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

Percentage Accurate: 68.7% → 99.8%

Time: 12.4s

Alternatives: 19

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: 68.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 68.8%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

   8. associate-+l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

   10. +-lowering-+.f64 68.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

3. Simplified 68.8%

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

   1. associate-/r* N/A

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

   2. associate-/r* N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + y}\right), \color{blue}{\left(x + \left(y + 1\right)\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + y\right)\right), \left(\color{blue}{x} + \left(y + 1\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y \cdot x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

   6. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

   7. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{1}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

   8. un-div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(\frac{x + y}{x}\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(x + y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right) \]

   14. +-lowering-+.f64 99.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

6. Applied egg-rr 99.8%

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

7. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \color{blue}{\left(1 + \frac{y}{x}\right)}\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right) \]
8. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(\frac{y}{x}\right)\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right) \]

   2. /-lowering-/.f64 99.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(y, x\right)\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right) \]

9. Simplified 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 90.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 3200000:\\ \;\;\;\;x \cdot \frac{\frac{1}{y + x}}{y + \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{y + x}}{y + x}}{\frac{y + x}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 9.5e-152)
   (/ (/ y (+ y x)) (+ x (+ y 1.0)))
   (if (<= y 3200000.0)
     (* x (/ (/ 1.0 (+ y x)) (+ y (+ 1.0 x))))
     (/ (/ (/ x (+ y x)) (+ y x)) (/ (+ y x) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 9.5e-152) {
		tmp = (y / (y + x)) / (x + (y + 1.0));
	} else if (y <= 3200000.0) {
		tmp = x * ((1.0 / (y + x)) / (y + (1.0 + x)));
	} else {
		tmp = ((x / (y + x)) / (y + x)) / ((y + x) / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 9.5d-152) then
        tmp = (y / (y + x)) / (x + (y + 1.0d0))
    else if (y <= 3200000.0d0) then
        tmp = x * ((1.0d0 / (y + x)) / (y + (1.0d0 + x)))
    else
        tmp = ((x / (y + x)) / (y + x)) / ((y + x) / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 9.5e-152) {
		tmp = (y / (y + x)) / (x + (y + 1.0));
	} else if (y <= 3200000.0) {
		tmp = x * ((1.0 / (y + x)) / (y + (1.0 + x)));
	} else {
		tmp = ((x / (y + x)) / (y + x)) / ((y + x) / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 9.5e-152:
		tmp = (y / (y + x)) / (x + (y + 1.0))
	elif y <= 3200000.0:
		tmp = x * ((1.0 / (y + x)) / (y + (1.0 + x)))
	else:
		tmp = ((x / (y + x)) / (y + x)) / ((y + x) / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 9.5e-152)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + Float64(y + 1.0)));
	elseif (y <= 3200000.0)
		tmp = Float64(x * Float64(Float64(1.0 / Float64(y + x)) / Float64(y + Float64(1.0 + x))));
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) / Float64(y + x)) / Float64(Float64(y + x) / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9.5e-152)
		tmp = (y / (y + x)) / (x + (y + 1.0));
	elseif (y <= 3200000.0)
		tmp = x * ((1.0 / (y + x)) / (y + (1.0 + x)));
	else
		tmp = ((x / (y + x)) / (y + x)) / ((y + x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 9.5e-152], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3200000.0], N[(x * N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 9.49999999999999925e-152

   1. Initial program 68.5%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 68.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 68.6%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + y}\right), \color{blue}{\left(x + \left(y + 1\right)\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + y\right)\right), \left(\color{blue}{x} + \left(y + 1\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y \cdot x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{1}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(\frac{x + y}{x}\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(x + y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right) \]

      14. +-lowering-+.f64 99.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{y}, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 62.8%

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

   if 9.49999999999999925e-152 < y < 3.2e6

   1. Initial program 86.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 86.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 86.6%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + y}\right), \color{blue}{\left(x + \left(y + 1\right)\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + y\right)\right), \left(\color{blue}{x} + \left(y + 1\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y \cdot x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{1}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(\frac{x + y}{x}\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(x + y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right) \]

      14. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{x}, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 43.7%

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

       1. div-inv N/A

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

       2. associate-/l* N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\frac{1}{x + y}}{x + \left(y + 1\right)}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{1}{x + y}\right), \color{blue}{\left(x + \left(y + 1\right)\right)}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(x + y\right)\right), \left(\color{blue}{x} + \left(y + 1\right)\right)\right)\right) \]

       6. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(y + x\right)\right), \left(x + \left(y + 1\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(y, x\right)\right), \left(x + \left(y + 1\right)\right)\right)\right) \]

       8. associate-+r+ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(y, x\right)\right), \left(\left(x + y\right) + \color{blue}{1}\right)\right)\right) \]

       9. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(y, x\right)\right), \left(\left(y + x\right) + 1\right)\right)\right) \]

       10. associate-+l+ N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(y, x\right)\right), \left(y + \color{blue}{\left(x + 1\right)}\right)\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(y, x\right)\right), \mathsf{+.f64}\left(y, \color{blue}{\left(x + 1\right)}\right)\right)\right) \]

       12. +-lowering-+.f64 75.4%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(y, x\right)\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right)\right) \]

