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

Percentage Accurate: 68.8% → 99.7%

Time: 13.8s

Alternatives: 23

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: 68.8% 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: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.2%

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

   1. +-commutative 68.2%

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

   2. +-commutative 68.2%

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

   3. +-commutative 68.2%

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

   4. *-commutative 68.2%

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

   5. distribute-rgt1-in 52.9%

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

   6. fma-define 68.2%

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

   7. +-commutative 68.2%

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

   8. +-commutative 68.2%

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

   9. cube-unmult 68.2%

      \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

   10. +-commutative 68.2%

       \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

3. Simplified 68.2%

   \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative 68.2%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

   2. fma-define 52.9%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   3. cube-mult 52.9%

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

   4. distribute-rgt1-in 68.2%

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

   5. *-commutative 68.2%

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

   6. associate-*l* 68.1%

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

   7. times-frac 92.3%

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

   8. associate-+r+ 92.3%

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

6. Applied egg-rr 92.3%

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

   1. clear-num 92.2%

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

   2. associate-/r* 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-+l+ 99.8%

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

   5. frac-times 99.7%

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

   6. metadata-eval 99.7%

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

   7. times-frac 99.7%

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

   8. *-un-lft-identity 99.7%

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

   9. *-un-lft-identity 99.7%

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

   10. +-commutative 99.7%

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

   11. +-commutative 99.7%

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

   12. +-commutative 99.7%

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

8. Applied egg-rr 99.7%

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

   1. div-inv 99.7%

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

   2. *-commutative 99.7%

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

   3. times-frac 99.7%

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

   4. +-commutative 99.7%

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

   5. +-commutative 99.7%

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

10. Applied egg-rr 99.7%

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

Alternative 2: 93.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1:\\ \;\;\;\;\frac{\frac{1}{x + y}}{\frac{x + y}{y}}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+19}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{x}{\left(x + 1\right) \cdot \left(x + y\right)}\\ \mathbf{elif}\;y \leq 10^{+146}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(y + \left(x + 1\right)\right) \cdot \frac{y}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -1.1)
   (/ (/ 1.0 (+ x y)) (/ (+ x y) y))
   (if (<= y 8e+19)
     (* (/ y (+ x y)) (/ x (* (+ x 1.0) (+ x y))))
     (if (<= y 1e+146)
       (/ x (* (+ x y) (+ x (+ y 1.0))))
       (/ (/ x (+ x y)) (* (+ y (+ x 1.0)) (/ y y)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -1.1) {
		tmp = (1.0 / (x + y)) / ((x + y) / y);
	} else if (y <= 8e+19) {
		tmp = (y / (x + y)) * (x / ((x + 1.0) * (x + y)));
	} else if (y <= 1e+146) {
		tmp = x / ((x + y) * (x + (y + 1.0)));
	} else {
		tmp = (x / (x + y)) / ((y + (x + 1.0)) * (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 <= (-1.1d0)) then
        tmp = (1.0d0 / (x + y)) / ((x + y) / y)
    else if (y <= 8d+19) then
        tmp = (y / (x + y)) * (x / ((x + 1.0d0) * (x + y)))
    else if (y <= 1d+146) then
        tmp = x / ((x + y) * (x + (y + 1.0d0)))
    else
        tmp = (x / (x + y)) / ((y + (x + 1.0d0)) * (y / y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= -1.1) {
		tmp = (1.0 / (x + y)) / ((x + y) / y);
	} else if (y <= 8e+19) {
		tmp = (y / (x + y)) * (x / ((x + 1.0) * (x + y)));
	} else if (y <= 1e+146) {
		tmp = x / ((x + y) * (x + (y + 1.0)));
	} else {
		tmp = (x / (x + y)) / ((y + (x + 1.0)) * (y / y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -1.1:
		tmp = (1.0 / (x + y)) / ((x + y) / y)
	elif y <= 8e+19:
		tmp = (y / (x + y)) * (x / ((x + 1.0) * (x + y)))
	elif y <= 1e+146:
		tmp = x / ((x + y) * (x + (y + 1.0)))
	else:
		tmp = (x / (x + y)) / ((y + (x + 1.0)) * (y / y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -1.1)
		tmp = Float64(Float64(1.0 / Float64(x + y)) / Float64(Float64(x + y) / y));
	elseif (y <= 8e+19)
		tmp = Float64(Float64(y / Float64(x + y)) * Float64(x / Float64(Float64(x + 1.0) * Float64(x + y))));
	elseif (y <= 1e+146)
		tmp = Float64(x / Float64(Float64(x + y) * Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(Float64(y + Float64(x + 1.0)) * 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 <= -1.1)
		tmp = (1.0 / (x + y)) / ((x + y) / y);
	elseif (y <= 8e+19)
		tmp = (y / (x + y)) * (x / ((x + 1.0) * (x + y)));
	elseif (y <= 1e+146)
		tmp = x / ((x + y) * (x + (y + 1.0)));
	else
		tmp = (x / (x + y)) / ((y + (x + 1.0)) * (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, -1.1], N[(N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e+19], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(x + 1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+146], N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] * N[(y / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.1000000000000001

   1. Initial program 59.1%

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

      1. +-commutative 59.1%

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

      2. +-commutative 59.1%

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

      3. +-commutative 59.1%

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

      4. *-commutative 59.1%

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

      5. distribute-rgt1-in 25.0%

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

      6. fma-define 59.1%

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

      7. +-commutative 59.1%

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

      8. +-commutative 59.1%

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

      9. cube-unmult 59.1%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 59.1%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 59.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 59.1%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 25.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 25.0%

