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

Percentage Accurate: 69.0% → 99.8%

Time: 21.0s

Alternatives: 18

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.0% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

   1. associate-/l* 82.7%

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

   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. associate-*r/ 70.6%

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

   2. associate-+r+ 70.6%

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

   3. times-frac 90.0%

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

   4. associate-*l/ 83.3%

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

   5. associate-+r+ 83.3%

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

   6. +-commutative 83.3%

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

   7. associate-+l+ 83.3%

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

   8. pow2 83.3%

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

   9. +-commutative 83.3%

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

6. Applied egg-rr 83.3%

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

   1. *-commutative 83.3%

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

   2. unpow2 83.3%

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

   3. times-frac 99.8%

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

   4. +-commutative 99.8%

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

8. Applied egg-rr 99.8%

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

   1. associate-*l/ 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 95.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + x\right) \cdot \left(y + x\right)\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{+163}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{+24}:\\ \;\;\;\;\frac{y}{t\_0}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{y}{t\_0 \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (+ y x) (+ y x))))
   (if (<= x -5.8e+163)
     (/ (/ y x) (+ y x))
     (if (<= x -5e+24)
       (/ y t_0)
       (if (<= x -8.2e-14)
         (* x (/ y (* t_0 (+ x (+ y 1.0)))))
         (* (/ x (+ y x)) (/ (/ y (+ y 1.0)) (+ y x))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + x) * (y + x);
	double tmp;
	if (x <= -5.8e+163) {
		tmp = (y / x) / (y + x);
	} else if (x <= -5e+24) {
		tmp = y / t_0;
	} else if (x <= -8.2e-14) {
		tmp = x * (y / (t_0 * (x + (y + 1.0))));
	} else {
		tmp = (x / (y + x)) * ((y / (y + 1.0)) / (y + x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y + x) * (y + x)
    if (x <= (-5.8d+163)) then
        tmp = (y / x) / (y + x)
    else if (x <= (-5d+24)) then
        tmp = y / t_0
    else if (x <= (-8.2d-14)) then
        tmp = x * (y / (t_0 * (x + (y + 1.0d0))))
    else
        tmp = (x / (y + x)) * ((y / (y + 1.0d0)) / (y + x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y + x) * (y + x);
	double tmp;
	if (x <= -5.8e+163) {
		tmp = (y / x) / (y + x);
	} else if (x <= -5e+24) {
		tmp = y / t_0;
	} else if (x <= -8.2e-14) {
		tmp = x * (y / (t_0 * (x + (y + 1.0))));
	} else {
		tmp = (x / (y + x)) * ((y / (y + 1.0)) / (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y + x) * (y + x)
	tmp = 0
	if x <= -5.8e+163:
		tmp = (y / x) / (y + x)
	elif x <= -5e+24:
		tmp = y / t_0
	elif x <= -8.2e-14:
		tmp = x * (y / (t_0 * (x + (y + 1.0))))
	else:
		tmp = (x / (y + x)) * ((y / (y + 1.0)) / (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y + x) * Float64(y + x))
	tmp = 0.0
	if (x <= -5.8e+163)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -5e+24)
		tmp = Float64(y / t_0);
	elseif (x <= -8.2e-14)
		tmp = Float64(x * Float64(y / Float64(t_0 * Float64(x + Float64(y + 1.0)))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(Float64(y / Float64(y + 1.0)) / Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (y + x) * (y + x);
	tmp = 0.0;
	if (x <= -5.8e+163)
		tmp = (y / x) / (y + x);
	elseif (x <= -5e+24)
		tmp = y / t_0;
	elseif (x <= -8.2e-14)
		tmp = x * (y / (t_0 * (x + (y + 1.0))));
	else
		tmp = (x / (y + x)) * ((y / (y + 1.0)) / (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.8e+163], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5e+24], N[(y / t$95$0), $MachinePrecision], If[LessEqual[x, -8.2e-14], N[(x * N[(y / N[(t$95$0 * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.79999999999999996e163