   11. Applied egg-rr 75.4%

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

   if 3.2e6 < y

   1. Initial program 60.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 60.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 60.6%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + y}\right), \color{blue}{\left(x + \left(y + 1\right)\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + y\right)\right), \left(\color{blue}{x} + \left(y + 1\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y \cdot x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{1}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(\frac{x + y}{x}\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(x + y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right) \]

      14. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

   6. Applied egg-rr 99.9%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. associate-*r/ N/A

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

      4. associate-/l/ N/A

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

      5. *-commutative N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x}{x + y}}{x + \left(y + 1\right)}\right), \color{blue}{\left(\frac{x + y}{y}\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(\frac{\color{blue}{x + y}}{y}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(\frac{\color{blue}{x} + y}{y}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(y + x\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(\frac{x + y}{y}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(y, x\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(\frac{x + y}{y}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(y, x\right)\right), \mathsf{+.f64}\left(x, \left(y + 1\right)\right)\right), \left(\frac{x + \color{blue}{y}}{y}\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(y, x\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right), \left(\frac{x + y}{y}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(y, x\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right), \mathsf{/.f64}\left(\left(x + y\right), \color{blue}{y}\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(y, x\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right), \mathsf{/.f64}\left(\left(y + x\right), y\right)\right) \]

      17. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(y, x\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(y, x\right), y\right)\right) \]

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(y, x\right)\right), \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(y, x\right), y\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 99.9%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 3200000:\\ \;\;\;\;x \cdot \frac{\frac{1}{y + x}}{y + \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{y + x}}{y + x}}{\frac{y + x}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -7.8 \cdot 10^{-105}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{-170}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+24}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

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 x))))
   (if (<= y -7.8e-105)
     t_0
     (if (<= y 2.05e-170) (/ y x) (if (<= y 2.4e+24) t_0 (/ (/ x y) y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x * x);
	double tmp;
	if (y <= -7.8e-105) {
		tmp = t_0;
	} else if (y <= 2.05e-170) {
		tmp = y / x;
	} else if (y <= 2.4e+24) {
		tmp = t_0;
	} else {
		tmp = (x / y) / 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 * x)
    if (y <= (-7.8d-105)) then
        tmp = t_0
    else if (y <= 2.05d-170) then
        tmp = y / x
    else if (y <= 2.4d+24) then
        tmp = t_0
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (x * x);
	double tmp;
	if (y <= -7.8e-105) {
		tmp = t_0;
	} else if (y <= 2.05e-170) {
		tmp = y / x;
	} else if (y <= 2.4e+24) {
		tmp = t_0;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (x * x)
	tmp = 0
	if y <= -7.8e-105:
		tmp = t_0
	elif y <= 2.05e-170:
		tmp = y / x
	elif y <= 2.4e+24:
		tmp = t_0
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(x * x))
	tmp = 0.0
	if (y <= -7.8e-105)
		tmp = t_0;
	elseif (y <= 2.05e-170)
		tmp = Float64(y / x);
	elseif (y <= 2.4e+24)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (x * x);
	tmp = 0.0;
	if (y <= -7.8e-105)
		tmp = t_0;
	elseif (y <= 2.05e-170)
		tmp = y / x;
	elseif (y <= 2.4e+24)
		tmp = t_0;
	else
		tmp = (x / y) / 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 * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.8e-105], t$95$0, If[LessEqual[y, 2.05e-170], N[(y / x), $MachinePrecision], If[LessEqual[y, 2.4e+24], t$95$0, N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.8e-105 or 2.04999999999999983e-170 < y < 2.4000000000000001e24

   1. Initial program 75.5%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 75.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 75.5%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + y}\right), \color{blue}{\left(x + \left(y + 1\right)\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + y\right)\right), \left(\color{blue}{x} + \left(y + 1\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y \cdot x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{1}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(\frac{x + y}{x}\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(x + y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right) \]

      14. +-lowering-+.f64 99.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 42.8%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   9. Simplified 42.8%

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

   if -7.8e-105 < y < 2.04999999999999983e-170

   1. Initial program 66.8%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 66.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 66.8%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(x \cdot \left(1 + x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \left(x + \color{blue}{1}\right)\right)\right) \]

      4. +-lowering-+.f64 84.6%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   7. Simplified 84.6%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 70.9%

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{x}\right) \]

   10. Simplified 70.9%

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

   if 2.4000000000000001e24 < y

   1. Initial program 60.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 60.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 60.0%

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

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 75.0%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   7. Simplified 75.0%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{y}\right), \color{blue}{y}\right) \]

      3. /-lowering-/.f64 74.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right) \]

   9. Applied egg-rr 74.8%

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

Alternative 4: 68.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -4.1 \cdot 10^{-103}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-168}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+23}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

