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

Percentage Accurate: 68.3% → 99.8%

Time: 14.1s

Alternatives: 18

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return ((1.0 / (x + (y + 1.0))) * (x / (x + y))) * (y / (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 = ((1.0d0 / (x + (y + 1.0d0))) * (x / (x + y))) * (y / (x + y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.7%

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

   1. +-commutative 68.7%

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

   2. distribute-rgt-in 68.7%

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

   3. +-commutative 68.7%

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

   4. +-commutative 68.7%

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

4. Applied egg-rr 68.7%

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

   1. times-frac 84.4%

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

   2. distribute-rgt-out 88.0%

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

   3. associate-/l/ 99.7%

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

   4. +-commutative 99.7%

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

   5. associate-+r+ 99.7%

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

   6. frac-times 94.1%

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

   7. +-commutative 94.1%

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

   8. *-commutative 94.1%

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

   9. +-commutative 94.1%

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

   10. times-frac 99.8%

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

   11. +-commutative 99.8%

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

   12. associate-+r+ 99.8%

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

   13. +-commutative 99.8%

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

   14. associate-+l+ 99.8%

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

   15. +-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

   1. div-inv 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 95.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -1.56 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 - \frac{y}{x}}{x + y} \cdot \frac{y}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -1.56e+154)
     (* (/ (- 1.0 (/ y x)) (+ x y)) (/ y t_0))
     (* (/ x (+ x y)) (/ y (* (+ x y) t_0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.56e+154) {
		tmp = ((1.0 - (y / x)) / (x + y)) * (y / t_0);
	} else {
		tmp = (x / (x + y)) * (y / ((x + y) * t_0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (x <= (-1.56d+154)) then
        tmp = ((1.0d0 - (y / x)) / (x + y)) * (y / t_0)
    else
        tmp = (x / (x + y)) * (y / ((x + y) * t_0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.56e+154) {
		tmp = ((1.0 - (y / x)) / (x + y)) * (y / t_0);
	} else {
		tmp = (x / (x + y)) * (y / ((x + y) * t_0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -1.56e+154:
		tmp = ((1.0 - (y / x)) / (x + y)) * (y / t_0)
	else:
		tmp = (x / (x + y)) * (y / ((x + y) * t_0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1.56e+154)
		tmp = Float64(Float64(Float64(1.0 - Float64(y / x)) / Float64(x + y)) * Float64(y / t_0));
	else
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(y / Float64(Float64(x + y) * t_0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -1.56e+154)
		tmp = ((1.0 - (y / x)) / (x + y)) * (y / t_0);
	else
		tmp = (x / (x + y)) * (y / ((x + y) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.56e+154], N[(N[(N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55999999999999993e154

   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. associate-*l* 57.8%

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

      2. times-frac 75.4%

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

      3. +-commutative 75.4%

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

      4. +-commutative 75.4%

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

      5. associate-+r+ 75.4%

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

      6. +-commutative 75.4%

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

      7. associate-+l+ 75.4%

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

   4. Applied egg-rr 75.4%

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

      1. frac-times 57.8%

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

      2. *-commutative 57.8%

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

      3. +-commutative 57.8%

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

   6. Applied egg-rr 57.8%

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

      1. associate-/r* 57.8%

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

      2. *-commutative 57.8%

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

      3. associate-*l/ 75.4%

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

      4. times-frac 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around inf 81.5%

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

       1. mul-1-neg 81.5%

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

       2. unsub-neg 81.5%

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

   11. Simplified 81.5%

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

   if -1.55999999999999993e154 < x

   1. Initial program 70.0%

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

      1. associate-*l* 70.1%

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

      2. times-frac 96.4%

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

      3. +-commutative 96.4%

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

      4. +-commutative 96.4%

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

      5. associate-+r+ 96.4%

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

      6. +-commutative 96.4%

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

      7. associate-+l+ 96.4%

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

   4. Applied egg-rr 96.4%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.56 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 - \frac{y}{x}}{x + y} \cdot \frac{y}{y + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.35e+154) {
		tmp = 1.0 / ((x + y) * ((x + (y + 1.0)) / y));
	} else {
		tmp = (x / (x + y)) * (y / ((x + y) * (y + (x + 1.0))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.35d+154)) then
        tmp = 1.0d0 / ((x + y) * ((x + (y + 1.0d0)) / y))
    else
        tmp = (x / (x + y)) * (y / ((x + y) * (y + (x + 1.0d0))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.35e+154) {
		tmp = 1.0 / ((x + y) * ((x + (y + 1.0)) / y));
	} else {
		tmp = (x / (x + y)) * (y / ((x + y) * (y + (x + 1.0))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.35e+154:
		tmp = 1.0 / ((x + y) * ((x + (y + 1.0)) / y))
	else:
		tmp = (x / (x + y)) * (y / ((x + y) * (y + (x + 1.0))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.35e+154)
		tmp = Float64(1.0 / Float64(Float64(x + y) * Float64(Float64(x + Float64(y + 1.0)) / y)));
	else
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(y / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.35e+154)
		tmp = 1.0 / ((x + y) * ((x + (y + 1.0)) / y));
	else
		tmp = (x / (x + y)) * (y / ((x + y) * (y + (x + 1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.35e+154], N[(1.0 / N[(N[(x + y), $MachinePrecision] * N[(N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.35000000000000003e154

