

      2. associate-+l+ 87.7%

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

   3. Simplified 87.7%

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

   if -1.59999999999999988e-162 < x

   1. Initial program 63.0%

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

      1. associate-*l* 63.0%

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

      2. times-frac 92.7%

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

      3. +-commutative 92.7%

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

      4. +-commutative 92.7%

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

      5. associate-+r+ 92.7%

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

      6. +-commutative 92.7%

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

      7. associate-+l+ 92.7%

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

   4. Applied egg-rr 92.7%

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

      1. *-commutative 92.7%

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

      2. clear-num 92.6%

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

      3. frac-times 87.9%

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

      4. *-rgt-identity 87.9%

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

      5. +-commutative 87.9%

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

   6. Applied egg-rr 87.9%

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

      1. associate-/r* 92.6%

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

      2. associate-/r* 99.7%

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

      3. associate-/r* 99.0%

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

      4. *-commutative 99.0%

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

      5. associate-/r* 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in y around inf 55.5%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+101}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -2 \cdot 10^{+111}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{t\_0}\\ \mathbf{elif}\;x \leq 3.65 \cdot 10^{-47}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(y + x\right) \cdot t\_0}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -2e+111)
     (/ (/ y (+ y x)) t_0)
     (if (<= x 3.65e-47)
       (* x (/ (/ y (* (+ y x) t_0)) (+ y x)))
       (/ (/ x y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -2e+111) {
		tmp = (y / (y + x)) / t_0;
	} else if (x <= 3.65e-47) {
		tmp = x * ((y / ((y + x) * t_0)) / (y + x));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (x <= (-2d+111)) then
        tmp = (y / (y + x)) / t_0
    else if (x <= 3.65d-47) then
        tmp = x * ((y / ((y + x) * t_0)) / (y + x))
    else
        tmp = (x / y) / t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -2e+111) {
		tmp = (y / (y + x)) / t_0;
	} else if (x <= 3.65e-47) {
		tmp = x * ((y / ((y + x) * t_0)) / (y + x));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -2e+111:
		tmp = (y / (y + x)) / t_0
	elif x <= 3.65e-47:
		tmp = x * ((y / ((y + x) * t_0)) / (y + x))
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -2e+111)
		tmp = Float64(Float64(y / Float64(y + x)) / t_0);
	elseif (x <= 3.65e-47)
		tmp = Float64(x * Float64(Float64(y / Float64(Float64(y + x) * t_0)) / Float64(y + x)));
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -2e+111)
		tmp = (y / (y + x)) / t_0;
	elseif (x <= 3.65e-47)
		tmp = x * ((y / ((y + x) * t_0)) / (y + x));
	else
		tmp = (x / y) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2e+111], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, 3.65e-47], N[(x * N[(N[(y / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.99999999999999991e111

   1. Initial program 42.1%

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

      1. associate-*l* 42.1%

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

      2. times-frac 78.6%

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

      3. +-commutative 78.6%

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

      4. +-commutative 78.6%

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

      5. associate-+r+ 78.6%

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

      6. +-commutative 78.6%

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

      7. associate-+l+ 78.6%

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

   4. Applied egg-rr 78.6%

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

      1. *-commutative 78.6%

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

      2. clear-num 78.6%

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

      3. frac-times 76.4%

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

      4. *-rgt-identity 76.4%

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

      5. +-commutative 76.4%

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

   6. Applied egg-rr 76.4%

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

      1. associate-/r* 78.6%

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

      2. associate-/r* 99.8%

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

      3. associate-/r* 99.0%

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

      4. *-commutative 99.0%

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

      5. associate-/r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in y around 0 78.5%