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 x))))
   (if (<= y -4.1e-103)
     t_0
     (if (<= y 5.4e-168) (/ y x) (if (<= y 1.7e+23) t_0 (/ x (* y y)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x * x);
	double tmp;
	if (y <= -4.1e-103) {
		tmp = t_0;
	} else if (y <= 5.4e-168) {
		tmp = y / x;
	} else if (y <= 1.7e+23) {
		tmp = t_0;
	} else {
		tmp = x / (y * 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 * x)
    if (y <= (-4.1d-103)) then
        tmp = t_0
    else if (y <= 5.4d-168) then
        tmp = y / x
    else if (y <= 1.7d+23) then
        tmp = t_0
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (x * x);
	double tmp;
	if (y <= -4.1e-103) {
		tmp = t_0;
	} else if (y <= 5.4e-168) {
		tmp = y / x;
	} else if (y <= 1.7e+23) {
		tmp = t_0;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (x * x)
	tmp = 0
	if y <= -4.1e-103:
		tmp = t_0
	elif y <= 5.4e-168:
		tmp = y / x
	elif y <= 1.7e+23:
		tmp = t_0
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(x * x))
	tmp = 0.0
	if (y <= -4.1e-103)
		tmp = t_0;
	elseif (y <= 5.4e-168)
		tmp = Float64(y / x);
	elseif (y <= 1.7e+23)
		tmp = t_0;
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (x * x);
	tmp = 0.0;
	if (y <= -4.1e-103)
		tmp = t_0;
	elseif (y <= 5.4e-168)
		tmp = y / x;
	elseif (y <= 1.7e+23)
		tmp = t_0;
	else
		tmp = x / (y * 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 * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.1e-103], t$95$0, If[LessEqual[y, 5.4e-168], N[(y / x), $MachinePrecision], If[LessEqual[y, 1.7e+23], t$95$0, N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.09999999999999996e-103 or 5.40000000000000031e-168 < y < 1.69999999999999996e23

   1. Initial program 75.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 75.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 75.7%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + y}\right), \color{blue}{\left(x + \left(y + 1\right)\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + y\right)\right), \left(\color{blue}{x} + \left(y + 1\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y \cdot x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{1}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(\frac{x + y}{x}\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(x + y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right) \]

      14. +-lowering-+.f64 99.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 43.1%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   9. Simplified 43.1%

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

   if -4.09999999999999996e-103 < y < 5.40000000000000031e-168

   1. Initial program 66.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 66.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 66.7%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(x \cdot \left(1 + x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \left(x + \color{blue}{1}\right)\right)\right) \]

      4. +-lowering-+.f64 83.5%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   7. Simplified 83.5%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 70.0%

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{x}\right) \]

   10. Simplified 70.0%

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

   if 1.69999999999999996e23 < y

   1. Initial program 60.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 60.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 60.0%

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

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 75.0%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   7. Simplified 75.0%

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

Alternative 5: 98.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 60.9%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 60.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 60.9%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + y}\right), \color{blue}{\left(x + \left(y + 1\right)\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + y\right)\right), \left(\color{blue}{x} + \left(y + 1\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y \cdot x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{1}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(\frac{x + y}{x}\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(x + y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right) \]

      14. +-lowering-+.f64 99.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

   6. Applied egg-rr 99.8%

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

      1. associate-/l/ N/A

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

      2. associate-/r/ N/A

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

      3. associate-*r/ N/A

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

      4. associate-/l/ N/A

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

      5. *-commutative N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x}{x + y}}{x + \left(y + 1\right)}\right), \color{blue}{\left(\frac{x + y}{y}\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(\frac{\color{blue}{x + y}}{y}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(\frac{\color{blue}{x} + y}{y}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(y + x\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(\frac{x + y}{y}\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(y, x\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(\frac{x + y}{y}\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(y, x\right)\right), \mathsf{+.f64}\left(x, \left(y + 1\right)\right)\right), \left(\frac{x + \color{blue}{y}}{y}\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(y, x\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right), \left(\frac{x + y}{y}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(y, x\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right), \mathsf{/.f64}\left(\left(x + y\right), \color{blue}{y}\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(y, x\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right), \mathsf{/.f64}\left(\left(y + x\right), y\right)\right) \]

      17. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(y, x\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(y, x\right), y\right)\right) \]

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(y, x\right)\right), \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(y, x\right), y\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 99.3%

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

   if -1 < x

   1. Initial program 72.3%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 72.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 72.3%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + y}\right), \color{blue}{\left(x + \left(y + 1\right)\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + y\right)\right), \left(\color{blue}{x} + \left(y + 1\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y \cdot x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{1}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(\frac{x + y}{x}\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(x + y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right) \]

      14. +-lowering-+.f64 99.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \color{blue}{\left(1 + \frac{y}{x}\right)}\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \left(\frac{y}{x}\right)\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right) \]

      2. /-lowering-/.f64 99.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(y, x\right)\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right) \]

   9. Simplified 99.8%

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

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(y, x\right)\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \color{blue}{\left(1 + y\right)}\right) \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(y, x\right)\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \left(y + \color{blue}{1}\right)\right) \]

       2. +-lowering-+.f64 86.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(y, x\right)\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right) \]