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

      4. distribute-rgt1-in 59.1%

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

      5. *-commutative 59.1%

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

      6. associate-*l* 59.1%

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

      7. times-frac 82.3%

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

      8. associate-+r+ 82.3%

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

   6. Applied egg-rr 82.3%

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

      1. clear-num 82.3%

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

      2. associate-/r* 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+l+ 99.6%

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

      5. frac-times 99.7%

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

      6. metadata-eval 99.7%

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

      7. times-frac 99.7%

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

      8. *-un-lft-identity 99.7%

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

      9. *-un-lft-identity 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. div-inv 99.7%

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

      2. *-commutative 99.7%

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

      3. times-frac 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

   10. Applied egg-rr 99.6%

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

   11. Taylor expanded in x around inf 35.8%

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

   if -1.1000000000000001 < y < 8e19

   1. Initial program 78.6%

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

   3. Taylor expanded in x around inf 77.5%

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

      1. *-commutative 77.5%

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

      2. associate-*l* 77.4%

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

      3. times-frac 98.8%

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

      4. +-commutative 98.8%

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

      5. +-commutative 98.8%

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

   5. Applied egg-rr 98.8%

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

   if 8e19 < y < 9.99999999999999934e145

   1. Initial program 69.0%

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

      1. +-commutative 69.0%

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

      2. +-commutative 69.0%

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

      3. +-commutative 69.0%

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

      4. *-commutative 69.0%

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

      5. distribute-rgt1-in 60.2%

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

      6. fma-define 69.0%

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

      7. +-commutative 69.0%

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

      8. +-commutative 69.0%

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

      9. cube-unmult 69.0%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 69.0%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 69.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 69.0%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 60.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 60.2%

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

      4. distribute-rgt1-in 69.0%

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

      5. *-commutative 69.0%

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

      6. associate-*l* 69.0%

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

      7. times-frac 94.3%

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

      8. associate-+r+ 94.3%

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

   6. Applied egg-rr 94.3%

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

   7. Taylor expanded in y around inf 82.9%

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

   if 9.99999999999999934e145 < y

   1. Initial program 44.7%

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

      1. +-commutative 44.7%

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

      2. +-commutative 44.7%

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

      3. +-commutative 44.7%

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

      4. *-commutative 44.7%

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

      5. distribute-rgt1-in 44.7%

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

      6. fma-define 44.7%

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

      7. +-commutative 44.7%

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

      8. +-commutative 44.7%

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

      9. cube-unmult 44.7%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 44.7%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 44.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 44.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 44.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 44.7%

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

      4. distribute-rgt1-in 44.7%

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

      5. *-commutative 44.7%

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

      6. associate-*l* 44.7%

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

      7. times-frac 78.9%

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

      8. associate-+r+ 78.9%

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

   6. Applied egg-rr 78.9%

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

      1. clear-num 78.9%

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

      2. associate-/r* 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l+ 99.9%

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

      5. frac-times 99.9%

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

      6. metadata-eval 99.9%

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

      7. times-frac 99.9%

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

      8. *-un-lft-identity 99.9%

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

      9. *-un-lft-identity 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

      12. +-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in y around inf 92.3%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1:\\ \;\;\;\;\frac{\frac{1}{x + y}}{\frac{x + y}{y}}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+19}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{x}{\left(x + 1\right) \cdot \left(x + y\right)}\\ \mathbf{elif}\;y \leq 10^{+146}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(y + \left(x + 1\right)\right) \cdot \frac{y}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;y \leq -520000:\\ \;\;\;\;\frac{\frac{1}{x + y}}{\frac{x + y}{y}}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+134}:\\ \;\;\;\;t\_0 \cdot \frac{y}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(y + \left(x + 1\right)\right) \cdot \frac{y}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))))
   (if (<= y -520000.0)
     (/ (/ 1.0 (+ x y)) (/ (+ x y) y))
     (if (<= y 1.05e+134)
       (* t_0 (/ y (* (+ x y) (+ x (+ y 1.0)))))
       (/ t_0 (* (+ y (+ x 1.0)) (/ y y)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= -520000.0) {
		tmp = (1.0 / (x + y)) / ((x + y) / y);
	} else if (y <= 1.05e+134) {
		tmp = t_0 * (y / ((x + y) * (x + (y + 1.0))));
	} else {
		tmp = t_0 / ((y + (x + 1.0)) * (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 = x / (x + y)
    if (y <= (-520000.0d0)) then
        tmp = (1.0d0 / (x + y)) / ((x + y) / y)
    else if (y <= 1.05d+134) then
        tmp = t_0 * (y / ((x + y) * (x + (y + 1.0d0))))
    else
        tmp = t_0 / ((y + (x + 1.0d0)) * (y / y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= -520000.0) {
		tmp = (1.0 / (x + y)) / ((x + y) / y);
	} else if (y <= 1.05e+134) {
		tmp = t_0 * (y / ((x + y) * (x + (y + 1.0))));
	} else {
		tmp = t_0 / ((y + (x + 1.0)) * (y / y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (x + y)
	tmp = 0
	if y <= -520000.0:
		tmp = (1.0 / (x + y)) / ((x + y) / y)
	elif y <= 1.05e+134:
		tmp = t_0 * (y / ((x + y) * (x + (y + 1.0))))
	else:
		tmp = t_0 / ((y + (x + 1.0)) * (y / y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	tmp = 0.0
	if (y <= -520000.0)
		tmp = Float64(Float64(1.0 / Float64(x + y)) / Float64(Float64(x + y) / y));
	elseif (y <= 1.05e+134)
		tmp = Float64(t_0 * Float64(y / Float64(Float64(x + y) * Float64(x + Float64(y + 1.0)))));
	else
		tmp = Float64(t_0 / Float64(Float64(y + Float64(x + 1.0)) * Float64(y / y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	tmp = 0.0;
	if (y <= -520000.0)
		tmp = (1.0 / (x + y)) / ((x + y) / y);
	elseif (y <= 1.05e+134)
		tmp = t_0 * (y / ((x + y) * (x + (y + 1.0))));
	else
		tmp = t_0 / ((y + (x + 1.0)) * (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[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -520000.0], N[(N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.05e+134], N[(t$95$0 * N[(y / N[(N[(x + y), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] * N[(y / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.2e5