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

   if -1.99999999999999991e111 < x < 3.65000000000000021e-47

   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* 82.7%

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

      2. associate-+l+ 82.7%

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

   3. Simplified 82.7%

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

      1. *-un-lft-identity 82.7%

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

      2. associate-+r+ 82.7%

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

      3. associate-*l* 82.7%

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

      4. times-frac 97.8%

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

      5. +-commutative 97.8%

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

      6. +-commutative 97.8%

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

      7. associate-+r+ 97.8%

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

      8. +-commutative 97.8%

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

      9. associate-+l+ 97.8%

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

   6. Applied egg-rr 97.8%

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

      1. associate-*l/ 97.8%

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

      2. *-lft-identity 97.8%

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

      3. +-commutative 97.8%

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

   8. Simplified 97.8%

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

   if 3.65000000000000021e-47 < x

   1. Initial program 61.3%

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

      1. associate-*l* 61.3%

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

      2. times-frac 85.6%

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

      3. +-commutative 85.6%

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

      4. +-commutative 85.6%

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

      5. associate-+r+ 85.6%

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

      6. +-commutative 85.6%

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

      7. associate-+l+ 85.6%

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

   4. Applied egg-rr 85.6%

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

      1. *-commutative 85.6%

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

      2. clear-num 85.6%

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

      3. frac-times 81.0%

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

      4. *-rgt-identity 81.0%

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

      5. +-commutative 81.0%

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

   6. Applied egg-rr 81.0%

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

      1. associate-/r* 85.6%

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

      2. associate-/r* 99.7%

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

      3. associate-/r* 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-/r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in y around inf 36.1%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+111}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 3.65 \cdot 10^{-47}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{t\_0}\\ \mathbf{elif}\;x \leq 3.65 \cdot 10^{-47}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -1.35e+141)
     (/ (/ y (+ y x)) t_0)
     (if (<= x 3.65e-47)
       (* (/ x (+ y x)) (/ y (* (+ y x) t_0)))
       (/ (/ x y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.35e+141) {
		tmp = (y / (y + x)) / t_0;
	} else if (x <= 3.65e-47) {
		tmp = (x / (y + x)) * (y / ((y + x) * t_0));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (x <= (-1.35d+141)) then
        tmp = (y / (y + x)) / t_0
    else if (x <= 3.65d-47) then
        tmp = (x / (y + x)) * (y / ((y + x) * t_0))
    else
        tmp = (x / y) / t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.35e+141) {
		tmp = (y / (y + x)) / t_0;
	} else if (x <= 3.65e-47) {
		tmp = (x / (y + x)) * (y / ((y + x) * t_0));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -1.35e+141:
		tmp = (y / (y + x)) / t_0
	elif x <= 3.65e-47:
		tmp = (x / (y + x)) * (y / ((y + x) * t_0))
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1.35e+141)
		tmp = Float64(Float64(y / Float64(y + x)) / t_0);
	elseif (x <= 3.65e-47)
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(Float64(y + x) * t_0)));
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -1.35e+141)
		tmp = (y / (y + x)) / t_0;
	elseif (x <= 3.65e-47)
		tmp = (x / (y + x)) * (y / ((y + x) * t_0));
	else
		tmp = (x / y) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e+141], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, 3.65e-47], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35e141

   1. Initial program 45.6%

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

      1. associate-*l* 45.6%

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

      2. times-frac 80.1%

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

      3. +-commutative 80.1%

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

      4. +-commutative 80.1%

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

      5. associate-+r+ 80.1%

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

      6. +-commutative 80.1%

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

      7. associate-+l+ 80.1%

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

   4. Applied egg-rr 80.1%

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

      1. *-commutative 80.1%

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

      2. clear-num 80.1%

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

      3. frac-times 80.1%

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

      4. *-rgt-identity 80.1%

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

      5. +-commutative 80.1%

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

   6. Applied egg-rr 80.1%

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

      1. associate-/r* 80.1%

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

      2. associate-/r* 99.9%

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

      3. associate-/r* 99.0%

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

      4. *-commutative 99.0%

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

      5. associate-/r* 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in y around 0 85.3%