   12. Simplified 86.6%

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

4. Final simplification 90.5%

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

Alternative 6: 82.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;y \leq 5.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{t\_0}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= y 5.5e-106)
     (/ (/ y (+ y x)) t_0)
     (if (<= y 6.5e+19) (/ (/ x y) t_0) (/ x (* (+ y x) (+ y x)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (y <= 5.5e-106) {
		tmp = (y / (y + x)) / t_0;
	} else if (y <= 6.5e+19) {
		tmp = (x / y) / t_0;
	} else {
		tmp = x / ((y + x) * (y + x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (y + 1.0d0)
    if (y <= 5.5d-106) then
        tmp = (y / (y + x)) / t_0
    else if (y <= 6.5d+19) then
        tmp = (x / y) / t_0
    else
        tmp = x / ((y + x) * (y + x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (y <= 5.5e-106) {
		tmp = (y / (y + x)) / t_0;
	} else if (y <= 6.5e+19) {
		tmp = (x / y) / t_0;
	} else {
		tmp = x / ((y + x) * (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x + (y + 1.0)
	tmp = 0
	if y <= 5.5e-106:
		tmp = (y / (y + x)) / t_0
	elif y <= 6.5e+19:
		tmp = (x / y) / t_0
	else:
		tmp = x / ((y + x) * (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (y <= 5.5e-106)
		tmp = Float64(Float64(y / Float64(y + x)) / t_0);
	elseif (y <= 6.5e+19)
		tmp = Float64(Float64(x / y) / t_0);
	else
		tmp = Float64(x / Float64(Float64(y + x) * Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x + (y + 1.0);
	tmp = 0.0;
	if (y <= 5.5e-106)
		tmp = (y / (y + x)) / t_0;
	elseif (y <= 6.5e+19)
		tmp = (x / y) / t_0;
	else
		tmp = x / ((y + x) * (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 5.5e-106], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 6.5e+19], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision], N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.5000000000000001e-106

   1. Initial program 69.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 69.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 69.4%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + y}\right), \color{blue}{\left(x + \left(y + 1\right)\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + y\right)\right), \left(\color{blue}{x} + \left(y + 1\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y \cdot x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{1}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(\frac{x + y}{x}\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(x + y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right) \]

      14. +-lowering-+.f64 99.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{y}, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 63.9%

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

   if 5.5000000000000001e-106 < y < 6.5e19

   1. Initial program 88.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 88.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 88.7%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + y}\right), \color{blue}{\left(x + \left(y + 1\right)\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + y\right)\right), \left(\color{blue}{x} + \left(y + 1\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y \cdot x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{1}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(\frac{x + y}{x}\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(x + y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right) \]

      14. +-lowering-+.f64 99.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{x}{y}\right)}, \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 48.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{+.f64}\left(\color{blue}{x}, \mathsf{+.f64}\left(y, 1\right)\right)\right) \]

   9. Simplified 48.0%

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

   if 6.5e19 < y

   1. Initial program 60.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 60.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 60.0%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \color{blue}{y}\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 59.4%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. *-inverses N/A

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

      5. div-inv N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. clear-num N/A

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

      9. associate-*r/ N/A

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

      10. clear-num N/A

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

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(x + y\right)}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(y + x\right), \left(\color{blue}{x} + y\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, x\right), \left(\color{blue}{x} + y\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, x\right), \left(y + \color{blue}{x}\right)\right)\right) \]

      16. +-lowering-+.f64 83.5%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{+.f64}\left(y, \color{blue}{x}\right)\right)\right) \]

   9. Applied egg-rr 83.5%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;y \leq 9.2 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= y 9.2e-106)
     (/ (/ y x) t_0)
     (if (<= y 6.5e+19) (/ (/ x y) t_0) (/ x (* (+ y x) (+ y x)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (y <= 9.2e-106) {
		tmp = (y / x) / t_0;
	} else if (y <= 6.5e+19) {
		tmp = (x / y) / t_0;
	} else {
		tmp = x / ((y + x) * (y + x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (y + 1.0d0)
    if (y <= 9.2d-106) then
        tmp = (y / x) / t_0
    else if (y <= 6.5d+19) then
        tmp = (x / y) / t_0
    else
        tmp = x / ((y + x) * (y + x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (y <= 9.2e-106) {
		tmp = (y / x) / t_0;
	} else if (y <= 6.5e+19) {
		tmp = (x / y) / t_0;
	} else {
		tmp = x / ((y + x) * (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x + (y + 1.0)
	tmp = 0
	if y <= 9.2e-106:
		tmp = (y / x) / t_0
	elif y <= 6.5e+19:
		tmp = (x / y) / t_0
	else:
		tmp = x / ((y + x) * (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (y <= 9.2e-106)
		tmp = Float64(Float64(y / x) / t_0);
	elseif (y <= 6.5e+19)
		tmp = Float64(Float64(x / y) / t_0);
	else
		tmp = Float64(x / Float64(Float64(y + x) * Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x + (y + 1.0);
	tmp = 0.0;
	if (y <= 9.2e-106)
		tmp = (y / x) / t_0;
	elseif (y <= 6.5e+19)
		tmp = (x / y) / t_0;
	else
		tmp = x / ((y + x) * (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 9.2e-106], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 6.5e+19], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision], N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 9.2000000000000004e-106

   1. Initial program 69.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 69.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 69.4%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + y}\right), \color{blue}{\left(x + \left(y + 1\right)\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + y\right)\right), \left(\color{blue}{x} + \left(y + 1\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y \cdot x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{1}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(\frac{x + y}{x}\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(x + y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right) \]