   1. Initial program 58.4%

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

      1. +-commutative 58.4%

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

      2. +-commutative 58.4%

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

      3. +-commutative 58.4%

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

      4. *-commutative 58.4%

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

      5. distribute-rgt1-in 23.7%

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

      6. fma-define 58.3%

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

      7. +-commutative 58.3%

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

      8. +-commutative 58.3%

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

      9. cube-unmult 58.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 58.4%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 58.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 58.4%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 23.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 23.7%

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

      4. distribute-rgt1-in 58.4%

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

      5. *-commutative 58.4%

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

      6. associate-*l* 58.4%

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

      7. times-frac 82.0%

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

      8. associate-+r+ 82.0%

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

   6. Applied egg-rr 82.0%

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

      1. clear-num 82.0%

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

      2. associate-/r* 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+l+ 99.6%

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

      5. frac-times 99.7%

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

      6. metadata-eval 99.7%

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

      7. times-frac 99.7%

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

      8. *-un-lft-identity 99.7%

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

      9. *-un-lft-identity 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. div-inv 99.7%

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

      2. *-commutative 99.7%

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

      3. times-frac 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

   10. Applied egg-rr 99.6%

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

   11. Taylor expanded in x around inf 34.6%

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

   if -5.2e5 < y < 1.05e134

   1. Initial program 77.9%

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

      1. +-commutative 77.9%

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

      2. +-commutative 77.9%

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

      3. +-commutative 77.9%

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

      4. *-commutative 77.9%

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

      5. distribute-rgt1-in 65.4%

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

      6. fma-define 77.9%

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

      7. +-commutative 77.9%

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

      8. +-commutative 77.9%

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

      9. cube-unmult 77.9%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 77.9%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 77.9%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-define 65.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      2. cube-mult 65.4%

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

      3. distribute-rgt1-in 77.9%

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

      4. *-commutative 77.9%

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

      5. associate-*l* 77.9%

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

      6. times-frac 98.7%

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

      7. associate-+r+ 98.7%

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

   6. Applied egg-rr 98.7%

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

   if 1.05e134 < y

   1. Initial program 43.3%

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

      1. +-commutative 43.3%

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

      2. +-commutative 43.3%

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

      3. +-commutative 43.3%

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

      4. *-commutative 43.3%

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

      5. distribute-rgt1-in 43.3%

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

      6. fma-define 43.3%

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

      7. +-commutative 43.3%

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

      8. +-commutative 43.3%

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

      9. cube-unmult 43.3%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 43.3%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 43.3%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 43.3%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 43.3%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 43.3%