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

   if -1.35e141 < x < 3.65000000000000021e-47

   1. Initial program 69.8%

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

      1. associate-*l* 69.8%

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

      2. times-frac 98.0%

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

      3. +-commutative 98.0%

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

      4. +-commutative 98.0%

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

      5. associate-+r+ 98.0%

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

      6. +-commutative 98.0%

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

      7. associate-+l+ 98.0%

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

   4. Applied egg-rr 98.0%

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

   if 3.65000000000000021e-47 < x

   1. Initial program 61.3%

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

      1. associate-*l* 61.3%

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

      2. times-frac 85.6%

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

      3. +-commutative 85.6%

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

      4. +-commutative 85.6%

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

      5. associate-+r+ 85.6%

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

      6. +-commutative 85.6%

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

      7. associate-+l+ 85.6%

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

   4. Applied egg-rr 85.6%

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

      1. *-commutative 85.6%

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

      2. clear-num 85.6%

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

      3. frac-times 81.0%

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

      4. *-rgt-identity 81.0%

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

      5. +-commutative 81.0%

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

   6. Applied egg-rr 81.0%

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

      1. associate-/r* 85.6%

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

      2. associate-/r* 99.7%

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

      3. associate-/r* 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-/r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in y around inf 36.1%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 3.65 \cdot 10^{-47}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{+72}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-55}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -1.4e+72)
     (/ (/ y x) t_0)
     (if (<= x -4.1e-55) (/ y (* (+ y x) t_0)) (/ (/ x y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.4e+72) {
		tmp = (y / x) / t_0;
	} else if (x <= -4.1e-55) {
		tmp = y / ((y + x) * t_0);
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (x <= (-1.4d+72)) then
        tmp = (y / x) / t_0
    else if (x <= (-4.1d-55)) then
        tmp = y / ((y + x) * t_0)
    else
        tmp = (x / y) / t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.4e+72) {
		tmp = (y / x) / t_0;
	} else if (x <= -4.1e-55) {
		tmp = y / ((y + x) * t_0);
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -1.4e+72:
		tmp = (y / x) / t_0
	elif x <= -4.1e-55:
		tmp = y / ((y + x) * t_0)
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1.4e+72)
		tmp = Float64(Float64(y / x) / t_0);
	elseif (x <= -4.1e-55)
		tmp = Float64(y / Float64(Float64(y + x) * t_0));
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -1.4e+72)
		tmp = (y / x) / t_0;
	elseif (x <= -4.1e-55)
		tmp = y / ((y + x) * t_0);
	else
		tmp = (x / y) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.4e+72], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, -4.1e-55], N[(y / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4e72

   1. Initial program 45.2%

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

      1. associate-*l* 45.2%

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

      2. times-frac 81.8%

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

      3. +-commutative 81.8%

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

      4. +-commutative 81.8%

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

      5. associate-+r+ 81.8%

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

      6. +-commutative 81.8%

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

      7. associate-+l+ 81.8%

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

   4. Applied egg-rr 81.8%

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

      1. *-commutative 81.8%

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

      2. clear-num 81.8%

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

      3. frac-times 80.0%

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

      4. *-rgt-identity 80.0%

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

      5. +-commutative 80.0%

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

   6. Applied egg-rr 80.0%

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

      1. associate-/r* 81.8%

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

      2. associate-/r* 99.7%

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

      3. associate-/r* 99.1%

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

      4. *-commutative 99.1%

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

      5. associate-/r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in y around 0 74.6%