   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. associate-*l* 57.8%

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

      2. times-frac 75.4%

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

      3. +-commutative 75.4%

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

      4. +-commutative 75.4%

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

      5. associate-+r+ 75.4%

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

      6. +-commutative 75.4%

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

      7. associate-+l+ 75.4%

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

   4. Applied egg-rr 75.4%

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

      1. frac-times 57.8%

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

      2. *-commutative 57.8%

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

      3. +-commutative 57.8%

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

   6. Applied egg-rr 57.8%

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

      1. associate-/r* 57.8%

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

      2. *-commutative 57.8%

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

      3. associate-*l/ 75.4%

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

      4. times-frac 99.7%

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

   8. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. clear-num 99.9%

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

      3. frac-times 99.8%

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

      4. *-un-lft-identity 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

      9. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in x around inf 81.1%

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

   if -1.35000000000000003e154 < x

   1. Initial program 70.0%

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

      1. associate-*l* 70.1%

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

      2. times-frac 96.4%

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

      3. +-commutative 96.4%

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

      4. +-commutative 96.4%

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

      5. associate-+r+ 96.4%

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

      6. +-commutative 96.4%

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

      7. associate-+l+ 96.4%

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

   4. Applied egg-rr 96.4%

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

4. Final simplification 94.7%

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

Alternative 4: 93.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{+118}:\\ \;\;\;\;\frac{y}{t\_0} \cdot \frac{1}{x + y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(x + y\right) \cdot t\_0}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -9.5e+118)
     (* (/ y t_0) (/ 1.0 (+ x y)))
     (* x (/ (/ y (* (+ x y) t_0)) (+ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -9.5e+118) {
		tmp = (y / t_0) * (1.0 / (x + y));
	} else {
		tmp = x * ((y / ((x + y) * t_0)) / (x + y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (x <= (-9.5d+118)) then
        tmp = (y / t_0) * (1.0d0 / (x + y))
    else
        tmp = x * ((y / ((x + y) * t_0)) / (x + y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -9.5e+118) {
		tmp = (y / t_0) * (1.0 / (x + y));
	} else {
		tmp = x * ((y / ((x + y) * t_0)) / (x + y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -9.5e+118:
		tmp = (y / t_0) * (1.0 / (x + y))
	else:
		tmp = x * ((y / ((x + y) * t_0)) / (x + y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -9.5e+118)
		tmp = Float64(Float64(y / t_0) * Float64(1.0 / Float64(x + y)));
	else
		tmp = Float64(x * Float64(Float64(y / Float64(Float64(x + y) * t_0)) / Float64(x + y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -9.5e+118)
		tmp = (y / t_0) * (1.0 / (x + y));
	else
		tmp = x * ((y / ((x + y) * t_0)) / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9.5e+118], N[(N[(y / t$95$0), $MachinePrecision] * N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.49999999999999974e118

   1. Initial program 49.8%

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

      1. associate-*l* 49.8%

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

      2. times-frac 80.2%

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

      3. +-commutative 80.2%

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

      4. +-commutative 80.2%

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

      5. associate-+r+ 80.2%

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

      6. +-commutative 80.2%

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

      7. associate-+l+ 80.2%

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

   4. Applied egg-rr 80.2%

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

      1. frac-times 49.8%

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

      2. *-commutative 49.8%

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

      3. +-commutative 49.8%

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

   6. Applied egg-rr 49.8%

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

      1. associate-/r* 60.4%

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

      2. *-commutative 60.4%

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

      3. associate-*l/ 80.2%

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

      4. times-frac 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around inf 76.7%