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

      4. distribute-rgt1-in 43.3%

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

      5. *-commutative 43.3%

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

      6. associate-*l* 43.3%

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

      7. times-frac 80.9%

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

      8. associate-+r+ 80.9%

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

   6. Applied egg-rr 80.9%

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

      1. clear-num 80.9%

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

      2. associate-/r* 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l+ 99.9%

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

      5. frac-times 99.9%

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

      6. metadata-eval 99.9%

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

      7. times-frac 99.9%

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

      8. *-un-lft-identity 99.9%

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

      9. *-un-lft-identity 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

      12. +-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in y around inf 93.1%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -520000:\\ \;\;\;\;\frac{\frac{1}{x + y}}{\frac{x + y}{y}}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+134}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(y + \left(x + 1\right)\right) \cdot \frac{y}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;y \leq 6.8 \cdot 10^{-23}:\\ \;\;\;\;\frac{t\_0}{\left(x + 1\right) \cdot \frac{x + y}{y}}\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{+77}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(y + \left(x + 1\right)\right) \cdot \frac{y}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))))
   (if (<= y 6.8e-23)
     (/ t_0 (* (+ x 1.0) (/ (+ x y) y)))
     (if (<= y 8.8e+77)
       (* x (/ y (* (+ x (+ y 1.0)) (* (+ x y) (+ x y)))))
       (/ t_0 (* (+ y (+ x 1.0)) (/ y y)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= 6.8e-23) {
		tmp = t_0 / ((x + 1.0) * ((x + y) / y));
	} else if (y <= 8.8e+77) {
		tmp = x * (y / ((x + (y + 1.0)) * ((x + y) * (x + y))));
	} else {
		tmp = t_0 / ((y + (x + 1.0)) * (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 = x / (x + y)
    if (y <= 6.8d-23) then
        tmp = t_0 / ((x + 1.0d0) * ((x + y) / y))
    else if (y <= 8.8d+77) then
        tmp = x * (y / ((x + (y + 1.0d0)) * ((x + y) * (x + y))))
    else
        tmp = t_0 / ((y + (x + 1.0d0)) * (y / y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= 6.8e-23) {
		tmp = t_0 / ((x + 1.0) * ((x + y) / y));
	} else if (y <= 8.8e+77) {
		tmp = x * (y / ((x + (y + 1.0)) * ((x + y) * (x + y))));
	} else {
		tmp = t_0 / ((y + (x + 1.0)) * (y / y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (x + y)
	tmp = 0
	if y <= 6.8e-23:
		tmp = t_0 / ((x + 1.0) * ((x + y) / y))
	elif y <= 8.8e+77:
		tmp = x * (y / ((x + (y + 1.0)) * ((x + y) * (x + y))))
	else:
		tmp = t_0 / ((y + (x + 1.0)) * (y / y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	tmp = 0.0
	if (y <= 6.8e-23)
		tmp = Float64(t_0 / Float64(Float64(x + 1.0) * Float64(Float64(x + y) / y)));
	elseif (y <= 8.8e+77)
		tmp = Float64(x * Float64(y / Float64(Float64(x + Float64(y + 1.0)) * Float64(Float64(x + y) * Float64(x + y)))));
	else
		tmp = Float64(t_0 / Float64(Float64(y + Float64(x + 1.0)) * Float64(y / y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	tmp = 0.0;
	if (y <= 6.8e-23)
		tmp = t_0 / ((x + 1.0) * ((x + y) / y));
	elseif (y <= 8.8e+77)
		tmp = x * (y / ((x + (y + 1.0)) * ((x + y) * (x + y))));
	else
		tmp = t_0 / ((y + (x + 1.0)) * (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[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 6.8e-23], N[(t$95$0 / N[(N[(x + 1.0), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.8e+77], N[(x * N[(y / N[(N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] * N[(y / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.8000000000000001e-23

   1. Initial program 71.5%

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

      1. +-commutative 71.5%

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

      2. +-commutative 71.5%

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

      3. +-commutative 71.5%

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

      4. *-commutative 71.5%

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

      5. distribute-rgt1-in 51.1%

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

      6. fma-define 71.5%

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

      7. +-commutative 71.5%

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

      8. +-commutative 71.5%

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

      9. cube-unmult 71.5%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 71.5%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 71.5%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 71.5%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 51.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 51.1%

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

      4. distribute-rgt1-in 71.5%

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

      5. *-commutative 71.5%

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

      6. associate-*l* 71.5%

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

      7. times-frac 94.3%

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

      8. associate-+r+ 94.3%

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

   6. Applied egg-rr 94.3%

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

      1. clear-num 94.3%

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

      2. associate-/r* 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+l+ 99.8%

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

      5. frac-times 99.7%

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

      6. metadata-eval 99.7%

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

      7. times-frac 99.7%

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

      8. *-un-lft-identity 99.7%

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

      9. *-un-lft-identity 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in y around 0 84.8%

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

       1. +-commutative 84.8%

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

   11. Simplified 84.8%

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

   if 6.8000000000000001e-23 < y < 8.8000000000000002e77

   1. Initial program 88.3%

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

      1. associate-/l* 92.2%

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

      2. associate-+l+ 92.2%

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

   3. Simplified 92.2%

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

   if 8.8000000000000002e77 < y

   1. Initial program 47.1%

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

      1. +-commutative 47.1%

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

      2. +-commutative 47.1%

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

      3. +-commutative 47.1%

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

      4. *-commutative 47.1%

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

      5. distribute-rgt1-in 47.1%

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

      6. fma-define 47.1%

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

      7. +-commutative 47.1%

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

      8. +-commutative 47.1%

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

      9. cube-unmult 47.1%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 47.1%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 47.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 47.1%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 47.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 47.1%