   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. Step-by-step derivation

      1. associate-/l* 83.2%

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

      2. associate-+l+ 83.2%

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

   3. Simplified 83.2%

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

      1. associate-*r/ 63.0%

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

      2. associate-+r+ 63.0%

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

      3. times-frac 83.2%

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

      4. associate-*l/ 83.2%

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

      5. associate-+r+ 83.2%

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

      6. +-commutative 83.2%

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

      7. associate-+l+ 83.2%

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

      8. pow2 83.2%

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

      9. +-commutative 83.2%

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

   6. Applied egg-rr 83.2%

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

      1. *-commutative 83.2%

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

      2. unpow2 83.2%

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

      3. times-frac 99.9%

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

      4. +-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. associate-*l/ 99.9%

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

   10. Applied egg-rr 99.9%

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

   11. Taylor expanded in x around inf 96.4%

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

   if -5.79999999999999996e163 < x < -5.00000000000000045e24

   1. Initial program 43.5%

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

      1. associate-/l* 55.3%

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

      2. associate-+l+ 55.3%

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

   3. Simplified 55.3%

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

      1. associate-*r/ 43.5%

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

      2. associate-+r+ 43.5%

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

      3. times-frac 93.1%

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

      4. associate-*l/ 93.0%

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

      5. associate-+r+ 93.0%

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

      6. +-commutative 93.0%

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

      7. associate-+l+ 93.0%

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

      8. pow2 93.0%

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

      9. +-commutative 93.0%

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

   6. Applied egg-rr 93.0%

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

      1. unpow2 93.0%

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

   8. Applied egg-rr 93.0%

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

   9. Taylor expanded in x around inf 86.5%

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

   if -5.00000000000000045e24 < x < -8.2000000000000004e-14

   1. Initial program 99.2%

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

      1. associate-/l* 99.2%

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

      2. associate-+l+ 99.2%

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

   3. Simplified 99.2%

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

   if -8.2000000000000004e-14 < x

   1. Initial program 74.5%

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

      1. associate-/l* 85.9%

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

      2. associate-+l+ 85.9%

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

   3. Simplified 85.9%

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

      1. associate-*r/ 74.5%

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

      2. associate-+r+ 74.5%

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

      3. times-frac 90.2%

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

      4. associate-*l/ 81.5%

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

      5. associate-+r+ 81.5%

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

      6. +-commutative 81.5%

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

      7. associate-+l+ 81.5%

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

      8. pow2 81.5%

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

      9. +-commutative 81.5%

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

   6. Applied egg-rr 81.5%

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

      1. *-commutative 81.5%

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

      2. unpow2 81.5%

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

      3. times-frac 99.8%

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

      4. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in x around 0 84.2%

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

       1. +-commutative 84.2%

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

   11. Simplified 84.2%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+163}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{+24}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{+100}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{+74}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-24}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ y x) (/ 1.0 x))))
   (if (<= x -1.7e+100)
     t_0
     (if (<= x -3.6e+93)
       (/ x (* y (+ y x)))
       (if (<= x -5e+74)
         t_0
         (if (<= x -6.5e-24) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0)))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y / x) * (1.0 / x);
	double tmp;
	if (x <= -1.7e+100) {
		tmp = t_0;
	} else if (x <= -3.6e+93) {
		tmp = x / (y * (y + x));
	} else if (x <= -5e+74) {
		tmp = t_0;
	} else if (x <= -6.5e-24) {
		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) :: t_0
    real(8) :: tmp
    t_0 = (y / x) * (1.0d0 / x)
    if (x <= (-1.7d+100)) then
        tmp = t_0
    else if (x <= (-3.6d+93)) then
        tmp = x / (y * (y + x))
    else if (x <= (-5d+74)) then
        tmp = t_0
    else if (x <= (-6.5d-24)) 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 t_0 = (y / x) * (1.0 / x);
	double tmp;
	if (x <= -1.7e+100) {
		tmp = t_0;
	} else if (x <= -3.6e+93) {
		tmp = x / (y * (y + x));
	} else if (x <= -5e+74) {
		tmp = t_0;
	} else if (x <= -6.5e-24) {
		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):
	t_0 = (y / x) * (1.0 / x)
	tmp = 0
	if x <= -1.7e+100:
		tmp = t_0
	elif x <= -3.6e+93:
		tmp = x / (y * (y + x))
	elif x <= -5e+74:
		tmp = t_0
	elif x <= -6.5e-24:
		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)
	t_0 = Float64(Float64(y / x) * Float64(1.0 / x))
	tmp = 0.0
	if (x <= -1.7e+100)
		tmp = t_0;
	elseif (x <= -3.6e+93)
		tmp = Float64(x / Float64(y * Float64(y + x)));
	elseif (x <= -5e+74)
		tmp = t_0;
	elseif (x <= -6.5e-24)
		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)
	t_0 = (y / x) * (1.0 / x);
	tmp = 0.0;
	if (x <= -1.7e+100)
		tmp = t_0;
	elseif (x <= -3.6e+93)
		tmp = x / (y * (y + x));
	elseif (x <= -5e+74)
		tmp = t_0;
	elseif (x <= -6.5e-24)
		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_] := Block[{t$95$0 = N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.7e+100], t$95$0, If[LessEqual[x, -3.6e+93], N[(x / N[(y * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5e+74], t$95$0, If[LessEqual[x, -6.5e-24], 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 4 regimes

2. if x < -1.69999999999999997e100 or -3.5999999999999999e93 < x < -4.99999999999999963e74

   1. Initial program 53.6%

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

      1. *-un-lft-identity 53.6%

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

      2. associate-*l* 53.6%

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

      3. times-frac 70.3%

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

      4. +-commutative 70.3%

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

      5. *-commutative 70.3%

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

      6. +-commutative 70.3%

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

      7. associate-+r+ 70.3%

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

      8. +-commutative 70.3%

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

      9. associate-+l+ 70.3%

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

   4. Applied egg-rr 70.3%

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

   5. Taylor expanded in x around inf 87.0%

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

   6. Taylor expanded in y around 0 86.8%

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

   if -1.69999999999999997e100 < x < -3.5999999999999999e93

   1. Initial program 4.2%

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

      1. associate-/l* 79.2%

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

      2. associate-+l+ 79.2%

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

   3. Simplified 79.2%

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

      1. *-un-lft-identity 79.2%

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

      2. associate-+r+ 79.2%

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

      3. associate-*l* 79.2%

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

      4. times-frac 79.2%

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

      5. +-commutative 79.2%

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

      6. +-commutative 79.2%

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

      7. associate-+r+ 79.2%

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

      8. +-commutative 79.2%

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

      9. associate-+l+ 79.2%

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

   6. Applied egg-rr 79.2%

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

      1. associate-*l/ 79.2%

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

      2. *-lft-identity 79.2%

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

      3. +-commutative 79.2%

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

   8. Simplified 79.2%

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

   9. Taylor expanded in y around inf 79.2%

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

       1. associate-/l/ 79.2%

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

       2. un-div-inv 79.2%

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

   11. Applied egg-rr 79.2%

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

   if -4.99999999999999963e74 < x < -6.5e-24

   1. Initial program 88.8%

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

      1. associate-/l* 88.6%

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

      2. associate-+l+ 88.6%

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

   3. Simplified 88.6%

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

   5. Taylor expanded in y around 0 73.5%

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

      1. +-commutative 73.5%

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

   7. Simplified 73.5%

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

   if -6.5e-24 < x

   1. Initial program 74.5%

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

      1. associate-/l* 86.0%

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

      2. associate-+l+ 86.0%

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

   3. Simplified 86.0%

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

   5. Taylor expanded in x around 0 60.3%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+100}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{+74}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-24}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+158}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{+26}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(y + \left(x + 1\right)\right) \cdot \left(y + x\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.4e+158)
   (/ (/ y x) (+ y x))
   (if (<= x -8.5e+26)
     (/ y (* (+ y x) (+ y x)))
     (* x (/ (/ y (* (+ y (+ x 1.0)) (+ y x))) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.4e+158) {
		tmp = (y / x) / (y + x);
	} else if (x <= -8.5e+26) {
		tmp = y / ((y + x) * (y + x));
	} else {
		tmp = x * ((y / ((y + (x + 1.0)) * (y + x))) / (y + x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.4d+158)) then
        tmp = (y / x) / (y + x)
    else if (x <= (-8.5d+26)) then
        tmp = y / ((y + x) * (y + x))
    else
        tmp = x * ((y / ((y + (x + 1.0d0)) * (y + x))) / (y + x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.4e+158) {
		tmp = (y / x) / (y + x);
	} else if (x <= -8.5e+26) {
		tmp = y / ((y + x) * (y + x));
	} else {
		tmp = x * ((y / ((y + (x + 1.0)) * (y + x))) / (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.4e+158:
		tmp = (y / x) / (y + x)
	elif x <= -8.5e+26:
		tmp = y / ((y + x) * (y + x))
	else:
		tmp = x * ((y / ((y + (x + 1.0)) * (y + x))) / (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.4e+158)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -8.5e+26)
		tmp = Float64(y / Float64(Float64(y + x) * Float64(y + x)));
	else
		tmp = Float64(x * Float64(Float64(y / Float64(Float64(y + Float64(x + 1.0)) * Float64(y + x))) / Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.4e+158)
		tmp = (y / x) / (y + x);
	elseif (x <= -8.5e+26)
		tmp = y / ((y + x) * (y + x));
	else
		tmp = x * ((y / ((y + (x + 1.0)) * (y + x))) / (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.4e+158], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8.5e+26], N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y / N[(N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.40000000000000008e158