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

   if -1.4e72 < x < -4.0999999999999998e-55

   1. Initial program 82.7%

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

      1. associate-*l* 82.7%

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

      2. times-frac 97.3%

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

      3. +-commutative 97.3%

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

      4. +-commutative 97.3%

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

      5. associate-+r+ 97.3%

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

      6. +-commutative 97.3%

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

      7. associate-+l+ 97.3%

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

   4. Applied egg-rr 97.3%

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

   5. Taylor expanded in x around inf 67.6%

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

   if -4.0999999999999998e-55 < x

   1. Initial program 66.1%

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

      1. associate-*l* 66.1%

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

      2. times-frac 93.4%

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

      3. +-commutative 93.4%

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

      4. +-commutative 93.4%

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

      5. associate-+r+ 93.4%

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

      6. +-commutative 93.4%

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

      7. associate-+l+ 93.4%

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

   4. Applied egg-rr 93.4%

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

      1. *-commutative 93.4%

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

      2. clear-num 93.4%

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

      3. frac-times 89.1%

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

      4. *-rgt-identity 89.1%

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

      5. +-commutative 89.1%

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

   6. Applied egg-rr 89.1%

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

      1. associate-/r* 93.4%

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

      2. associate-/r* 99.7%

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

      3. associate-/r* 99.1%

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

      4. *-commutative 99.1%

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

      5. associate-/r* 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in y around inf 58.8%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+72}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-55}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{t\_0}\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -1.35e+141)
     (/ (/ y (+ y x)) t_0)
     (if (<= x -1.2e-54) (/ y (* (+ y x) t_0)) (/ (/ x y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.35e+141) {
		tmp = (y / (y + x)) / t_0;
	} else if (x <= -1.2e-54) {
		tmp = y / ((y + x) * t_0);
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (x <= (-1.35d+141)) then
        tmp = (y / (y + x)) / t_0
    else if (x <= (-1.2d-54)) then
        tmp = y / ((y + x) * t_0)
    else
        tmp = (x / y) / t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.35e+141) {
		tmp = (y / (y + x)) / t_0;
	} else if (x <= -1.2e-54) {
		tmp = y / ((y + x) * t_0);
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -1.35e+141:
		tmp = (y / (y + x)) / t_0
	elif x <= -1.2e-54:
		tmp = y / ((y + x) * t_0)
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1.35e+141)
		tmp = Float64(Float64(y / Float64(y + x)) / t_0);
	elseif (x <= -1.2e-54)
		tmp = Float64(y / Float64(Float64(y + x) * t_0));
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -1.35e+141)
		tmp = (y / (y + x)) / t_0;
	elseif (x <= -1.2e-54)
		tmp = y / ((y + x) * t_0);
	else
		tmp = (x / y) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e+141], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, -1.2e-54], N[(y / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35e141

   1. Initial program 45.6%

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

      1. associate-*l* 45.6%

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

      2. times-frac 80.1%

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

      3. +-commutative 80.1%

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

      4. +-commutative 80.1%

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

      5. associate-+r+ 80.1%

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

      6. +-commutative 80.1%

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

      7. associate-+l+ 80.1%

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

   4. Applied egg-rr 80.1%

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

      1. *-commutative 80.1%

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

      2. clear-num 80.1%

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

      3. frac-times 80.1%

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

      4. *-rgt-identity 80.1%

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

      5. +-commutative 80.1%

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

   6. Applied egg-rr 80.1%

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

      1. associate-/r* 80.1%

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

      2. associate-/r* 99.9%

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

      3. associate-/r* 99.0%

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

      4. *-commutative 99.0%

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

      5. associate-/r* 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in y around 0 85.3%

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

   if -1.35e141 < x < -1.20000000000000007e-54

   1. Initial program 69.3%

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

      1. associate-*l* 69.3%

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

      2. times-frac 93.5%

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

      3. +-commutative 93.5%

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

      4. +-commutative 93.5%

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

      5. associate-+r+ 93.5%

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

      6. +-commutative 93.5%

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

      7. associate-+l+ 93.5%

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

   4. Applied egg-rr 93.5%

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

   5. Taylor expanded in x around inf 64.8%

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

   if -1.20000000000000007e-54 < x

   1. Initial program 66.1%

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

      1. associate-*l* 66.1%

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

      2. times-frac 93.4%

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

      3. +-commutative 93.4%

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

      4. +-commutative 93.4%

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

      5. associate-+r+ 93.4%

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

      6. +-commutative 93.4%

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

      7. associate-+l+ 93.4%

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

   4. Applied egg-rr 93.4%

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

      1. *-commutative 93.4%

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

      2. clear-num 93.4%

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

      3. frac-times 89.1%

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

      4. *-rgt-identity 89.1%

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

      5. +-commutative 89.1%

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

   6. Applied egg-rr 89.1%

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

      1. associate-/r* 93.4%

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

      2. associate-/r* 99.7%

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

      3. associate-/r* 99.1%

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

      4. *-commutative 99.1%

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

      5. associate-/r* 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in y around inf 58.8%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