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

   if -9.49999999999999974e118 < x

   1. Initial program 71.7%

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

      1. associate-/l* 83.6%

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

      2. associate-+l+ 83.6%

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

   3. Simplified 83.6%

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

      1. *-un-lft-identity 83.6%

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

      2. associate-+r+ 83.6%

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

      3. associate-*l* 83.6%

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

      4. times-frac 94.6%

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

      5. +-commutative 94.6%

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

      6. +-commutative 94.6%

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

      7. associate-+r+ 94.6%

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

      8. +-commutative 94.6%

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

      9. associate-+l+ 94.6%

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

   6. Applied egg-rr 94.6%

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

      1. associate-*l/ 94.6%

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

      2. *-lft-identity 94.6%

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

      3. +-commutative 94.6%

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

   8. Simplified 94.6%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+118}:\\ \;\;\;\;\frac{y}{y + \left(x + 1\right)} \cdot \frac{1}{x + y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\left(x + y\right) \cdot \frac{x + \left(y + 1\right)}{y}}\\ \mathbf{elif}\;x \leq -9.8 \cdot 10^{-68}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.4e+154)
   (/ 1.0 (* (+ x y) (/ (+ x (+ y 1.0)) y)))
   (if (<= x -9.8e-68)
     (/ y (* (+ x y) (+ y (+ x 1.0))))
     (/ (/ x (+ x y)) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.4e+154) {
		tmp = 1.0 / ((x + y) * ((x + (y + 1.0)) / y));
	} else if (x <= -9.8e-68) {
		tmp = y / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = (x / (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 <= (-1.4d+154)) then
        tmp = 1.0d0 / ((x + y) * ((x + (y + 1.0d0)) / y))
    else if (x <= (-9.8d-68)) then
        tmp = y / ((x + y) * (y + (x + 1.0d0)))
    else
        tmp = (x / (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 <= -1.4e+154) {
		tmp = 1.0 / ((x + y) * ((x + (y + 1.0)) / y));
	} else if (x <= -9.8e-68) {
		tmp = y / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.4e+154:
		tmp = 1.0 / ((x + y) * ((x + (y + 1.0)) / y))
	elif x <= -9.8e-68:
		tmp = y / ((x + y) * (y + (x + 1.0)))
	else:
		tmp = (x / (x + y)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.4e+154)
		tmp = Float64(1.0 / Float64(Float64(x + y) * Float64(Float64(x + Float64(y + 1.0)) / y)));
	elseif (x <= -9.8e-68)
		tmp = Float64(y / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.4e+154)
		tmp = 1.0 / ((x + y) * ((x + (y + 1.0)) / y));
	elseif (x <= -9.8e-68)
		tmp = y / ((x + y) * (y + (x + 1.0)));
	else
		tmp = (x / (x + y)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.4e+154], N[(1.0 / N[(N[(x + y), $MachinePrecision] * N[(N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -9.8e-68], N[(y / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4e154

   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. associate-*l* 57.8%

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

      2. times-frac 75.4%

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

      3. +-commutative 75.4%

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

      4. +-commutative 75.4%

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

      5. associate-+r+ 75.4%

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

      6. +-commutative 75.4%

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

      7. associate-+l+ 75.4%

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

   4. Applied egg-rr 75.4%

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

      1. frac-times 57.8%

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

      2. *-commutative 57.8%

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

      3. +-commutative 57.8%

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

   6. Applied egg-rr 57.8%

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

      1. associate-/r* 57.8%

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

      2. *-commutative 57.8%

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

      3. associate-*l/ 75.4%

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

      4. times-frac 99.7%

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

   8. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. clear-num 99.9%

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

      3. frac-times 99.8%

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

      4. *-un-lft-identity 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

      9. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in x around inf 81.1%

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

   if -1.4e154 < x < -9.79999999999999954e-68

   1. Initial program 69.1%

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

      1. associate-*l* 69.1%

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

      2. times-frac 96.0%

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

      3. +-commutative 96.0%

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

      4. +-commutative 96.0%

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

      5. associate-+r+ 96.0%

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

      6. +-commutative 96.0%

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

      7. associate-+l+ 96.0%

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

   4. Applied egg-rr 96.0%

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

   5. Taylor expanded in x around inf 72.7%

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

   if -9.79999999999999954e-68 < x

   1. Initial program 70.3%

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

      1. associate-*l* 70.3%

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

      2. times-frac 96.6%

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

      3. +-commutative 96.6%

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

      4. +-commutative 96.6%

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

      5. associate-+r+ 96.6%

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

      6. +-commutative 96.6%

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

      7. associate-+l+ 96.6%

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

   4. Applied egg-rr 96.6%

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

      1. frac-times 70.3%

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

      2. *-commutative 70.3%

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

      3. +-commutative 70.3%

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

   6. Applied egg-rr 70.3%

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

      1. associate-/r* 74.3%

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

      2. *-commutative 74.3%

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

      3. associate-*l/ 96.6%

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

      4. times-frac 99.7%

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

   8. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. clear-num 99.7%

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

      3. frac-times 99.5%

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

      4. *-un-lft-identity 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+r+ 99.5%