      14. +-lowering-+.f64 99.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y}{x}\right)}, \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 63.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \mathsf{+.f64}\left(\color{blue}{x}, \mathsf{+.f64}\left(y, 1\right)\right)\right) \]

   9. Simplified 63.2%

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

   if 9.2000000000000004e-106 < y < 6.5e19

   1. Initial program 88.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 88.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 88.7%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + y}\right), \color{blue}{\left(x + \left(y + 1\right)\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + y\right)\right), \left(\color{blue}{x} + \left(y + 1\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y \cdot x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{1}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(\frac{x + y}{x}\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(x + y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right) \]

      14. +-lowering-+.f64 99.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{x}{y}\right)}, \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 48.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{+.f64}\left(\color{blue}{x}, \mathsf{+.f64}\left(y, 1\right)\right)\right) \]

   9. Simplified 48.0%

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

   if 6.5e19 < y

   1. Initial program 60.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 60.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 60.0%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \color{blue}{y}\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 59.4%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. *-inverses N/A

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

      5. div-inv N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. clear-num N/A

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

      9. associate-*r/ N/A

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

      10. clear-num N/A

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

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(x + y\right)}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(y + x\right), \left(\color{blue}{x} + y\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, x\right), \left(\color{blue}{x} + y\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, x\right), \left(y + \color{blue}{x}\right)\right)\right) \]

      16. +-lowering-+.f64 83.5%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{+.f64}\left(y, \color{blue}{x}\right)\right)\right) \]

   9. Applied egg-rr 83.5%

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

Alternative 8: 82.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{y}{x}}{1 + x}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 9.2e-106)
   (/ (/ y x) (+ 1.0 x))
   (if (<= y 6.5e+19) (/ (/ x y) (+ x (+ y 1.0))) (/ x (* (+ y x) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 9.2e-106) {
		tmp = (y / x) / (1.0 + x);
	} else if (y <= 6.5e+19) {
		tmp = (x / y) / (x + (y + 1.0));
	} else {
		tmp = x / ((y + x) * (y + x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 9.2d-106) then
        tmp = (y / x) / (1.0d0 + x)
    else if (y <= 6.5d+19) then
        tmp = (x / y) / (x + (y + 1.0d0))
    else
        tmp = x / ((y + x) * (y + x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 9.2e-106) {
		tmp = (y / x) / (1.0 + x);
	} else if (y <= 6.5e+19) {
		tmp = (x / y) / (x + (y + 1.0));
	} else {
		tmp = x / ((y + x) * (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 9.2e-106:
		tmp = (y / x) / (1.0 + x)
	elif y <= 6.5e+19:
		tmp = (x / y) / (x + (y + 1.0))
	else:
		tmp = x / ((y + x) * (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 9.2e-106)
		tmp = Float64(Float64(y / x) / Float64(1.0 + x));
	elseif (y <= 6.5e+19)
		tmp = Float64(Float64(x / y) / Float64(x + Float64(y + 1.0)));
	else
		tmp = Float64(x / Float64(Float64(y + x) * Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9.2e-106)
		tmp = (y / x) / (1.0 + x);
	elseif (y <= 6.5e+19)
		tmp = (x / y) / (x + (y + 1.0));
	else
		tmp = x / ((y + x) * (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 9.2e-106], N[(N[(y / x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.5e+19], N[(N[(x / y), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 9.2000000000000004e-106

   1. Initial program 69.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 69.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 69.4%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(x \cdot \left(1 + x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \left(x + \color{blue}{1}\right)\right)\right) \]

      4. +-lowering-+.f64 63.0%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   7. Simplified 63.0%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{x}\right), \color{blue}{\left(x + 1\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\color{blue}{x} + 1\right)\right) \]

      4. +-lowering-+.f64 63.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right) \]

   9. Applied egg-rr 63.1%

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

   if 9.2000000000000004e-106 < y < 6.5e19

   1. Initial program 88.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 88.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 88.7%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + y}\right), \color{blue}{\left(x + \left(y + 1\right)\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + y\right)\right), \left(\color{blue}{x} + \left(y + 1\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y \cdot x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{1}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(\frac{x + y}{x}\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(x + y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right) \]

      14. +-lowering-+.f64 99.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{x}{y}\right)}, \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 48.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{+.f64}\left(\color{blue}{x}, \mathsf{+.f64}\left(y, 1\right)\right)\right) \]

   9. Simplified 48.0%

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

   if 6.5e19 < y

   1. Initial program 60.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 60.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 60.0%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \color{blue}{y}\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 59.4%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. *-inverses N/A

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

      5. div-inv N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. clear-num N/A

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

      9. associate-*r/ N/A

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

      10. clear-num N/A

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

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(x + y\right)}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(y + x\right), \left(\color{blue}{x} + y\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, x\right), \left(\color{blue}{x} + y\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, x\right), \left(y + \color{blue}{x}\right)\right)\right) \]

      16. +-lowering-+.f64 83.5%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{+.f64}\left(y, \color{blue}{x}\right)\right)\right) \]