   1. Initial program 60.7%

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

      1. associate-/l* 80.4%

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

      2. associate-+l+ 80.4%

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

   3. Simplified 80.4%

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

      1. associate-*r/ 60.7%

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

      2. associate-+r+ 60.7%

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

      3. times-frac 80.4%

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

      4. associate-*l/ 80.4%

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

      5. associate-+r+ 80.4%

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

      6. +-commutative 80.4%

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

      7. associate-+l+ 80.4%

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

      8. pow2 80.4%

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

      9. +-commutative 80.4%

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

   6. Applied egg-rr 80.4%

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

      1. *-commutative 80.4%

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

      2. unpow2 80.4%

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

      3. times-frac 99.8%

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

      4. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.9%

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

   10. Applied egg-rr 99.9%

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

   11. Taylor expanded in x around inf 93.1%

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

   if -2.40000000000000008e158 < x < -8.5e26

   1. Initial program 45.1%

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

      1. associate-/l* 57.2%

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

      2. associate-+l+ 57.2%

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

   3. Simplified 57.2%

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

      1. associate-*r/ 45.1%

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

      2. associate-+r+ 45.1%

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

      3. times-frac 96.5%

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

      4. associate-*l/ 96.3%

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

      5. associate-+r+ 96.3%

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

      6. +-commutative 96.3%

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

      7. associate-+l+ 96.3%

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

      8. pow2 96.3%

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

      9. +-commutative 96.3%

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

   6. Applied egg-rr 96.3%

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

      1. unpow2 96.3%

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

   8. Applied egg-rr 96.3%

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

   9. Taylor expanded in x around inf 89.5%

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

   if -8.5e26 < x

   1. Initial program 75.3%

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

      1. associate-/l* 86.3%

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

      2. associate-+l+ 86.3%

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

   3. Simplified 86.3%

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

      1. *-un-lft-identity 86.3%

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

      2. associate-+r+ 86.3%

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

      3. associate-*l* 86.3%

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

      4. times-frac 95.8%

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

      5. +-commutative 95.8%

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

      6. +-commutative 95.8%

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

      7. associate-+r+ 95.8%

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

      8. +-commutative 95.8%

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

      9. associate-+l+ 95.8%

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

   6. Applied egg-rr 95.8%

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

      1. associate-*l/ 95.8%

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

      2. *-lft-identity 95.8%

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

      3. +-commutative 95.8%

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

   8. Simplified 95.8%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+158}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{+26}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(y + \left(x + 1\right)\right) \cdot \left(y + x\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+163}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -5000000:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5.4e+163)
   (/ (/ y x) (+ y x))
   (if (<= x -5000000.0)
     (/ y (* (+ y x) (+ y x)))
     (* (/ x (+ y x)) (/ (/ y (+ y 1.0)) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5.4e+163) {
		tmp = (y / x) / (y + x);
	} else if (x <= -5000000.0) {
		tmp = y / ((y + x) * (y + x));
	} else {
		tmp = (x / (y + x)) * ((y / (y + 1.0)) / (y + x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.4d+163)) then
        tmp = (y / x) / (y + x)
    else if (x <= (-5000000.0d0)) then
        tmp = y / ((y + x) * (y + x))
    else
        tmp = (x / (y + x)) * ((y / (y + 1.0d0)) / (y + x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -5.4e+163) {
		tmp = (y / x) / (y + x);
	} else if (x <= -5000000.0) {
		tmp = y / ((y + x) * (y + x));
	} else {
		tmp = (x / (y + x)) * ((y / (y + 1.0)) / (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -5.4e+163:
		tmp = (y / x) / (y + x)
	elif x <= -5000000.0:
		tmp = y / ((y + x) * (y + x))
	else:
		tmp = (x / (y + x)) * ((y / (y + 1.0)) / (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5.4e+163)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -5000000.0)
		tmp = Float64(y / Float64(Float64(y + x) * Float64(y + x)));
	else
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(Float64(y / Float64(y + 1.0)) / Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.4e+163)
		tmp = (y / x) / (y + x);
	elseif (x <= -5000000.0)
		tmp = y / ((y + x) * (y + x));
	else
		tmp = (x / (y + x)) * ((y / (y + 1.0)) / (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -5.4e+163], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5000000.0], N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.39999999999999998e163