   1. associate-*l* 63.6%

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

   2. times-frac 91.5%

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

   3. +-commutative 91.5%

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

   4. +-commutative 91.5%

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

   5. associate-+r+ 91.5%

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

   6. +-commutative 91.5%

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

   7. associate-+l+ 91.5%

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

4. Applied egg-rr 91.5%

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

   1. clear-num 91.4%

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

   2. associate-/r* 99.7%

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

   3. frac-times 98.9%

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

   4. *-un-lft-identity 98.9%

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

   5. +-commutative 98.9%

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

6. Applied egg-rr 98.9%

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

7. Final simplification 98.9%

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

Alternative 8: 82.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-54}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + \left(x + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.3e-54) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y (+ x 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.3e-54) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (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 (x <= (-2.3d-54)) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / y) / (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 (x <= -2.3e-54) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + (x + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.3e-54:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / y) / (y + (x + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.3e-54)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / y) / 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 (x <= -2.3e-54)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / y) / (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[x, -2.3e-54], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2999999999999999e-54

   1. Initial program 57.7%

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

      1. associate-/l* 76.2%

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

      2. associate-+l+ 76.2%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in y around 0 61.5%

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

      1. associate-/r* 63.6%

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

      2. +-commutative 63.6%

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

   7. Simplified 63.6%

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

   if -2.2999999999999999e-54 < x

   1. Initial program 66.1%

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

      1. associate-*l* 66.1%

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

      2. times-frac 93.4%

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

      3. +-commutative 93.4%

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

      4. +-commutative 93.4%

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

      5. associate-+r+ 93.4%

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

      6. +-commutative 93.4%

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

      7. associate-+l+ 93.4%

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

   4. Applied egg-rr 93.4%

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

      1. *-commutative 93.4%

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

      2. clear-num 93.4%

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

      3. frac-times 89.1%

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

      4. *-rgt-identity 89.1%

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

      5. +-commutative 89.1%

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

   6. Applied egg-rr 89.1%

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

      1. associate-/r* 93.4%

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

      2. associate-/r* 99.7%

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

      3. associate-/r* 99.1%

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

      4. *-commutative 99.1%

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

      5. associate-/r* 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in y around inf 58.8%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-54}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -7.2e-55) (/ (/ y x) t_0) (/ (/ x y) t_0))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -7.2e-55) {
		tmp = (y / x) / t_0;
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (x <= (-7.2d-55)) then
        tmp = (y / x) / t_0
    else
        tmp = (x / y) / t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -7.2e-55) {
		tmp = (y / x) / t_0;
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -7.2e-55:
		tmp = (y / x) / t_0
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -7.2e-55)
		tmp = Float64(Float64(y / x) / t_0);
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -7.2e-55)
		tmp = (y / x) / t_0;
	else
		tmp = (x / y) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.2e-55], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.2000000000000001e-55

   1. Initial program 57.7%

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

      1. associate-*l* 57.7%

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

      2. times-frac 87.0%

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

      3. +-commutative 87.0%

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

      4. +-commutative 87.0%

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

      5. associate-+r+ 87.0%

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

      6. +-commutative 87.0%

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

      7. associate-+l+ 87.0%

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

   4. Applied egg-rr 87.0%

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

      1. *-commutative 87.0%

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

      2. clear-num 86.9%

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

      3. frac-times 83.4%

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

      4. *-rgt-identity 83.4%

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

      5. +-commutative 83.4%

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

   6. Applied egg-rr 83.4%

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

      1. associate-/r* 86.8%

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

      2. associate-/r* 99.7%

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

      3. associate-/r* 98.5%

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

      4. *-commutative 98.5%

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

      5. associate-/r* 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in y around 0 64.6%