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

      7. +-commutative 99.5%

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

      8. associate-+l+ 99.5%

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

      9. +-commutative 99.5%

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

   10. Applied egg-rr 99.5%

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

   11. Taylor expanded in x around 0 61.3%

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

       1. +-commutative 61.3%

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

   13. Simplified 61.3%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\left(x + y\right) \cdot \frac{x + \left(y + 1\right)}{y}}\\ \mathbf{elif}\;x \leq -9.8 \cdot 10^{-68}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{t\_0} \cdot \frac{1}{x + y}\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-68}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -1.35e+154)
     (* (/ y t_0) (/ 1.0 (+ x y)))
     (if (<= x -1.15e-68) (/ y (* (+ x y) t_0)) (/ (/ x (+ x y)) (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.35e+154) {
		tmp = (y / t_0) * (1.0 / (x + y));
	} else if (x <= -1.15e-68) {
		tmp = y / ((x + y) * t_0);
	} else {
		tmp = (x / (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)
    if (x <= (-1.35d+154)) then
        tmp = (y / t_0) * (1.0d0 / (x + y))
    else if (x <= (-1.15d-68)) then
        tmp = y / ((x + y) * t_0)
    else
        tmp = (x / (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);
	double tmp;
	if (x <= -1.35e+154) {
		tmp = (y / t_0) * (1.0 / (x + y));
	} else if (x <= -1.15e-68) {
		tmp = y / ((x + y) * t_0);
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -1.35e+154:
		tmp = (y / t_0) * (1.0 / (x + y))
	elif x <= -1.15e-68:
		tmp = y / ((x + y) * t_0)
	else:
		tmp = (x / (x + y)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1.35e+154)
		tmp = Float64(Float64(y / t_0) * Float64(1.0 / Float64(x + y)));
	elseif (x <= -1.15e-68)
		tmp = Float64(y / Float64(Float64(x + y) * t_0));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -1.35e+154)
		tmp = (y / t_0) * (1.0 / (x + y));
	elseif (x <= -1.15e-68)
		tmp = y / ((x + y) * t_0);
	else
		tmp = (x / (x + y)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e+154], N[(N[(y / t$95$0), $MachinePrecision] * N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.15e-68], N[(y / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35000000000000003e154

   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. associate-*l* 57.8%

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

      2. times-frac 75.4%

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

      3. +-commutative 75.4%

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

      4. +-commutative 75.4%

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

      5. associate-+r+ 75.4%

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

      6. +-commutative 75.4%

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

      7. associate-+l+ 75.4%

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

   4. Applied egg-rr 75.4%

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

      1. frac-times 57.8%

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

      2. *-commutative 57.8%

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

      3. +-commutative 57.8%

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

   6. Applied egg-rr 57.8%

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

      1. associate-/r* 57.8%

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

      2. *-commutative 57.8%

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

      3. associate-*l/ 75.4%

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

      4. times-frac 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around inf 81.0%

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

   if -1.35000000000000003e154 < x < -1.14999999999999998e-68

   1. Initial program 69.7%

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

      1. associate-*l* 69.7%

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

      2. times-frac 96.0%

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

      3. +-commutative 96.0%

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

      4. +-commutative 96.0%

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

      5. associate-+r+ 96.0%

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

      6. +-commutative 96.0%

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

      7. associate-+l+ 96.0%

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

   4. Applied egg-rr 96.0%

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

   5. Taylor expanded in x around inf 71.4%

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

   if -1.14999999999999998e-68 < x

   1. Initial program 70.1%

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

      1. associate-*l* 70.2%

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

      2. times-frac 96.5%

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

      3. +-commutative 96.5%

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

      4. +-commutative 96.5%

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

      5. associate-+r+ 96.5%

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

      6. +-commutative 96.5%

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

      7. associate-+l+ 96.5%

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

   4. Applied egg-rr 96.5%

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

      1. frac-times 70.2%

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

      2. *-commutative 70.2%

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

      3. +-commutative 70.2%

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

   6. Applied egg-rr 70.2%

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

      1. associate-/r* 74.2%

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

      2. *-commutative 74.2%

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

      3. associate-*l/ 96.5%

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

      4. times-frac 99.7%

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

   8. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. clear-num 99.7%

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

      3. frac-times 99.5%

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

      4. *-un-lft-identity 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+r+ 99.5%