   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. Step-by-step derivation

      1. associate-/l* 83.2%

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

      2. associate-+l+ 83.2%

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

   3. Simplified 83.2%

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

      1. associate-*r/ 63.0%

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

      2. associate-+r+ 63.0%

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

      3. times-frac 83.2%

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

      4. associate-*l/ 83.2%

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

      5. associate-+r+ 83.2%

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

      6. +-commutative 83.2%

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

      7. associate-+l+ 83.2%

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

      8. pow2 83.2%

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

      9. +-commutative 83.2%

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

   6. Applied egg-rr 83.2%

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

      1. *-commutative 83.2%

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

      2. unpow2 83.2%

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

      3. times-frac 99.9%

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

      4. +-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. associate-*l/ 99.9%

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

   10. Applied egg-rr 99.9%

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

   11. Taylor expanded in x around inf 96.4%

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

   if -5.39999999999999998e163 < x < -5e6

   1. Initial program 52.2%

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

      1. associate-/l* 62.2%

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

      2. associate-+l+ 62.2%

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

   3. Simplified 62.2%

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

      1. associate-*r/ 52.2%

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

      2. associate-+r+ 52.2%

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

      3. times-frac 94.0%

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

      4. associate-*l/ 94.0%

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

      5. associate-+r+ 94.0%

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

      6. +-commutative 94.0%

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

      7. associate-+l+ 94.0%

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

      8. pow2 94.0%

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

      9. +-commutative 94.0%

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

   6. Applied egg-rr 94.0%

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

      1. unpow2 94.0%

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

   8. Applied egg-rr 94.0%

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

   9. Taylor expanded in x around inf 86.8%

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

   if -5e6 < x

   1. Initial program 74.7%

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

      1. associate-/l* 86.0%

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

      2. associate-+l+ 86.0%

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

   3. Simplified 86.0%

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

      1. associate-*r/ 74.7%

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

      2. associate-+r+ 74.7%

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

      3. times-frac 90.3%

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

      4. associate-*l/ 81.6%

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

      5. associate-+r+ 81.6%

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

      6. +-commutative 81.6%

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

      7. associate-+l+ 81.6%

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

      8. pow2 81.6%

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

      9. +-commutative 81.6%

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

   6. Applied egg-rr 81.6%

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

      1. *-commutative 81.6%

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

      2. unpow2 81.6%

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

      3. times-frac 99.8%

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

      4. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in x around 0 83.8%

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

       1. +-commutative 83.8%

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

   11. Simplified 83.8%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+163}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -5000000:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+158}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -1350:\\ \;\;\;\;\frac{y \cdot \frac{x}{x + 1}}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.4e+158)
   (/ (/ y x) (+ y x))
   (if (<= x -1350.0)
     (/ (* y (/ x (+ x 1.0))) (* (+ y x) (+ y x)))
     (* (/ x (+ y x)) (/ (/ y (+ y 1.0)) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.4e+158) {
		tmp = (y / x) / (y + x);
	} else if (x <= -1350.0) {
		tmp = (y * (x / (x + 1.0))) / ((y + x) * (y + x));
	} else {
		tmp = (x / (y + x)) * ((y / (y + 1.0)) / (y + x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.4d+158)) then
        tmp = (y / x) / (y + x)
    else if (x <= (-1350.0d0)) then
        tmp = (y * (x / (x + 1.0d0))) / ((y + x) * (y + x))
    else
        tmp = (x / (y + x)) * ((y / (y + 1.0d0)) / (y + x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.4e+158) {
		tmp = (y / x) / (y + x);
	} else if (x <= -1350.0) {
		tmp = (y * (x / (x + 1.0))) / ((y + x) * (y + x));
	} else {
		tmp = (x / (y + x)) * ((y / (y + 1.0)) / (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.4e+158:
		tmp = (y / x) / (y + x)
	elif x <= -1350.0:
		tmp = (y * (x / (x + 1.0))) / ((y + x) * (y + x))
	else:
		tmp = (x / (y + x)) * ((y / (y + 1.0)) / (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.4e+158)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -1350.0)
		tmp = Float64(Float64(y * Float64(x / Float64(x + 1.0))) / Float64(Float64(y + x) * Float64(y + x)));
	else
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(Float64(y / Float64(y + 1.0)) / Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.4e+158)
		tmp = (y / x) / (y + x);
	elseif (x <= -1350.0)
		tmp = (y * (x / (x + 1.0))) / ((y + x) * (y + x));
	else
		tmp = (x / (y + x)) * ((y / (y + 1.0)) / (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.4e+158], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1350.0], N[(N[(y * N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.40000000000000008e158

   1. Initial program 60.7%

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

      1. associate-/l* 80.4%

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

      2. associate-+l+ 80.4%

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

   3. Simplified 80.4%

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

      1. associate-*r/ 60.7%

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

      2. associate-+r+ 60.7%

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

      3. times-frac 80.4%

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

      4. associate-*l/ 80.4%

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

      5. associate-+r+ 80.4%

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

      6. +-commutative 80.4%

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

      7. associate-+l+ 80.4%

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

      8. pow2 80.4%

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

      9. +-commutative 80.4%

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

   6. Applied egg-rr 80.4%

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

      1. *-commutative 80.4%

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

      2. unpow2 80.4%

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

      3. times-frac 99.8%

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

      4. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.9%

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

   10. Applied egg-rr 99.9%

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

   11. Taylor expanded in x around inf 93.1%

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

   if -2.40000000000000008e158 < x < -1350

   1. Initial program 53.9%

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

      1. associate-/l* 64.0%

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

      2. associate-+l+ 64.0%

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

   3. Simplified 64.0%

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

      1. associate-*r/ 53.9%

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

      2. associate-+r+ 53.9%

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

      3. times-frac 96.9%

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

      4. associate-*l/ 96.9%

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

      5. associate-+r+ 96.9%

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

      6. +-commutative 96.9%

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

      7. associate-+l+ 96.9%

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

      8. pow2 96.9%

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

      9. +-commutative 96.9%

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

   6. Applied egg-rr 96.9%

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

      1. unpow2 96.9%

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

   8. Applied egg-rr 96.9%

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

   9. Taylor expanded in y around 0 71.8%

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

       1. div-inv 71.8%

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

       2. *-commutative 71.8%

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

       3. associate-*l* 91.1%

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

       4. +-commutative 91.1%

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

   11. Applied egg-rr 91.1%

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

       1. associate-*r* 71.8%

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

       2. associate-*r/ 71.8%

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

       3. *-rgt-identity 71.8%

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

       4. associate-*r/ 91.2%

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

   13. Simplified 91.2%

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

   if -1350 < x

   1. Initial program 74.7%

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

      1. associate-/l* 86.0%

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

      2. associate-+l+ 86.0%

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

   3. Simplified 86.0%

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

      1. associate-*r/ 74.7%

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

      2. associate-+r+ 74.7%

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

      3. times-frac 90.3%

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

      4. associate-*l/ 81.6%

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

      5. associate-+r+ 81.6%

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

      6. +-commutative 81.6%

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

      7. associate-+l+ 81.6%

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

      8. pow2 81.6%

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

      9. +-commutative 81.6%

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

   6. Applied egg-rr 81.6%

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

      1. *-commutative 81.6%

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

      2. unpow2 81.6%

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

      3. times-frac 99.8%

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

      4. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in x around 0 83.8%

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

       1. +-commutative 83.8%

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

   11. Simplified 83.8%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+158}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -1350:\\ \;\;\;\;\frac{y \cdot \frac{x}{x + 1}}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.18 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -9 \cdot 10^{+26}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.18e+160)
   (/ (/ y x) (+ y x))
   (if (<= x -9e+26)
     (/ y (* (+ y x) (+ y x)))
     (if (<= x -6.2e-48)
       (/ (/ y (+ x 1.0)) (+ y x))
       (/ (/ x (+ y 1.0)) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.18e+160) {
		tmp = (y / x) / (y + x);
	} else if (x <= -9e+26) {
		tmp = y / ((y + x) * (y + x));
	} else if (x <= -6.2e-48) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.18d+160)) then
        tmp = (y / x) / (y + x)
    else if (x <= (-9d+26)) then
        tmp = y / ((y + x) * (y + x))
    else if (x <= (-6.2d-48)) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.18e+160) {
		tmp = (y / x) / (y + x);
	} else if (x <= -9e+26) {
		tmp = y / ((y + x) * (y + x));
	} else if (x <= -6.2e-48) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.18e+160:
		tmp = (y / x) / (y + x)
	elif x <= -9e+26:
		tmp = y / ((y + x) * (y + x))
	elif x <= -6.2e-48:
		tmp = (y / (x + 1.0)) / (y + x)
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.18e+160)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -9e+26)
		tmp = Float64(y / Float64(Float64(y + x) * Float64(y + x)));
	elseif (x <= -6.2e-48)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.18e+160)
		tmp = (y / x) / (y + x);
	elseif (x <= -9e+26)
		tmp = y / ((y + x) * (y + x));
	elseif (x <= -6.2e-48)
		tmp = (y / (x + 1.0)) / (y + x);
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.18e+160], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -9e+26], N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6.2e-48], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.17999999999999999e160