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

      4. distribute-rgt1-in 47.1%

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

      5. *-commutative 47.1%

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

      6. associate-*l* 47.1%

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

      7. times-frac 83.5%

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

      8. associate-+r+ 83.5%

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

   6. Applied egg-rr 83.5%

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

      1. clear-num 83.5%

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

      2. associate-/r* 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l+ 99.9%

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

      5. frac-times 99.9%

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

      6. metadata-eval 99.9%

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

      7. times-frac 99.9%

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

      8. *-un-lft-identity 99.9%

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

      9. *-un-lft-identity 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

      12. +-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in y around inf 91.2%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(x + 1\right) \cdot \frac{x + y}{y}}\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{+77}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(y + \left(x + 1\right)\right) \cdot \frac{y}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2100:\\ \;\;\;\;\frac{\frac{1}{x + y}}{\frac{x + y}{y}}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{x}{\left(x + 1\right) \cdot \left(x + y\right)}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -2100.0)
   (/ (/ 1.0 (+ x y)) (/ (+ x y) y))
   (if (<= y 8.2e+19)
     (* (/ y (+ x y)) (/ x (* (+ x 1.0) (+ x y))))
     (if (<= y 3.4e+149)
       (/ x (* (+ x y) (+ x (+ y 1.0))))
       (/ (/ x (+ x y)) (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -2100.0) {
		tmp = (1.0 / (x + y)) / ((x + y) / y);
	} else if (y <= 8.2e+19) {
		tmp = (y / (x + y)) * (x / ((x + 1.0) * (x + y)));
	} else if (y <= 3.4e+149) {
		tmp = x / ((x + y) * (x + (y + 1.0)));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2100.0d0)) then
        tmp = (1.0d0 / (x + y)) / ((x + y) / y)
    else if (y <= 8.2d+19) then
        tmp = (y / (x + y)) * (x / ((x + 1.0d0) * (x + y)))
    else if (y <= 3.4d+149) then
        tmp = x / ((x + y) * (x + (y + 1.0d0)))
    else
        tmp = (x / (x + y)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= -2100.0) {
		tmp = (1.0 / (x + y)) / ((x + y) / y);
	} else if (y <= 8.2e+19) {
		tmp = (y / (x + y)) * (x / ((x + 1.0) * (x + y)));
	} else if (y <= 3.4e+149) {
		tmp = x / ((x + y) * (x + (y + 1.0)));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -2100.0:
		tmp = (1.0 / (x + y)) / ((x + y) / y)
	elif y <= 8.2e+19:
		tmp = (y / (x + y)) * (x / ((x + 1.0) * (x + y)))
	elif y <= 3.4e+149:
		tmp = x / ((x + y) * (x + (y + 1.0)))
	else:
		tmp = (x / (x + y)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -2100.0)
		tmp = Float64(Float64(1.0 / Float64(x + y)) / Float64(Float64(x + y) / y));
	elseif (y <= 8.2e+19)
		tmp = Float64(Float64(y / Float64(x + y)) * Float64(x / Float64(Float64(x + 1.0) * Float64(x + y))));
	elseif (y <= 3.4e+149)
		tmp = Float64(x / Float64(Float64(x + y) * Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2100.0)
		tmp = (1.0 / (x + y)) / ((x + y) / y);
	elseif (y <= 8.2e+19)
		tmp = (y / (x + y)) * (x / ((x + 1.0) * (x + y)));
	elseif (y <= 3.4e+149)
		tmp = x / ((x + y) * (x + (y + 1.0)));
	else
		tmp = (x / (x + y)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -2100.0], N[(N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.2e+19], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(x + 1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.4e+149], N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2100

   1. Initial program 58.4%

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

      1. +-commutative 58.4%

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

      2. +-commutative 58.4%

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

      3. +-commutative 58.4%

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

      4. *-commutative 58.4%

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

      5. distribute-rgt1-in 23.7%

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

      6. fma-define 58.3%

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

      7. +-commutative 58.3%

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

      8. +-commutative 58.3%

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

      9. cube-unmult 58.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 58.4%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 58.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 58.4%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 23.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 23.7%

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

      4. distribute-rgt1-in 58.4%

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

      5. *-commutative 58.4%

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

      6. associate-*l* 58.4%

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

      7. times-frac 82.0%

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

      8. associate-+r+ 82.0%

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

   6. Applied egg-rr 82.0%

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

      1. clear-num 82.0%

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

      2. associate-/r* 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+l+ 99.6%

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

      5. frac-times 99.7%

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

      6. metadata-eval 99.7%

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

      7. times-frac 99.7%

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

      8. *-un-lft-identity 99.7%

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

      9. *-un-lft-identity 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. div-inv 99.7%

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

      2. *-commutative 99.7%

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

      3. times-frac 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

   10. Applied egg-rr 99.6%

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

   11. Taylor expanded in x around inf 34.6%

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

   if -2100 < y < 8.2e19

   1. Initial program 78.8%

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

   3. Taylor expanded in x around inf 77.6%

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

      1. *-commutative 77.6%

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

      2. associate-*l* 77.6%

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

      3. times-frac 98.8%

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

      4. +-commutative 98.8%

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

      5. +-commutative 98.8%

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

   5. Applied egg-rr 98.8%

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

   if 8.2e19 < y < 3.3999999999999998e149

   1. Initial program 69.0%

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

      1. +-commutative 69.0%

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

      2. +-commutative 69.0%

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

      3. +-commutative 69.0%

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

      4. *-commutative 69.0%

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

      5. distribute-rgt1-in 60.2%

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

      6. fma-define 69.0%

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

      7. +-commutative 69.0%

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

      8. +-commutative 69.0%

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

      9. cube-unmult 69.0%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 69.0%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 69.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 69.0%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 60.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 60.2%

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

      4. distribute-rgt1-in 69.0%

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

      5. *-commutative 69.0%

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

      6. associate-*l* 69.0%

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

      7. times-frac 94.3%

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

      8. associate-+r+ 94.3%

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

   6. Applied egg-rr 94.3%

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

   7. Taylor expanded in y around inf 82.9%

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

   if 3.3999999999999998e149 < y

   1. Initial program 44.7%

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

      1. +-commutative 44.7%

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

      2. +-commutative 44.7%

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

      3. +-commutative 44.7%

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

      4. *-commutative 44.7%

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

      5. distribute-rgt1-in 44.7%

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

      6. fma-define 44.7%

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

      7. +-commutative 44.7%

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

      8. +-commutative 44.7%

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

      9. cube-unmult 44.7%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 44.7%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 44.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 44.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 44.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 44.7%