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

      7. +-commutative 99.5%

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

      8. associate-+l+ 99.5%

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

      9. +-commutative 99.5%

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

   10. Applied egg-rr 99.5%

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

   11. Taylor expanded in x around 0 61.0%

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

       1. +-commutative 61.0%

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

   13. Simplified 61.0%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{y + \left(x + 1\right)} \cdot \frac{1}{x + y}\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-68}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-69}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.25e+154)
   (* (/ y (+ x y)) (/ 1.0 x))
   (if (<= x -6e-69)
     (/ y (* (+ x y) (+ y (+ x 1.0))))
     (/ (/ x (+ x y)) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.25e+154) {
		tmp = (y / (x + y)) * (1.0 / x);
	} else if (x <= -6e-69) {
		tmp = y / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = (x / (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 <= (-1.25d+154)) then
        tmp = (y / (x + y)) * (1.0d0 / x)
    else if (x <= (-6d-69)) then
        tmp = y / ((x + y) * (y + (x + 1.0d0)))
    else
        tmp = (x / (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 <= -1.25e+154) {
		tmp = (y / (x + y)) * (1.0 / x);
	} else if (x <= -6e-69) {
		tmp = y / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.25e+154:
		tmp = (y / (x + y)) * (1.0 / x)
	elif x <= -6e-69:
		tmp = y / ((x + y) * (y + (x + 1.0)))
	else:
		tmp = (x / (x + y)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.25e+154)
		tmp = Float64(Float64(y / Float64(x + y)) * Float64(1.0 / x));
	elseif (x <= -6e-69)
		tmp = Float64(y / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.25e+154)
		tmp = (y / (x + y)) * (1.0 / x);
	elseif (x <= -6e-69)
		tmp = y / ((x + y) * (y + (x + 1.0)));
	else
		tmp = (x / (x + y)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.25e+154], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6e-69], N[(y / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25000000000000001e154

   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. +-commutative 57.8%

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

      2. distribute-rgt-in 57.8%

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

      3. +-commutative 57.8%

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

      4. +-commutative 57.8%

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

   4. Applied egg-rr 57.8%

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

      1. times-frac 69.0%

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

      2. distribute-rgt-out 75.4%

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

      3. associate-/l/ 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-+r+ 99.7%

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

      6. frac-times 75.4%

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

      7. +-commutative 75.4%

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

      8. *-commutative 75.4%

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

      9. +-commutative 75.4%

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

      10. times-frac 99.7%

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

      11. +-commutative 99.7%

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

      12. associate-+r+ 99.7%

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

      13. +-commutative 99.7%

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

      14. associate-+l+ 99.7%

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

      15. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around inf 80.6%

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

   if -1.25000000000000001e154 < x < -5.99999999999999978e-69

   1. Initial program 69.7%

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

      1. associate-*l* 69.7%

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

      2. times-frac 96.0%

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

      3. +-commutative 96.0%

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

      4. +-commutative 96.0%

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

      5. associate-+r+ 96.0%

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

      6. +-commutative 96.0%

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

      7. associate-+l+ 96.0%

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

   4. Applied egg-rr 96.0%

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

   5. Taylor expanded in x around inf 71.4%

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

   if -5.99999999999999978e-69 < x

   1. Initial program 70.1%

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

      1. associate-*l* 70.2%

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

      2. times-frac 96.5%

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

      3. +-commutative 96.5%

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

      4. +-commutative 96.5%

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

      5. associate-+r+ 96.5%

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

      6. +-commutative 96.5%

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

      7. associate-+l+ 96.5%

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

   4. Applied egg-rr 96.5%

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

      1. frac-times 70.2%

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

      2. *-commutative 70.2%

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

      3. +-commutative 70.2%

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

   6. Applied egg-rr 70.2%

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

      1. associate-/r* 74.2%

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

      2. *-commutative 74.2%

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

      3. associate-*l/ 96.5%

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

      4. times-frac 99.7%

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

   8. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. clear-num 99.7%

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

      3. frac-times 99.5%

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

      4. *-un-lft-identity 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+r+ 99.5%

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

      7. +-commutative 99.5%

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

      8. associate-+l+ 99.5%

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

      9. +-commutative 99.5%

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

   10. Applied egg-rr 99.5%

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

   11. Taylor expanded in x around 0 61.0%

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

       1. +-commutative 61.0%

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

   13. Simplified 61.0%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-69}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2200:\\ \;\;\;\;\frac{y}{y + \left(x + 1\right)} \cdot \frac{1}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y} \cdot \frac{y}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2200.0)
   (* (/ y (+ y (+ x 1.0))) (/ 1.0 (+ x y)))
   (* (/ (/ x (+ x y)) (+ x y)) (/ y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2200.0) {
		tmp = (y / (y + (x + 1.0))) * (1.0 / (x + y));
	} else {
		tmp = ((x / (x + y)) / (x + y)) * (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 <= (-2200.0d0)) then
        tmp = (y / (y + (x + 1.0d0))) * (1.0d0 / (x + y))
    else
        tmp = ((x / (x + y)) / (x + y)) * (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 <= -2200.0) {
		tmp = (y / (y + (x + 1.0))) * (1.0 / (x + y));
	} else {
		tmp = ((x / (x + y)) / (x + y)) * (y / (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2200.0:
		tmp = (y / (y + (x + 1.0))) * (1.0 / (x + y))
	else:
		tmp = ((x / (x + y)) / (x + y)) * (y / (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2200.0)
		tmp = Float64(Float64(y / Float64(y + Float64(x + 1.0))) * Float64(1.0 / Float64(x + y)));
	else
		tmp = Float64(Float64(Float64(x / Float64(x + y)) / Float64(x + y)) * 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 <= -2200.0)
		tmp = (y / (y + (x + 1.0))) * (1.0 / (x + y));
	else
		tmp = ((x / (x + y)) / (x + y)) * (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, -2200.0], N[(N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2200