   1. Initial program 60.7%

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

      1. associate-/l* 80.4%

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

      2. associate-+l+ 80.4%

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

   3. Simplified 80.4%

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

      1. associate-*r/ 60.7%

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

      2. associate-+r+ 60.7%

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

      3. times-frac 80.4%

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

      4. associate-*l/ 80.4%

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

      5. associate-+r+ 80.4%

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

      6. +-commutative 80.4%

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

      7. associate-+l+ 80.4%

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

      8. pow2 80.4%

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

      9. +-commutative 80.4%

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

   6. Applied egg-rr 80.4%

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

      1. *-commutative 80.4%

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

      2. unpow2 80.4%

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

      3. times-frac 99.8%

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

      4. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.9%

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

   10. Applied egg-rr 99.9%

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

   11. Taylor expanded in x around inf 93.1%

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

   if -1.17999999999999999e160 < x < -8.99999999999999957e26

   1. Initial program 45.1%

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

      1. associate-/l* 57.2%

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

      2. associate-+l+ 57.2%

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

   3. Simplified 57.2%

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

      1. associate-*r/ 45.1%

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

      2. associate-+r+ 45.1%

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

      3. times-frac 96.5%

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

      4. associate-*l/ 96.3%

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

      5. associate-+r+ 96.3%

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

      6. +-commutative 96.3%

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

      7. associate-+l+ 96.3%

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

      8. pow2 96.3%

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

      9. +-commutative 96.3%

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

   6. Applied egg-rr 96.3%

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

      1. unpow2 96.3%

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

   8. Applied egg-rr 96.3%

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

   9. Taylor expanded in x around inf 89.5%

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

   if -8.99999999999999957e26 < x < -6.20000000000000033e-48

   1. Initial program 91.1%

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

      1. associate-/l* 94.6%

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

      2. associate-+l+ 94.6%

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

   3. Simplified 94.6%

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

      1. associate-*r/ 91.1%

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

      2. associate-+r+ 91.1%

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

      3. times-frac 99.4%

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

      4. associate-*l/ 96.1%

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

      5. associate-+r+ 96.1%

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

      6. +-commutative 96.1%

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

      7. associate-+l+ 96.1%

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

      8. pow2 96.1%

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

      9. +-commutative 96.1%

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

   6. Applied egg-rr 96.1%

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

      1. *-commutative 96.1%

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

      2. unpow2 96.1%

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

      3. times-frac 99.8%

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

      4. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in y around 0 62.4%

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

       1. +-commutative 62.4%

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

   13. Simplified 62.4%

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

   if -6.20000000000000033e-48 < x

   1. Initial program 74.2%

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

      1. *-un-lft-identity 74.2%

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

      2. associate-*l* 74.2%

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

      3. times-frac 77.8%

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

      4. +-commutative 77.8%

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

      5. *-commutative 77.8%

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

      6. +-commutative 77.8%

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

      7. associate-+r+ 77.8%

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

      8. +-commutative 77.8%

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

      9. associate-+l+ 77.8%

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

   4. Applied egg-rr 77.8%

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

   5. Taylor expanded in x around 0 61.9%

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

      1. +-commutative 61.9%

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

   7. Simplified 61.9%

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

      1. associate-*l/ 62.1%

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

      2. *-un-lft-identity 62.1%

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

   9. Applied egg-rr 62.1%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.18 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -9 \cdot 10^{+26}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+100} \lor \neg \left(x \leq -3.6 \cdot 10^{+93}\right) \land x \leq -820000000:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.7e+100) (and (not (<= x -3.6e+93)) (<= x -820000000.0)))
   (* (/ y x) (/ 1.0 x))
   (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if ((x <= -1.7e+100) || (!(x <= -3.6e+93) && (x <= -820000000.0))) {
		tmp = (y / x) * (1.0 / x);
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.7d+100)) .or. (.not. (x <= (-3.6d+93))) .and. (x <= (-820000000.0d0))) then
        tmp = (y / x) * (1.0d0 / x)
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.7e+100) || (!(x <= -3.6e+93) && (x <= -820000000.0))) {
		tmp = (y / x) * (1.0 / x);
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if (x <= -1.7e+100) or (not (x <= -3.6e+93) and (x <= -820000000.0)):
		tmp = (y / x) * (1.0 / x)
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if ((x <= -1.7e+100) || (!(x <= -3.6e+93) && (x <= -820000000.0)))
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.7e+100) || (~((x <= -3.6e+93)) && (x <= -820000000.0)))
		tmp = (y / x) * (1.0 / x);
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[Or[LessEqual[x, -1.7e+100], And[N[Not[LessEqual[x, -3.6e+93]], $MachinePrecision], LessEqual[x, -820000000.0]]], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.69999999999999997e100 or -3.5999999999999999e93 < x < -8.2e8

   1. Initial program 61.0%

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

      1. *-un-lft-identity 61.0%

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

      2. associate-*l* 61.0%

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

      3. times-frac 74.3%

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

      4. +-commutative 74.3%

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

      5. *-commutative 74.3%

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

      6. +-commutative 74.3%

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

      7. associate-+r+ 74.3%

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

      8. +-commutative 74.3%

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

      9. associate-+l+ 74.3%

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

   4. Applied egg-rr 74.3%

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

   5. Taylor expanded in x around inf 83.3%

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

   6. Taylor expanded in y around 0 83.1%

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

   if -1.69999999999999997e100 < x < -3.5999999999999999e93 or -8.2e8 < x

   1. Initial program 73.3%

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

      1. associate-/l* 85.8%

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

      2. associate-+l+ 85.8%

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

   3. Simplified 85.8%

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

   5. Taylor expanded in x around 0 60.1%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+100} \lor \neg \left(x \leq -3.6 \cdot 10^{+93}\right) \land x \leq -820000000:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+100}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{+92} \lor \neg \left(x \leq -1.55 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.7e+100)
   (* (/ y x) (/ 1.0 x))
   (if (or (<= x -4.5e+92) (not (<= x -1.55e-23)))
     (/ (/ x y) (+ y 1.0))
     (/ y (* x (+ x 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.7e+100) {
		tmp = (y / x) * (1.0 / x);
	} else if ((x <= -4.5e+92) || !(x <= -1.55e-23)) {
		tmp = (x / y) / (y + 1.0);
	} else {
		tmp = y / (x * (x + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.7d+100)) then
        tmp = (y / x) * (1.0d0 / x)
    else if ((x <= (-4.5d+92)) .or. (.not. (x <= (-1.55d-23)))) then
        tmp = (x / y) / (y + 1.0d0)
    else
        tmp = y / (x * (x + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.7e+100) {
		tmp = (y / x) * (1.0 / x);
	} else if ((x <= -4.5e+92) || !(x <= -1.55e-23)) {
		tmp = (x / y) / (y + 1.0);
	} else {
		tmp = y / (x * (x + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.7e+100:
		tmp = (y / x) * (1.0 / x)
	elif (x <= -4.5e+92) or not (x <= -1.55e-23):
		tmp = (x / y) / (y + 1.0)
	else:
		tmp = y / (x * (x + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.7e+100)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	elseif ((x <= -4.5e+92) || !(x <= -1.55e-23))
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	else
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.7e+100)
		tmp = (y / x) * (1.0 / x);
	elseif ((x <= -4.5e+92) || ~((x <= -1.55e-23)))
		tmp = (x / y) / (y + 1.0);
	else
		tmp = y / (x * (x + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.7e+100], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -4.5e+92], N[Not[LessEqual[x, -1.55e-23]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.69999999999999997e100