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

      4. distribute-rgt1-in 44.7%

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

      5. *-commutative 44.7%

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

      6. associate-*l* 44.7%

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

      7. times-frac 78.9%

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

      8. associate-+r+ 78.9%

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

   6. Applied egg-rr 78.9%

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

      1. clear-num 78.9%

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

      2. associate-/r* 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l+ 99.9%

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

      5. frac-times 99.9%

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

      6. metadata-eval 99.9%

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

      7. times-frac 99.9%

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

      8. *-un-lft-identity 99.9%

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

      9. *-un-lft-identity 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

      12. +-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around 0 92.1%

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

       1. +-commutative 92.1%

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

   11. Simplified 92.1%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2100:\\ \;\;\;\;\frac{\frac{1}{x + y}}{\frac{x + y}{y}}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{x}{\left(x + 1\right) \cdot \left(x + y\right)}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;y \leq -41:\\ \;\;\;\;\frac{\frac{1}{x + y}}{\frac{x + y}{y}}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+19}:\\ \;\;\;\;t\_0 \cdot \frac{y}{\left(x + 1\right) \cdot \left(x + y\right)}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+148}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))))
   (if (<= y -41.0)
     (/ (/ 1.0 (+ x y)) (/ (+ x y) y))
     (if (<= y 8e+19)
       (* t_0 (/ y (* (+ x 1.0) (+ x y))))
       (if (<= y 6e+148)
         (/ x (* (+ x y) (+ x (+ y 1.0))))
         (/ t_0 (+ y 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= -41.0) {
		tmp = (1.0 / (x + y)) / ((x + y) / y);
	} else if (y <= 8e+19) {
		tmp = t_0 * (y / ((x + 1.0) * (x + y)));
	} else if (y <= 6e+148) {
		tmp = x / ((x + y) * (x + (y + 1.0)));
	} else {
		tmp = t_0 / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (x + y)
    if (y <= (-41.0d0)) then
        tmp = (1.0d0 / (x + y)) / ((x + y) / y)
    else if (y <= 8d+19) then
        tmp = t_0 * (y / ((x + 1.0d0) * (x + y)))
    else if (y <= 6d+148) then
        tmp = x / ((x + y) * (x + (y + 1.0d0)))
    else
        tmp = t_0 / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= -41.0) {
		tmp = (1.0 / (x + y)) / ((x + y) / y);
	} else if (y <= 8e+19) {
		tmp = t_0 * (y / ((x + 1.0) * (x + y)));
	} else if (y <= 6e+148) {
		tmp = x / ((x + y) * (x + (y + 1.0)));
	} else {
		tmp = t_0 / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (x + y)
	tmp = 0
	if y <= -41.0:
		tmp = (1.0 / (x + y)) / ((x + y) / y)
	elif y <= 8e+19:
		tmp = t_0 * (y / ((x + 1.0) * (x + y)))
	elif y <= 6e+148:
		tmp = x / ((x + y) * (x + (y + 1.0)))
	else:
		tmp = t_0 / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	tmp = 0.0
	if (y <= -41.0)
		tmp = Float64(Float64(1.0 / Float64(x + y)) / Float64(Float64(x + y) / y));
	elseif (y <= 8e+19)
		tmp = Float64(t_0 * Float64(y / Float64(Float64(x + 1.0) * Float64(x + y))));
	elseif (y <= 6e+148)
		tmp = Float64(x / Float64(Float64(x + y) * Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(t_0 / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	tmp = 0.0;
	if (y <= -41.0)
		tmp = (1.0 / (x + y)) / ((x + y) / y);
	elseif (y <= 8e+19)
		tmp = t_0 * (y / ((x + 1.0) * (x + y)));
	elseif (y <= 6e+148)
		tmp = x / ((x + y) * (x + (y + 1.0)));
	else
		tmp = t_0 / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -41.0], N[(N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e+19], N[(t$95$0 * N[(y / N[(N[(x + 1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e+148], N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -41

   1. Initial program 58.4%

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

      1. +-commutative 58.4%

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

      2. +-commutative 58.4%

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

      3. +-commutative 58.4%

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

      4. *-commutative 58.4%

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

      5. distribute-rgt1-in 23.7%

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

      6. fma-define 58.3%

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

      7. +-commutative 58.3%

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

      8. +-commutative 58.3%

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

      9. cube-unmult 58.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 58.4%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 58.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 58.4%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 23.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 23.7%

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

      4. distribute-rgt1-in 58.4%

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

      5. *-commutative 58.4%

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

      6. associate-*l* 58.4%

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

      7. times-frac 82.0%

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

      8. associate-+r+ 82.0%

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

   6. Applied egg-rr 82.0%

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

      1. clear-num 82.0%

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

      2. associate-/r* 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+l+ 99.6%

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

      5. frac-times 99.7%

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

      6. metadata-eval 99.7%

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

      7. times-frac 99.7%

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

      8. *-un-lft-identity 99.7%

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

      9. *-un-lft-identity 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. div-inv 99.7%

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

      2. *-commutative 99.7%

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

      3. times-frac 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

   10. Applied egg-rr 99.6%

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

   11. Taylor expanded in x around inf 34.6%

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

   if -41 < y < 8e19

   1. Initial program 78.8%

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

   3. Taylor expanded in x around inf 77.6%

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

      1. associate-*l* 77.6%

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

      2. times-frac 98.8%

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

      3. +-commutative 98.8%

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

      4. +-commutative 98.8%

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

   5. Applied egg-rr 98.8%

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

   if 8e19 < y < 6.00000000000000029e148

   1. Initial program 69.0%

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

      1. +-commutative 69.0%

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

      2. +-commutative 69.0%

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

      3. +-commutative 69.0%

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

      4. *-commutative 69.0%

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

      5. distribute-rgt1-in 60.2%

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

      6. fma-define 69.0%

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

      7. +-commutative 69.0%

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

      8. +-commutative 69.0%

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

      9. cube-unmult 69.0%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 69.0%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 69.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 69.0%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 60.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 60.2%