   1. Initial program 58.7%

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

      1. associate-*l* 58.7%

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

      2. times-frac 85.0%

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

      3. +-commutative 85.0%

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

      4. +-commutative 85.0%

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

      5. associate-+r+ 85.0%

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

      6. +-commutative 85.0%

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

      7. associate-+l+ 85.0%

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

   4. Applied egg-rr 85.0%

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

      1. frac-times 58.7%

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

      2. *-commutative 58.7%

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

      3. +-commutative 58.7%

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

   6. Applied egg-rr 58.7%

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

      1. associate-/r* 69.8%

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

      2. *-commutative 69.8%

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

      3. associate-*l/ 85.0%

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

      4. times-frac 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around inf 74.9%

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

   if -2200 < x

   1. Initial program 71.7%

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

      1. associate-*l* 71.7%

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

      2. times-frac 96.8%

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

      3. +-commutative 96.8%

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

      4. +-commutative 96.8%

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

      5. associate-+r+ 96.8%

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

      6. +-commutative 96.8%

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

      7. associate-+l+ 96.8%

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

   4. Applied egg-rr 96.8%

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

      1. frac-times 71.7%

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

      2. *-commutative 71.7%

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

      3. +-commutative 71.7%

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

   6. Applied egg-rr 71.7%

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

      1. associate-/r* 76.0%

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

      2. *-commutative 76.0%

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

      3. associate-*l/ 96.8%

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

      4. times-frac 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around 0 82.0%

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

       1. +-commutative 82.0%

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

   11. Simplified 82.0%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2200:\\ \;\;\;\;\frac{y}{y + \left(x + 1\right)} \cdot \frac{1}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y} \cdot \frac{y}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.7%

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

   1. +-commutative 68.7%

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

   2. distribute-rgt-in 68.7%

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

   3. +-commutative 68.7%

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

   4. +-commutative 68.7%

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

4. Applied egg-rr 68.7%

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

   1. times-frac 84.4%

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

   2. distribute-rgt-out 88.0%

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

   3. associate-/l/ 99.7%

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

   4. +-commutative 99.7%

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

   5. associate-+r+ 99.7%

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

   6. frac-times 94.1%

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

   7. +-commutative 94.1%

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

   8. *-commutative 94.1%

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

   9. +-commutative 94.1%

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

   10. times-frac 99.8%

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

   11. +-commutative 99.8%

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

   12. associate-+r+ 99.8%

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

   13. +-commutative 99.8%

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

   14. associate-+l+ 99.8%

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

   15. +-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 10: 83.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-68}:\\ \;\;\;\;\frac{y}{y + \left(x + 1\right)} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3e-68)
   (* (/ y (+ y (+ x 1.0))) (/ 1.0 x))
   (/ (/ x (+ x y)) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3e-68) {
		tmp = (y / (y + (x + 1.0))) * (1.0 / x);
	} else {
		tmp = (x / (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 <= (-3d-68)) then
        tmp = (y / (y + (x + 1.0d0))) * (1.0d0 / x)
    else
        tmp = (x / (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 <= -3e-68) {
		tmp = (y / (y + (x + 1.0))) * (1.0 / x);
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3e-68:
		tmp = (y / (y + (x + 1.0))) * (1.0 / x)
	else:
		tmp = (x / (x + y)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3e-68)
		tmp = Float64(Float64(y / Float64(y + Float64(x + 1.0))) * Float64(1.0 / x));
	else
		tmp = Float64(Float64(x / 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 <= -3e-68)
		tmp = (y / (y + (x + 1.0))) * (1.0 / x);
	else
		tmp = (x / (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, -3e-68], N[(N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3e-68