   1. Initial program 50.2%

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

      1. *-un-lft-identity 50.2%

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

      2. associate-*l* 50.2%

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

      3. times-frac 68.2%

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

      4. +-commutative 68.2%

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

      5. *-commutative 68.2%

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

      6. +-commutative 68.2%

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

      7. associate-+r+ 68.2%

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

      8. +-commutative 68.2%

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

      9. associate-+l+ 68.2%

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

   4. Applied egg-rr 68.2%

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

   5. Taylor expanded in x around inf 86.1%

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

   6. Taylor expanded in y around 0 85.9%

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

   if -1.69999999999999997e100 < x < -4.4999999999999999e92 or -1.5499999999999999e-23 < x

   1. Initial program 73.0%

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

      1. associate-/l* 85.9%

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

      2. associate-+l+ 85.9%

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

   3. Simplified 85.9%

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

   5. Taylor expanded in x around 0 60.7%

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

      1. associate-/r* 62.6%

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

      2. +-commutative 62.6%

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

   7. Simplified 62.6%

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

   if -4.4999999999999999e92 < x < -1.5499999999999999e-23

   1. Initial program 90.6%

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

      1. associate-/l* 80.2%

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

      2. associate-+l+ 80.2%

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

   3. Simplified 80.2%

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

   5. Taylor expanded in y around 0 77.7%

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

      1. +-commutative 77.7%

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

   7. Simplified 77.7%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+100}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{+92} \lor \neg \left(x \leq -1.55 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+100}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{+91} \lor \neg \left(x \leq -1.75 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.7e+100)
   (/ (/ y x) (+ y x))
   (if (or (<= x -5.8e+91) (not (<= x -1.75e-25)))
     (/ (/ x y) (+ y 1.0))
     (/ y (* x (+ x 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.7e+100) {
		tmp = (y / x) / (y + x);
	} else if ((x <= -5.8e+91) || !(x <= -1.75e-25)) {
		tmp = (x / y) / (y + 1.0);
	} else {
		tmp = y / (x * (x + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.7d+100)) then
        tmp = (y / x) / (y + x)
    else if ((x <= (-5.8d+91)) .or. (.not. (x <= (-1.75d-25)))) then
        tmp = (x / y) / (y + 1.0d0)
    else
        tmp = y / (x * (x + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.7e+100) {
		tmp = (y / x) / (y + x);
	} else if ((x <= -5.8e+91) || !(x <= -1.75e-25)) {
		tmp = (x / y) / (y + 1.0);
	} else {
		tmp = y / (x * (x + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.7e+100:
		tmp = (y / x) / (y + x)
	elif (x <= -5.8e+91) or not (x <= -1.75e-25):
		tmp = (x / y) / (y + 1.0)
	else:
		tmp = y / (x * (x + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.7e+100)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif ((x <= -5.8e+91) || !(x <= -1.75e-25))
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	else
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.7e+100)
		tmp = (y / x) / (y + x);
	elseif ((x <= -5.8e+91) || ~((x <= -1.75e-25)))
		tmp = (x / y) / (y + 1.0);
	else
		tmp = y / (x * (x + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.7e+100], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -5.8e+91], N[Not[LessEqual[x, -1.75e-25]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.69999999999999997e100

   1. Initial program 50.2%

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

      1. associate-/l* 68.6%

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

      2. associate-+l+ 68.6%

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

   3. Simplified 68.6%

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

      1. associate-*r/ 50.2%

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

      2. associate-+r+ 50.2%

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

      3. times-frac 86.5%

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

      4. associate-*l/ 86.5%

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

      5. associate-+r+ 86.5%

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

      6. +-commutative 86.5%

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

      7. associate-+l+ 86.5%

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

      8. pow2 86.5%

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

      9. +-commutative 86.5%

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

   6. Applied egg-rr 86.5%

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

      1. *-commutative 86.5%

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

      2. unpow2 86.5%

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

      3. times-frac 99.8%

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

      4. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in x around inf 86.1%

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

   if -1.69999999999999997e100 < x < -5.80000000000000028e91 or -1.7500000000000001e-25 < x

   1. Initial program 73.0%

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

      1. associate-/l* 85.9%

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

      2. associate-+l+ 85.9%

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

   3. Simplified 85.9%

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

   5. Taylor expanded in x around 0 60.7%

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

      1. associate-/r* 62.6%

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

      2. +-commutative 62.6%

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

   7. Simplified 62.6%

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

   if -5.80000000000000028e91 < x < -1.7500000000000001e-25

   1. Initial program 90.6%

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

      1. associate-/l* 80.2%

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

      2. associate-+l+ 80.2%

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

   3. Simplified 80.2%

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

   5. Taylor expanded in y around 0 77.7%

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

      1. +-commutative 77.7%

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

   7. Simplified 77.7%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+100}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{+91} \lor \neg \left(x \leq -1.75 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{if}\;x \leq -1.95 \cdot 10^{+100}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -3200000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ y x) (/ 1.0 x))))
   (if (<= x -1.95e+100)
     t_0
     (if (<= x -1.25e+93)
       (/ x (* y (+ y x)))
       (if (<= x -3200000.0) t_0 (/ x (* y (+ y 1.0))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y / x) * (1.0 / x);
	double tmp;
	if (x <= -1.95e+100) {
		tmp = t_0;
	} else if (x <= -1.25e+93) {
		tmp = x / (y * (y + x));
	} else if (x <= -3200000.0) {
		tmp = t_0;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y / x) * (1.0d0 / x)
    if (x <= (-1.95d+100)) then
        tmp = t_0
    else if (x <= (-1.25d+93)) then
        tmp = x / (y * (y + x))
    else if (x <= (-3200000.0d0)) then
        tmp = t_0
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y / x) * (1.0 / x);
	double tmp;
	if (x <= -1.95e+100) {
		tmp = t_0;
	} else if (x <= -1.25e+93) {
		tmp = x / (y * (y + x));
	} else if (x <= -3200000.0) {
		tmp = t_0;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y / x) * (1.0 / x)
	tmp = 0
	if x <= -1.95e+100:
		tmp = t_0
	elif x <= -1.25e+93:
		tmp = x / (y * (y + x))
	elif x <= -3200000.0:
		tmp = t_0
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y / x) * Float64(1.0 / x))
	tmp = 0.0
	if (x <= -1.95e+100)
		tmp = t_0;
	elseif (x <= -1.25e+93)
		tmp = Float64(x / Float64(y * Float64(y + x)));
	elseif (x <= -3200000.0)
		tmp = t_0;
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (y / x) * (1.0 / x);
	tmp = 0.0;
	if (x <= -1.95e+100)
		tmp = t_0;
	elseif (x <= -1.25e+93)
		tmp = x / (y * (y + x));
	elseif (x <= -3200000.0)
		tmp = t_0;
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.95e+100], t$95$0, If[LessEqual[x, -1.25e+93], N[(x / N[(y * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3200000.0], t$95$0, N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.95e100 or -1.25e93 < x < -3.2e6