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

      4. distribute-rgt1-in 69.0%

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

      5. *-commutative 69.0%

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

      6. associate-*l* 69.0%

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

      7. times-frac 94.3%

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

      8. associate-+r+ 94.3%

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

   6. Applied egg-rr 94.3%

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

   7. Taylor expanded in y around inf 82.9%

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

   if 6.00000000000000029e148 < y

   1. Initial program 44.7%

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

      1. +-commutative 44.7%

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

      2. +-commutative 44.7%

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

      3. +-commutative 44.7%

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

      4. *-commutative 44.7%

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

      5. distribute-rgt1-in 44.7%

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

      6. fma-define 44.7%

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

      7. +-commutative 44.7%

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

      8. +-commutative 44.7%

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

      9. cube-unmult 44.7%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 44.7%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 44.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 44.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 44.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 44.7%

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

      4. distribute-rgt1-in 44.7%

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

      5. *-commutative 44.7%

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

      6. associate-*l* 44.7%

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

      7. times-frac 78.9%

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

      8. associate-+r+ 78.9%

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

   6. Applied egg-rr 78.9%

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

      1. clear-num 78.9%

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

      2. associate-/r* 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l+ 99.9%

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

      5. frac-times 99.9%

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

      6. metadata-eval 99.9%

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

      7. times-frac 99.9%

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

      8. *-un-lft-identity 99.9%

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

      9. *-un-lft-identity 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

      12. +-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around 0 92.1%

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

       1. +-commutative 92.1%

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

   11. Simplified 92.1%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -41:\\ \;\;\;\;\frac{\frac{1}{x + y}}{\frac{x + y}{y}}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+19}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{\left(x + 1\right) \cdot \left(x + y\right)}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+148}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-148}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4e-148)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 8.5e+19)
     (* x (/ y (* (+ x 1.0) (* (+ x y) (+ x y)))))
     (if (<= y 4.2e+149)
       (/ x (* (+ x y) (+ x (+ y 1.0))))
       (/ (/ x (+ x y)) (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4e-148) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 8.5e+19) {
		tmp = x * (y / ((x + 1.0) * ((x + y) * (x + y))));
	} else if (y <= 4.2e+149) {
		tmp = x / ((x + y) * (x + (y + 1.0)));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4d-148) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 8.5d+19) then
        tmp = x * (y / ((x + 1.0d0) * ((x + y) * (x + y))))
    else if (y <= 4.2d+149) then
        tmp = x / ((x + y) * (x + (y + 1.0d0)))
    else
        tmp = (x / (x + y)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 4e-148) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 8.5e+19) {
		tmp = x * (y / ((x + 1.0) * ((x + y) * (x + y))));
	} else if (y <= 4.2e+149) {
		tmp = x / ((x + y) * (x + (y + 1.0)));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 4e-148:
		tmp = (y / x) / (x + 1.0)
	elif y <= 8.5e+19:
		tmp = x * (y / ((x + 1.0) * ((x + y) * (x + y))))
	elif y <= 4.2e+149:
		tmp = x / ((x + y) * (x + (y + 1.0)))
	else:
		tmp = (x / (x + y)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4e-148)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 8.5e+19)
		tmp = Float64(x * Float64(y / Float64(Float64(x + 1.0) * Float64(Float64(x + y) * Float64(x + y)))));
	elseif (y <= 4.2e+149)
		tmp = Float64(x / Float64(Float64(x + y) * Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4e-148)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 8.5e+19)
		tmp = x * (y / ((x + 1.0) * ((x + y) * (x + y))));
	elseif (y <= 4.2e+149)
		tmp = x / ((x + y) * (x + (y + 1.0)));
	else
		tmp = (x / (x + y)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 4e-148], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.5e+19], N[(x * N[(y / N[(N[(x + 1.0), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.2e+149], N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 3.99999999999999974e-148

   1. Initial program 67.1%

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

      1. +-commutative 67.1%

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

      2. +-commutative 67.1%

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

      3. +-commutative 67.1%

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

      4. *-commutative 67.1%

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

      5. distribute-rgt1-in 45.1%

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

      6. fma-define 67.1%

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

      7. +-commutative 67.1%

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

      8. +-commutative 67.1%

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

      9. cube-unmult 67.1%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 67.1%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 67.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 67.1%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 45.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 45.1%

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

      4. distribute-rgt1-in 67.1%

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

      5. *-commutative 67.1%

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

      6. associate-*l* 67.1%

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

      7. times-frac 92.9%

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

      8. associate-+r+ 92.9%

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

   6. Applied egg-rr 92.9%

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

      1. clear-num 92.8%

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

      2. associate-/r* 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+l+ 99.8%

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

      5. frac-times 99.8%

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

      6. metadata-eval 99.8%

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

      7. times-frac 99.8%

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

      8. *-un-lft-identity 99.8%

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

      9. *-un-lft-identity 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in y around 0 59.2%

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

       1. associate-/r* 61.9%

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

       2. +-commutative 61.9%

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

   11. Simplified 61.9%

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

   if 3.99999999999999974e-148 < y < 8.5e19

   1. Initial program 90.6%

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

      1. associate-/l* 93.0%

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

      2. associate-+l+ 93.0%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in y around 0 89.7%