   1. Initial program 65.4%

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

      1. associate-*l* 65.4%

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

      2. times-frac 88.6%

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

      3. +-commutative 88.6%

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

      4. +-commutative 88.6%

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

      5. associate-+r+ 88.6%

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

      6. +-commutative 88.6%

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

      7. associate-+l+ 88.6%

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

   4. Applied egg-rr 88.6%

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

      1. frac-times 65.4%

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

      2. *-commutative 65.4%

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

      3. +-commutative 65.4%

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

   6. Applied egg-rr 65.4%

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

      1. associate-/r* 75.5%

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

      2. *-commutative 75.5%

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

      3. associate-*l/ 88.6%

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

      4. times-frac 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in x around inf 66.1%

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

   if -3e-68 < x

   1. Initial program 70.1%

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

      1. associate-*l* 70.2%

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

      2. times-frac 96.5%

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

      3. +-commutative 96.5%

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

      4. +-commutative 96.5%

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

      5. associate-+r+ 96.5%

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

      6. +-commutative 96.5%

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

      7. associate-+l+ 96.5%

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

   4. Applied egg-rr 96.5%

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

      1. frac-times 70.2%

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

      2. *-commutative 70.2%

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

      3. +-commutative 70.2%

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

   6. Applied egg-rr 70.2%

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

      1. associate-/r* 74.2%

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

      2. *-commutative 74.2%

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

      3. associate-*l/ 96.5%

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

      4. times-frac 99.7%

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

   8. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. clear-num 99.7%

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

      3. frac-times 99.5%

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

      4. *-un-lft-identity 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+r+ 99.5%

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

      7. +-commutative 99.5%

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

      8. associate-+l+ 99.5%

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

      9. +-commutative 99.5%

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

   10. Applied egg-rr 99.5%

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

   11. Taylor expanded in x around 0 61.0%

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

       1. +-commutative 61.0%

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

   13. Simplified 61.0%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-68}:\\ \;\;\;\;\frac{y}{y + \left(x + 1\right)} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 82.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -6.2e-66) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ x y)) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -6.2e-66) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (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 <= (-6.2d-66)) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (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 <= -6.2e-66) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -6.2e-66:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (x + y)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -6.2e-66)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / 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 <= -6.2e-66)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (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, -6.2e-66], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.1999999999999995e-66

   1. Initial program 65.0%

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

      1. associate-/l* 78.1%

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

      2. associate-+l+ 78.1%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in y around 0 64.6%

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

      1. associate-/r* 66.5%

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

      2. +-commutative 66.5%

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

   7. Simplified 66.5%

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

   if -6.1999999999999995e-66 < x

   1. Initial program 70.3%

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

      1. associate-*l* 70.3%

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

      2. times-frac 96.6%

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

      3. +-commutative 96.6%

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

      4. +-commutative 96.6%

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

      5. associate-+r+ 96.6%

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

      6. +-commutative 96.6%

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

      7. associate-+l+ 96.6%

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

   4. Applied egg-rr 96.6%

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

      1. frac-times 70.3%

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

      2. *-commutative 70.3%

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

      3. +-commutative 70.3%

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

   6. Applied egg-rr 70.3%

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

      1. associate-/r* 74.3%

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

      2. *-commutative 74.3%

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

      3. associate-*l/ 96.6%

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

      4. times-frac 99.7%

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

   8. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. clear-num 99.7%

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

      3. frac-times 99.5%

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

      4. *-un-lft-identity 99.5%

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

      5. +-commutative 99.5%

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

      6. associate-+r+ 99.5%

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

      7. +-commutative 99.5%

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

      8. associate-+l+ 99.5%

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

      9. +-commutative 99.5%

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

   10. Applied egg-rr 99.5%

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

   11. Taylor expanded in x around 0 61.3%

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

       1. +-commutative 61.3%

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

   13. Simplified 61.3%

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

Alternative 12: 83.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -8.8e-66) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -8.8e-66) {
		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 <= (-8.8d-66)) 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 <= -8.8e-66) {
		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 <= -8.8e-66:
		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 <= -8.8e-66)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8.8e-66)
		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, -8.8e-66], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.8000000000000004e-66

   1. Initial program 65.0%

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

      1. associate-/l* 78.1%

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

      2. associate-+l+ 78.1%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in y around 0 64.6%

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

      1. associate-/r* 66.5%

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

      2. +-commutative 66.5%

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

   7. Simplified 66.5%

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

   if -8.8000000000000004e-66 < x

   1. Initial program 70.3%

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

      1. associate-/l* 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. Taylor expanded in x around 0 61.2%

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

      1. associate-/r* 61.2%

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

      2. +-commutative 61.2%

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

   7. Simplified 61.2%

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

      1. associate-/l/ 61.2%

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

      2. *-commutative 61.2%

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

      3. div-inv 61.3%

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

      4. associate-/r* 60.9%

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

   9. Applied egg-rr 60.9%

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

Alternative 13: 80.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-66}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.1e-66) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.1e-66) {
		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 <= (-3.1d-66)) 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 <= -3.1e-66) {
		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 <= -3.1e-66:
		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 <= -3.1e-66)
		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 <= -3.1e-66)
		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, -3.1e-66], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.0999999999999997e-66