   9. Applied egg-rr 83.5%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{y}{x}}{1 + x}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{y}{x}}{1 + x}\\ \mathbf{elif}\;y \leq 5500000:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 9.2e-106)
   (/ (/ y x) (+ 1.0 x))
   (if (<= y 5500000.0) (/ x (* y (+ y 1.0))) (/ x (* (+ y x) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 9.2e-106) {
		tmp = (y / x) / (1.0 + x);
	} else if (y <= 5500000.0) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = x / ((y + x) * (y + x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 9.2d-106) then
        tmp = (y / x) / (1.0d0 + x)
    else if (y <= 5500000.0d0) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = x / ((y + x) * (y + x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 9.2e-106) {
		tmp = (y / x) / (1.0 + x);
	} else if (y <= 5500000.0) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = x / ((y + x) * (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 9.2e-106:
		tmp = (y / x) / (1.0 + x)
	elif y <= 5500000.0:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = x / ((y + x) * (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 9.2e-106)
		tmp = Float64(Float64(y / x) / Float64(1.0 + x));
	elseif (y <= 5500000.0)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(x / Float64(Float64(y + x) * Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9.2e-106)
		tmp = (y / x) / (1.0 + x);
	elseif (y <= 5500000.0)
		tmp = x / (y * (y + 1.0));
	else
		tmp = x / ((y + x) * (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 9.2e-106], N[(N[(y / x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5500000.0], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 9.2000000000000004e-106

   1. Initial program 69.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 69.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 69.4%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(x \cdot \left(1 + x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \left(x + \color{blue}{1}\right)\right)\right) \]

      4. +-lowering-+.f64 63.0%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   7. Simplified 63.0%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{x}\right), \color{blue}{\left(x + 1\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\color{blue}{x} + 1\right)\right) \]

      4. +-lowering-+.f64 63.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right) \]

   9. Applied egg-rr 63.1%

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

   if 9.2000000000000004e-106 < y < 5.5e6

   1. Initial program 88.2%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 88.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 88.2%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(y \cdot \left(1 + y\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + y\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \left(y + \color{blue}{1}\right)\right)\right) \]

      4. +-lowering-+.f64 49.4%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

   7. Simplified 49.4%

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

   if 5.5e6 < y

   1. Initial program 60.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 60.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 60.6%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \color{blue}{y}\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 58.6%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. *-inverses N/A

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

      5. div-inv N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. clear-num N/A

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

      9. associate-*r/ N/A

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

      10. clear-num N/A

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

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(x + y\right)}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(y + x\right), \left(\color{blue}{x} + y\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, x\right), \left(\color{blue}{x} + y\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, x\right), \left(y + \color{blue}{x}\right)\right)\right) \]

      16. +-lowering-+.f64 82.3%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{+.f64}\left(y, \color{blue}{x}\right)\right)\right) \]

   9. Applied egg-rr 82.3%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{y}{x}}{1 + x}\\ \mathbf{elif}\;y \leq 5500000:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{1}{y + x}}{y + \left(1 + x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3e-145)
   (/ (/ y (+ y x)) (+ x (+ y 1.0)))
   (* x (/ (/ 1.0 (+ y x)) (+ y (+ 1.0 x))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3e-145) {
		tmp = (y / (y + x)) / (x + (y + 1.0));
	} else {
		tmp = x * ((1.0 / (y + x)) / (y + (1.0 + x)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3d-145) then
        tmp = (y / (y + x)) / (x + (y + 1.0d0))
    else
        tmp = x * ((1.0d0 / (y + x)) / (y + (1.0d0 + x)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3e-145) {
		tmp = (y / (y + x)) / (x + (y + 1.0));
	} else {
		tmp = x * ((1.0 / (y + x)) / (y + (1.0 + x)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3e-145:
		tmp = (y / (y + x)) / (x + (y + 1.0))
	else:
		tmp = x * ((1.0 / (y + x)) / (y + (1.0 + x)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3e-145)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + Float64(y + 1.0)));
	else
		tmp = Float64(x * Float64(Float64(1.0 / Float64(y + x)) / Float64(y + Float64(1.0 + x))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3e-145)
		tmp = (y / (y + x)) / (x + (y + 1.0));
	else
		tmp = x * ((1.0 / (y + x)) / (y + (1.0 + x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3e-145], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.99999999999999992e-145

   1. Initial program 68.5%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 68.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 68.6%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + y}\right), \color{blue}{\left(x + \left(y + 1\right)\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + y\right)\right), \left(\color{blue}{x} + \left(y + 1\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y \cdot x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{1}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(\frac{x + y}{x}\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(x + y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right) \]

      14. +-lowering-+.f64 99.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{y}, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 62.8%

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

   if 2.99999999999999992e-145 < y

   1. Initial program 69.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 69.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 69.1%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + y}\right), \color{blue}{\left(x + \left(y + 1\right)\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + y\right)\right), \left(\color{blue}{x} + \left(y + 1\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y \cdot x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{x}{x + y}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{1}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y}{\frac{x + y}{x}}\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(\frac{x + y}{x}\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(x + y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \left(x + \left(y + 1\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right) \]

      14. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, y\right), x\right)\right), \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{x}, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 64.4%

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

       1. div-inv N/A

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

       2. associate-/l* N/A

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

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\frac{1}{x + y}}{x + \left(y + 1\right)}\right)}\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{1}{x + y}\right), \color{blue}{\left(x + \left(y + 1\right)\right)}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(x + y\right)\right), \left(\color{blue}{x} + \left(y + 1\right)\right)\right)\right) \]