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

   if -7.2000000000000001e-55 < x

   1. Initial program 66.1%

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

      1. associate-*l* 66.1%

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

      2. times-frac 93.4%

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

      3. +-commutative 93.4%

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

      4. +-commutative 93.4%

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

      5. associate-+r+ 93.4%

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

      6. +-commutative 93.4%

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

      7. associate-+l+ 93.4%

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

   4. Applied egg-rr 93.4%

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

      1. *-commutative 93.4%

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

      2. clear-num 93.4%

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

      3. frac-times 89.1%

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

      4. *-rgt-identity 89.1%

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

      5. +-commutative 89.1%

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

   6. Applied egg-rr 89.1%

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

      1. associate-/r* 93.4%

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

      2. associate-/r* 99.7%

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

      3. associate-/r* 99.1%

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

      4. *-commutative 99.1%

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

      5. associate-/r* 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in y around inf 58.8%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.3e-54) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.3e-54) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.3d-54)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.3e-54) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.3e-54:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.3e-54)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.3e-54)
		tmp = y / (x * (x + 1.0));
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.3e-54], 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 x < -2.2999999999999999e-54

   1. Initial program 57.7%

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

      1. associate-/l* 76.2%

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

      2. associate-+l+ 76.2%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in y around 0 61.5%

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

   if -2.2999999999999999e-54 < x

   1. Initial program 66.1%

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

      1. associate-/l* 81.0%

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

      2. associate-+l+ 81.0%

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

   3. Simplified 81.0%

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

   5. Taylor expanded in x around 0 57.3%

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

      1. +-commutative 57.3%

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

   7. Simplified 57.3%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.3e-54) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.3e-54) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.3d-54)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.3e-54) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.3e-54:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.3e-54)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.3e-54)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.3e-54], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2999999999999999e-54

   1. Initial program 57.7%

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

      1. associate-/l* 76.2%

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

      2. associate-+l+ 76.2%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in y around 0 61.5%

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

   if -2.2999999999999999e-54 < x

   1. Initial program 66.1%

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

      1. associate-/l* 81.0%

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

      2. associate-+l+ 81.0%

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

   3. Simplified 81.0%

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

   5. Taylor expanded in x around 0 57.2%

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

      1. +-commutative 57.2%

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

      2. div-inv 57.3%

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

      3. associate-/r* 58.6%

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

   7. Applied egg-rr 58.6%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 82.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-54}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.75e-54) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.75e-54) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.75d-54)) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.75e-54) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.75e-54:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.75e-54)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.75e-54)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.75e-54], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.74999999999999991e-54

   1. Initial program 57.7%

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

      1. associate-/l* 76.2%

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

      2. associate-+l+ 76.2%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in y around 0 61.5%

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

      1. associate-/r* 63.6%

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

      2. +-commutative 63.6%

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

   7. Simplified 63.6%

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

   if -1.74999999999999991e-54 < x

   1. Initial program 66.1%

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

      1. associate-/l* 81.0%

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

      2. associate-+l+ 81.0%

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

   3. Simplified 81.0%

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

   5. Taylor expanded in x around 0 57.2%

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

      1. +-commutative 57.2%

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

      2. div-inv 57.3%

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

      3. associate-/r* 58.6%

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

   7. Applied egg-rr 58.6%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-54}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 49.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

   1. associate-/l* 79.6%

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

   2. associate-+l+ 79.6%

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

3. Simplified 79.6%

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

5. Taylor expanded in x around 0 51.1%

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

   1. +-commutative 51.1%

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

7. Simplified 51.1%

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

8. Final simplification 51.1%

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

Alternative 14: 26.5% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

   1. associate-/l* 79.6%

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

   2. associate-+l+ 79.6%

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

3. Simplified 79.6%

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

5. Taylor expanded in x around 0 51.1%

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

   1. +-commutative 51.1%

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

7. Simplified 51.1%

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

8. Taylor expanded in y around 0 21.7%

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

9. Final simplification 21.7%

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

Alternative 15: 3.4% accurate, 8.5× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return -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 = -x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

   1. associate-/l* 79.6%

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

   2. associate-+l+ 79.6%

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

3. Simplified 79.6%

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

5. Taylor expanded in x around 0 51.0%

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

6. Taylor expanded in y around 0 15.4%

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

   1. neg-mul-1 15.4%

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

8. Simplified 15.4%

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

9. Taylor expanded in y around inf 3.4%

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

    1. neg-mul-1 3.4%

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

11. Simplified 3.4%

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

12. Final simplification 3.4%

    \[\leadsto -x \]
13. 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 2024071 
(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))))