   1. Initial program 65.0%

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

      1. associate-/l* 78.1%

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

      2. associate-+l+ 78.1%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in y around 0 64.6%

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

   if -3.0999999999999997e-66 < x

   1. Initial program 70.3%

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

      1. associate-/l* 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. Taylor expanded in x around 0 61.2%

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

      1. associate-/r* 61.2%

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

      2. +-commutative 61.2%

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

   7. Simplified 61.2%

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

      1. associate-/l/ 61.2%

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

      2. *-commutative 61.2%

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

      3. div-inv 61.3%

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

      4. associate-/r* 60.9%

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

   9. Applied egg-rr 60.9%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-66}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-66}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.1e-66) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.1e-66) {
		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 <= (-3.1d-66)) 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 <= -3.1e-66) {
		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 <= -3.1e-66:
		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 <= -3.1e-66)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.1e-66)
		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, -3.1e-66], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.0999999999999997e-66

   1. Initial program 65.0%

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

      1. associate-/l* 78.1%

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

      2. associate-+l+ 78.1%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in y around 0 64.6%

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

   if -3.0999999999999997e-66 < x

   1. Initial program 70.3%

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

      1. associate-/l* 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. Taylor expanded in x around 0 61.3%

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

      1. +-commutative 61.3%

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

   7. Simplified 61.3%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-66}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 28.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2250:\\ \;\;\;\;\frac{1}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2250.0) (/ 1.0 (+ x 1.0)) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2250.0) {
		tmp = 1.0 / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2250.0d0)) then
        tmp = 1.0d0 / (x + 1.0d0)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2250.0) {
		tmp = 1.0 / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2250.0:
		tmp = 1.0 / (x + 1.0)
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2250.0)
		tmp = Float64(1.0 / Float64(x + 1.0));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2250.0)
		tmp = 1.0 / (x + 1.0);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2250.0], N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2250

   1. Initial program 58.7%

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

      1. associate-*l* 58.7%

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

      2. times-frac 85.0%

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

      3. +-commutative 85.0%

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

      4. +-commutative 85.0%

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

      5. associate-+r+ 85.0%

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

      6. +-commutative 85.0%

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

      7. associate-+l+ 85.0%

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

   4. Applied egg-rr 85.0%

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

   5. Taylor expanded in x around 0 33.4%

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

   6. Taylor expanded in y around 0 6.2%

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

      1. +-commutative 6.2%

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

   8. Simplified 6.2%

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

   if -2250 < x

   1. Initial program 71.7%

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

      1. associate-/l* 84.0%

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

      2. associate-+l+ 84.0%

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

   3. Simplified 84.0%

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

   5. Taylor expanded in x around 0 61.0%

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

      1. +-commutative 61.0%

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

   7. Simplified 61.0%

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

   8. Taylor expanded in y around 0 40.1%

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

Alternative 16: 49.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.7%

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

   1. associate-/l* 81.6%

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

   2. associate-+l+ 81.6%

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

3. Simplified 81.6%

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

5. Taylor expanded in x around 0 52.3%

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

   1. +-commutative 52.3%

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

7. Simplified 52.3%

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

Alternative 17: 27.0% 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 68.7%

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

   1. associate-/l* 81.6%

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

   2. associate-+l+ 81.6%

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

3. Simplified 81.6%

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

5. Taylor expanded in x around 0 52.3%

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

   1. +-commutative 52.3%

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

7. Simplified 52.3%

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

8. Taylor expanded in y around 0 31.4%

   \[\leadsto \frac{x}{\color{blue}{y}} \]
9. Add Preprocessing

Alternative 18: 3.0% accurate, 5.7× speedup

Math

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

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 68.7%

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

   1. associate-*l* 68.7%

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

   2. times-frac 94.1%

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

   3. +-commutative 94.1%

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

   4. +-commutative 94.1%

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

   5. associate-+r+ 94.1%

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

   6. +-commutative 94.1%

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

   7. associate-+l+ 94.1%

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

4. Applied egg-rr 94.1%

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

5. Taylor expanded in x around 0 61.9%

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

6. Taylor expanded in y around 0 5.1%

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

   1. +-commutative 5.1%

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

8. Simplified 5.1%

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

9. Taylor expanded in x around 0 3.2%

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

    1. neg-mul-1 3.2%

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

    2. sub-neg 3.2%

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

11. Simplified 3.2%

    \[\leadsto \color{blue}{1 - x} \]
12. Add Preprocessing

Developer target: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024088 
(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))))