       6. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(y + x\right)\right), \left(x + \left(y + 1\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(y, x\right)\right), \left(x + \left(y + 1\right)\right)\right)\right) \]

       8. associate-+r+ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(y, x\right)\right), \left(\left(x + y\right) + \color{blue}{1}\right)\right)\right) \]

       9. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(y, x\right)\right), \left(\left(y + x\right) + 1\right)\right)\right) \]

       10. associate-+l+ N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(y, x\right)\right), \left(y + \color{blue}{\left(x + 1\right)}\right)\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(y, x\right)\right), \mathsf{+.f64}\left(y, \color{blue}{\left(x + 1\right)}\right)\right)\right) \]

       12. +-lowering-+.f64 80.0%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(y, x\right)\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right)\right) \]

   11. Applied egg-rr 80.0%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{1}{y + x}}{y + \left(1 + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-102}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ (/ y x) x)
   (if (<= x -4.6e-102) (/ y x) (/ x (* y (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) / x;
	} else if (x <= -4.6e-102) {
		tmp = y / x;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (y / x) / x
    else if (x <= (-4.6d-102)) then
        tmp = y / x
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) / x;
	} else if (x <= -4.6e-102) {
		tmp = y / x;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (y / x) / x
	elif x <= -4.6e-102:
		tmp = y / x
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -4.6e-102)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (y / x) / x;
	elseif (x <= -4.6e-102)
		tmp = y / x;
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -4.6e-102], N[(y / x), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 60.9%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 60.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 60.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{x}\right), \color{blue}{x}\right) \]

      4. /-lowering-/.f64 79.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), x\right) \]

   7. Simplified 79.4%

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

   if -1 < x < -4.59999999999999973e-102

   1. Initial program 81.5%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 81.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 81.5%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(x \cdot \left(1 + x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \left(x + \color{blue}{1}\right)\right)\right) \]

      4. +-lowering-+.f64 39.3%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   7. Simplified 39.3%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 39.3%

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{x}\right) \]

   10. Simplified 39.3%

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

   if -4.59999999999999973e-102 < x

   1. Initial program 71.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 71.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 71.4%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(y \cdot \left(1 + y\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + y\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \left(y + \color{blue}{1}\right)\right)\right) \]

      4. +-lowering-+.f64 63.0%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

   7. Simplified 63.0%

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

Alternative 12: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.8%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

   8. associate-+l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

   10. +-lowering-+.f64 68.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

3. Simplified 68.8%

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

   1. times-frac N/A

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

   2. associate-*r/ N/A

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

   3. *-commutative N/A

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

   4. times-frac N/A

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

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{x}{x + y}}{x + \left(y + 1\right)}\right), \color{blue}{\left(\frac{y}{x + y}\right)}\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{x}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(\frac{\color{blue}{y}}{x + y}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(\frac{y}{x + y}\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(\frac{y}{x + y}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \left(y + 1\right)\right)\right), \left(\frac{y}{x + y}\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right), \left(\frac{y}{x + y}\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(x + y\right)}\right)\right) \]

   12. +-lowering-+.f64 99.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right), \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 13: 69.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -4.9 \cdot 10^{-219}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) (/ (/ y x) x) (if (<= x -4.9e-219) (/ y x) (/ (/ x y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) / x;
	} else if (x <= -4.9e-219) {
		tmp = y / x;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (y / x) / x
    else if (x <= (-4.9d-219)) then
        tmp = y / x
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) / x;
	} else if (x <= -4.9e-219) {
		tmp = y / x;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (y / x) / x
	elif x <= -4.9e-219:
		tmp = y / x
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -4.9e-219)
		tmp = Float64(y / x);
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (y / x) / x;
	elseif (x <= -4.9e-219)
		tmp = y / x;
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -4.9e-219], N[(y / x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 60.9%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 60.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 60.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{x}\right), \color{blue}{x}\right) \]

      4. /-lowering-/.f64 79.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), x\right) \]

   7. Simplified 79.4%

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

   if -1 < x < -4.8999999999999999e-219

   1. Initial program 72.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 72.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 72.1%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(x \cdot \left(1 + x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \left(x + \color{blue}{1}\right)\right)\right) \]

      4. +-lowering-+.f64 30.7%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   7. Simplified 30.7%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 30.7%

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{x}\right) \]

   10. Simplified 30.7%

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

   if -4.8999999999999999e-219 < x

   1. Initial program 72.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 72.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 72.4%

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

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 44.8%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   7. Simplified 44.8%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{y}\right), \color{blue}{y}\right) \]

      3. /-lowering-/.f64 44.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right) \]

   9. Applied egg-rr 44.8%

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

Alternative 14: 80.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{y}{x}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 9.2e-106) (/ (/ y x) (+ 1.0 x)) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 9.2e-106) {
		tmp = (y / x) / (1.0 + x);
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 9.2d-106) then
        tmp = (y / x) / (1.0d0 + x)
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 9.2e-106) {
		tmp = (y / x) / (1.0 + x);
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 9.2e-106:
		tmp = (y / x) / (1.0 + x)
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 9.2e-106)
		tmp = Float64(Float64(y / x) / Float64(1.0 + x));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9.2e-106)
		tmp = (y / x) / (1.0 + x);
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 9.2e-106], N[(N[(y / x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.2000000000000004e-106