   1. Initial program 61.0%

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

      1. *-un-lft-identity 61.0%

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

      2. associate-*l* 61.0%

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

      3. times-frac 74.3%

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

      4. +-commutative 74.3%

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

      5. *-commutative 74.3%

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

      6. +-commutative 74.3%

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

      7. associate-+r+ 74.3%

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

      8. +-commutative 74.3%

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

      9. associate-+l+ 74.3%

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

   4. Applied egg-rr 74.3%

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

   5. Taylor expanded in x around inf 83.3%

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

   6. Taylor expanded in y around 0 83.1%

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

   if -1.95e100 < x < -1.25e93

   1. Initial program 4.2%

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

      1. associate-/l* 79.2%

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

      2. associate-+l+ 79.2%

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

   3. Simplified 79.2%

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

      1. *-un-lft-identity 79.2%

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

      2. associate-+r+ 79.2%

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

      3. associate-*l* 79.2%

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

      4. times-frac 79.2%

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

      5. +-commutative 79.2%

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

      6. +-commutative 79.2%

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

      7. associate-+r+ 79.2%

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

      8. +-commutative 79.2%

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

      9. associate-+l+ 79.2%

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

   6. Applied egg-rr 79.2%

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

      1. associate-*l/ 79.2%

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

      2. *-lft-identity 79.2%

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

      3. +-commutative 79.2%

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

   8. Simplified 79.2%

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

   9. Taylor expanded in y around inf 79.2%

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

       1. associate-/l/ 79.2%

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

       2. un-div-inv 79.2%

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

   11. Applied egg-rr 79.2%

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

   if -3.2e6 < x

   1. Initial program 74.7%

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

      1. associate-/l* 86.0%

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

      2. associate-+l+ 86.0%

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

   3. Simplified 86.0%

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

   5. Taylor expanded in x around 0 59.7%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+100}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -3200000:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 82.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{+26}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(y + x\right)}{y}}\\ \mathbf{elif}\;x \leq -1.16 \cdot 10^{-23}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -9e+148)
   (/ (/ y x) (+ y x))
   (if (<= x -9.5e+26)
     (/ 1.0 (/ (* x (+ y x)) y))
     (if (<= x -1.16e-23) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -9e+148) {
		tmp = (y / x) / (y + x);
	} else if (x <= -9.5e+26) {
		tmp = 1.0 / ((x * (y + x)) / y);
	} else if (x <= -1.16e-23) {
		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 <= (-9d+148)) then
        tmp = (y / x) / (y + x)
    else if (x <= (-9.5d+26)) then
        tmp = 1.0d0 / ((x * (y + x)) / y)
    else if (x <= (-1.16d-23)) 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 <= -9e+148) {
		tmp = (y / x) / (y + x);
	} else if (x <= -9.5e+26) {
		tmp = 1.0 / ((x * (y + x)) / y);
	} else if (x <= -1.16e-23) {
		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 <= -9e+148:
		tmp = (y / x) / (y + x)
	elif x <= -9.5e+26:
		tmp = 1.0 / ((x * (y + x)) / y)
	elif x <= -1.16e-23:
		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 <= -9e+148)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -9.5e+26)
		tmp = Float64(1.0 / Float64(Float64(x * Float64(y + x)) / y));
	elseif (x <= -1.16e-23)
		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 <= -9e+148)
		tmp = (y / x) / (y + x);
	elseif (x <= -9.5e+26)
		tmp = 1.0 / ((x * (y + x)) / y);
	elseif (x <= -1.16e-23)
		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, -9e+148], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -9.5e+26], N[(1.0 / N[(N[(x * N[(y + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.16e-23], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.99999999999999987e148

   1. Initial program 60.2%

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

      1. associate-/l* 78.6%

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

      2. associate-+l+ 78.6%

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

   3. Simplified 78.6%

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

      1. associate-*r/ 60.2%

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

      2. associate-+r+ 60.2%

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

      3. times-frac 81.8%

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

      4. associate-*l/ 81.8%

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

      5. associate-+r+ 81.8%

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

      6. +-commutative 81.8%

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

      7. associate-+l+ 81.8%

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

      8. pow2 81.8%

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

      9. +-commutative 81.8%

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

   6. Applied egg-rr 81.8%

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

      1. *-commutative 81.8%

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

      2. unpow2 81.8%

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

      3. times-frac 99.8%

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

      4. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.9%

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

   10. Applied egg-rr 99.9%

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

   11. Taylor expanded in x around inf 93.6%

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

   if -8.99999999999999987e148 < x < -9.50000000000000054e26

   1. Initial program 44.5%

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

      1. *-un-lft-identity 44.5%

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

      2. associate-*l* 44.5%

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

      3. times-frac 71.2%

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

      4. +-commutative 71.2%

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

      5. *-commutative 71.2%

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

      6. +-commutative 71.2%

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

      7. associate-+r+ 71.2%

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

      8. +-commutative 71.2%

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

      9. associate-+l+ 71.2%

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

   4. Applied egg-rr 71.2%

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

   5. Taylor expanded in x around inf 60.1%

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

      1. frac-times 84.0%

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

      2. *-un-lft-identity 84.0%

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

      3. clear-num 83.4%

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

   7. Applied egg-rr 83.4%

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

   if -9.50000000000000054e26 < x < -1.1599999999999999e-23

   1. Initial program 92.6%

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

      1. associate-/l* 92.4%

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

      2. associate-+l+ 92.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in y around 0 78.0%