   1. Initial program 69.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 69.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 69.4%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(x \cdot \left(1 + x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \left(x + \color{blue}{1}\right)\right)\right) \]

      4. +-lowering-+.f64 63.0%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   7. Simplified 63.0%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{x}\right), \color{blue}{\left(x + 1\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\color{blue}{x} + 1\right)\right) \]

      4. +-lowering-+.f64 63.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right) \]

   9. Applied egg-rr 63.1%

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

   if 9.2000000000000004e-106 < y

   1. Initial program 67.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 67.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 67.7%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(y \cdot \left(1 + y\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + y\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \left(y + \color{blue}{1}\right)\right)\right) \]

      4. +-lowering-+.f64 67.6%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

   7. Simplified 67.6%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{y}{x}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 78.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-106}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 9.2e-106) (/ y (* x (+ 1.0 x))) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 9.2e-106) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 9.2d-106) then
        tmp = y / (x * (1.0d0 + x))
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 9.2e-106) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 9.2e-106:
		tmp = y / (x * (1.0 + x))
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 9.2e-106)
		tmp = Float64(y / Float64(x * Float64(1.0 + x)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9.2e-106)
		tmp = y / (x * (1.0 + x));
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 9.2e-106], N[(y / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.2000000000000004e-106

   1. Initial program 69.4%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 69.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 69.4%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(x \cdot \left(1 + x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \left(x + \color{blue}{1}\right)\right)\right) \]

      4. +-lowering-+.f64 63.0%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   7. Simplified 63.0%

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

   if 9.2000000000000004e-106 < y

   1. Initial program 67.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 67.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 67.7%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(y \cdot \left(1 + y\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + y\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \left(y + \color{blue}{1}\right)\right)\right) \]

      4. +-lowering-+.f64 67.6%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

   7. Simplified 67.6%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-106}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 59.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 1.2e-36) (/ y x) (/ x (* y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.2e-36) {
		tmp = y / x;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.2d-36) then
        tmp = y / x
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.2e-36) {
		tmp = y / x;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.2e-36:
		tmp = y / x
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.2e-36)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.2e-36)
		tmp = y / x;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.2e-36], N[(y / x), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.2e-36

   1. Initial program 70.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 70.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 70.7%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(x \cdot \left(1 + x\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \left(x + \color{blue}{1}\right)\right)\right) \]

      4. +-lowering-+.f64 62.3%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   7. Simplified 62.3%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 36.9%

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{x}\right) \]

   10. Simplified 36.9%

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

   if 1.2e-36 < y

   1. Initial program 64.5%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 64.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 64.5%

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

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 66.4%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   7. Simplified 66.4%

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

Alternative 17: 25.6% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.8%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

   8. associate-+l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

   10. +-lowering-+.f64 68.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

3. Simplified 68.8%

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

5. Taylor expanded in y around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(x \cdot \left(1 + x\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x\right)}\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \left(x + \color{blue}{1}\right)\right)\right) \]

   4. +-lowering-+.f64 49.3%

      \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

7. Simplified 49.3%

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

8. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 26.1%

      \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{x}\right) \]

10. Simplified 26.1%

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

Alternative 18: 4.3% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.8%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

   8. associate-+l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

   10. +-lowering-+.f64 68.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

3. Simplified 68.8%

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

5. Taylor expanded in y around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \color{blue}{y}\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 35.2%

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

8. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 4.1%

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{x}\right) \]

10. Simplified 4.1%

    \[\leadsto \color{blue}{\frac{1}{x}} \]
11. Add Preprocessing

Alternative 19: 3.7% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.8%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

   8. associate-+l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

   10. +-lowering-+.f64 68.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

3. Simplified 68.8%

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

   1. times-frac N/A

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

   2. associate-*r/ N/A

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

   3. *-commutative N/A

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

   4. times-frac N/A

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

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{x}{x + y}}{x + \left(y + 1\right)}\right), \color{blue}{\left(\frac{y}{x + y}\right)}\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{x}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(\frac{\color{blue}{y}}{x + y}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(x + y\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(\frac{y}{x + y}\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(\frac{y}{x + y}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \left(y + 1\right)\right)\right), \left(\frac{y}{x + y}\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right), \left(\frac{y}{x + y}\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right), \mathsf{/.f64}\left(y, \color{blue}{\left(x + y\right)}\right)\right) \]

   12. +-lowering-+.f64 99.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right), \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around inf

   \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(1 + -1 \cdot \frac{y}{x}\right)}, \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right), \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right) \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{y}{x}\right)\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right), \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right) \]

   2. unsub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(1 - \frac{y}{x}\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right), \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right) \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{y}{x}\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right), \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right) \]

   4. /-lowering-/.f64 51.3%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(y, x\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right), \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right) \]

9. Simplified 51.3%

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

10. Taylor expanded in y around inf

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

    1. /-lowering-/.f64 4.0%

       \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{x}\right) \]

12. Simplified 4.0%

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

Developer Target 1: 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 2024138 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (/ (/ x (+ (+ y 1) x)) (+ y x)) (/ 1 (/ y (+ y x)))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))