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

      1. +-commutative 78.0%

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

   7. Simplified 78.0%

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

   if -1.1599999999999999e-23 < x

   1. Initial program 74.5%

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

      1. associate-/l* 86.0%

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

      2. associate-+l+ 86.0%

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

   3. Simplified 86.0%

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

   5. Taylor expanded in x around 0 60.3%

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

      1. associate-/r* 61.8%

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

      2. +-commutative 61.8%

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

   7. Simplified 61.8%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{+26}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(y + x\right)}{y}}\\ \mathbf{elif}\;x \leq -1.16 \cdot 10^{-23}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 82.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -1550000000000:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-22}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.8e+161)
   (/ (/ y x) (+ y x))
   (if (<= x -1550000000000.0)
     (/ y (* (+ y x) (+ y x)))
     (if (<= x -1.3e-22) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.8e+161) {
		tmp = (y / x) / (y + x);
	} else if (x <= -1550000000000.0) {
		tmp = y / ((y + x) * (y + x));
	} else if (x <= -1.3e-22) {
		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 <= (-2.8d+161)) then
        tmp = (y / x) / (y + x)
    else if (x <= (-1550000000000.0d0)) then
        tmp = y / ((y + x) * (y + x))
    else if (x <= (-1.3d-22)) 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 <= -2.8e+161) {
		tmp = (y / x) / (y + x);
	} else if (x <= -1550000000000.0) {
		tmp = y / ((y + x) * (y + x));
	} else if (x <= -1.3e-22) {
		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 <= -2.8e+161:
		tmp = (y / x) / (y + x)
	elif x <= -1550000000000.0:
		tmp = y / ((y + x) * (y + x))
	elif x <= -1.3e-22:
		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 <= -2.8e+161)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -1550000000000.0)
		tmp = Float64(y / Float64(Float64(y + x) * Float64(y + x)));
	elseif (x <= -1.3e-22)
		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 <= -2.8e+161)
		tmp = (y / x) / (y + x);
	elseif (x <= -1550000000000.0)
		tmp = y / ((y + x) * (y + x));
	elseif (x <= -1.3e-22)
		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, -2.8e+161], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1550000000000.0], N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.3e-22], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.80000000000000021e161

   1. Initial program 60.7%

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

      1. associate-/l* 80.4%

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

      2. associate-+l+ 80.4%

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

   3. Simplified 80.4%

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

      1. associate-*r/ 60.7%

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

      2. associate-+r+ 60.7%

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

      3. times-frac 80.4%

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

      4. associate-*l/ 80.4%

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

      5. associate-+r+ 80.4%

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

      6. +-commutative 80.4%

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

      7. associate-+l+ 80.4%

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

      8. pow2 80.4%

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

      9. +-commutative 80.4%

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

   6. Applied egg-rr 80.4%

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

      1. *-commutative 80.4%

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

      2. unpow2 80.4%

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

      3. times-frac 99.8%

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

      4. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.9%

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

   10. Applied egg-rr 99.9%

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

   11. Taylor expanded in x around inf 93.1%

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

   if -2.80000000000000021e161 < x < -1.55e12

   1. Initial program 50.8%

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

      1. associate-/l* 61.6%

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

      2. associate-+l+ 61.6%

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

   3. Simplified 61.6%

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

      1. associate-*r/ 50.8%

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

      2. associate-+r+ 50.8%

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

      3. times-frac 96.8%

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

      4. associate-*l/ 96.7%

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

      5. associate-+r+ 96.7%

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

      6. +-commutative 96.7%

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

      7. associate-+l+ 96.7%

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

      8. pow2 96.7%

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

      9. +-commutative 96.7%

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

   6. Applied egg-rr 96.7%

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

      1. unpow2 96.7%

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

   8. Applied egg-rr 96.7%

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

   9. Taylor expanded in x around inf 90.6%

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

   if -1.55e12 < x < -1.3e-22

   1. Initial program 89.2%

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

      1. associate-/l* 88.9%

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

      2. associate-+l+ 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in y around 0 83.4%

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

      1. +-commutative 83.4%

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

   7. Simplified 83.4%

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

   if -1.3e-22 < x

   1. Initial program 74.5%

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

      1. associate-/l* 86.0%

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

      2. associate-+l+ 86.0%

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

   3. Simplified 86.0%

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

   5. Taylor expanded in x around 0 60.3%

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

      1. associate-/r* 61.8%

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

      2. +-commutative 61.8%

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

   7. Simplified 61.8%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -1550000000000:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-22}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

   1. associate-/l* 82.7%

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

   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. associate-*r/ 70.6%

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

   2. associate-+r+ 70.6%

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

   3. times-frac 90.0%

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

   4. associate-*l/ 83.3%

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

   5. associate-+r+ 83.3%

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

   6. +-commutative 83.3%

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

   7. associate-+l+ 83.3%

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

   8. pow2 83.3%

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

   9. +-commutative 83.3%

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

6. Applied egg-rr 83.3%

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

   1. *-commutative 83.3%

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

   2. unpow2 83.3%

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

   3. times-frac 99.8%

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

   4. +-commutative 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 15: 61.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 57.8%

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

      1. *-un-lft-identity 57.8%

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

      2. associate-*l* 57.8%

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

      3. times-frac 70.1%

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

      4. +-commutative 70.1%

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

      5. *-commutative 70.1%

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

      6. +-commutative 70.1%

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

      7. associate-+r+ 70.1%

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

      8. +-commutative 70.1%

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

      9. associate-+l+ 70.1%

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

   4. Applied egg-rr 70.1%

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

   5. Taylor expanded in x around inf 77.0%

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

   6. Taylor expanded in y around 0 76.8%

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

   if -1 < x

   1. Initial program 74.5%

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

      1. associate-/l* 85.9%

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

      2. associate-+l+ 85.9%

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

   3. Simplified 85.9%

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

   5. Taylor expanded in x around 0 60.0%

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

   6. Taylor expanded in y around 0 37.6%

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

4. Final simplification 46.8%

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

Alternative 16: 4.3% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

   1. *-un-lft-identity 70.6%

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

   2. associate-*l* 70.6%

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

   3. times-frac 76.4%

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

   4. +-commutative 76.4%

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

   5. *-commutative 76.4%

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

   6. +-commutative 76.4%

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

   7. associate-+r+ 76.4%

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

   8. +-commutative 76.4%

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

   9. associate-+l+ 76.4%

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

4. Applied egg-rr 76.4%

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

5. Taylor expanded in x around inf 37.1%

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

6. Taylor expanded in y around inf 4.0%

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

7. Final simplification 4.0%

   \[\leadsto \frac{1}{x} \]
8. Add Preprocessing

Alternative 17: 25.8% 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 70.6%

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

   1. associate-/l* 82.7%

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

   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. Taylor expanded in x around 0 50.8%

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

6. Taylor expanded in y around 0 29.3%

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

7. Final simplification 29.3%

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

Alternative 18: 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 70.6%

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

   1. associate-/l* 82.7%

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

   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. Taylor expanded in x around 0 50.8%

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

6. Taylor expanded in y around 0 16.6%

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

   1. +-commutative 16.6%

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

   2. neg-mul-1 16.6%

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

   3. unsub-neg 16.6%

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

8. Simplified 16.6%

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

9. Taylor expanded in y around inf 3.8%

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

    1. neg-mul-1 3.8%

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

11. Simplified 3.8%

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

12. Final simplification 3.8%

    \[\leadsto -x \]
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 2024053 
(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))))