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

      1. +-commutative 89.7%

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

   7. Simplified 89.7%

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

   if 8.5e19 < y < 4.2000000000000003e149

   1. Initial program 69.0%

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

      1. +-commutative 69.0%

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

      2. +-commutative 69.0%

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

      3. +-commutative 69.0%

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

      4. *-commutative 69.0%

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

      5. distribute-rgt1-in 60.2%

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

      6. fma-define 69.0%

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

      7. +-commutative 69.0%

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

      8. +-commutative 69.0%

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

      9. cube-unmult 69.0%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 69.0%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 69.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 69.0%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 60.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 60.2%

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

      4. distribute-rgt1-in 69.0%

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

      5. *-commutative 69.0%

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

      6. associate-*l* 69.0%

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

      7. times-frac 94.3%

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

      8. associate-+r+ 94.3%

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

   6. Applied egg-rr 94.3%

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

   7. Taylor expanded in y around inf 82.9%

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

   if 4.2000000000000003e149 < y

   1. Initial program 44.7%

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

      1. +-commutative 44.7%

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

      2. +-commutative 44.7%

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

      3. +-commutative 44.7%

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

      4. *-commutative 44.7%

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

      5. distribute-rgt1-in 44.7%

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

      6. fma-define 44.7%

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

      7. +-commutative 44.7%

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

      8. +-commutative 44.7%

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

      9. cube-unmult 44.7%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 44.7%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 44.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 44.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 44.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 44.7%

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

      4. distribute-rgt1-in 44.7%

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

      5. *-commutative 44.7%

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

      6. associate-*l* 44.7%

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

      7. times-frac 78.9%

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

      8. associate-+r+ 78.9%

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

   6. Applied egg-rr 78.9%

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

      1. clear-num 78.9%

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

      2. associate-/r* 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l+ 99.9%

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

      5. frac-times 99.9%

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

      6. metadata-eval 99.9%

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

      7. times-frac 99.9%

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

      8. *-un-lft-identity 99.9%

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

      9. *-un-lft-identity 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

      12. +-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around 0 92.1%

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

       1. +-commutative 92.1%

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

   11. Simplified 92.1%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-148}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;y \leq 8 \cdot 10^{+20}:\\ \;\;\;\;\frac{t\_0}{\left(x + 1\right) \cdot \frac{x + y}{y}}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(y + \left(x + 1\right)\right) \cdot \frac{y}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))))
   (if (<= y 8e+20)
     (/ t_0 (* (+ x 1.0) (/ (+ x y) y)))
     (if (<= y 4.2e+149)
       (/ x (* (+ x y) (+ x (+ y 1.0))))
       (/ t_0 (* (+ y (+ x 1.0)) (/ y y)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= 8e+20) {
		tmp = t_0 / ((x + 1.0) * ((x + y) / y));
	} else if (y <= 4.2e+149) {
		tmp = x / ((x + y) * (x + (y + 1.0)));
	} else {
		tmp = t_0 / ((y + (x + 1.0)) * (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 = x / (x + y)
    if (y <= 8d+20) then
        tmp = t_0 / ((x + 1.0d0) * ((x + y) / y))
    else if (y <= 4.2d+149) then
        tmp = x / ((x + y) * (x + (y + 1.0d0)))
    else
        tmp = t_0 / ((y + (x + 1.0d0)) * (y / y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= 8e+20) {
		tmp = t_0 / ((x + 1.0) * ((x + y) / y));
	} else if (y <= 4.2e+149) {
		tmp = x / ((x + y) * (x + (y + 1.0)));
	} else {
		tmp = t_0 / ((y + (x + 1.0)) * (y / y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (x + y)
	tmp = 0
	if y <= 8e+20:
		tmp = t_0 / ((x + 1.0) * ((x + y) / y))
	elif y <= 4.2e+149:
		tmp = x / ((x + y) * (x + (y + 1.0)))
	else:
		tmp = t_0 / ((y + (x + 1.0)) * (y / y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	tmp = 0.0
	if (y <= 8e+20)
		tmp = Float64(t_0 / Float64(Float64(x + 1.0) * Float64(Float64(x + y) / y)));
	elseif (y <= 4.2e+149)
		tmp = Float64(x / Float64(Float64(x + y) * Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(t_0 / Float64(Float64(y + Float64(x + 1.0)) * Float64(y / y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	tmp = 0.0;
	if (y <= 8e+20)
		tmp = t_0 / ((x + 1.0) * ((x + y) / y));
	elseif (y <= 4.2e+149)
		tmp = x / ((x + y) * (x + (y + 1.0)));
	else
		tmp = t_0 / ((y + (x + 1.0)) * (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[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 8e+20], N[(t$95$0 / N[(N[(x + 1.0), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.2e+149], N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] * N[(y / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 8e20

   1. Initial program 72.7%

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

      1. +-commutative 72.7%

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

      2. +-commutative 72.7%

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

      3. +-commutative 72.7%

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

      4. *-commutative 72.7%

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

      5. distribute-rgt1-in 53.2%

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

      6. fma-define 72.7%

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

      7. +-commutative 72.7%

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

      8. +-commutative 72.7%

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

      9. cube-unmult 72.7%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 72.7%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 72.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 72.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 53.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 53.2%

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

      4. distribute-rgt1-in 72.7%

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

      5. *-commutative 72.7%

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

      6. associate-*l* 72.7%

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

      7. times-frac 94.6%

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

      8. associate-+r+ 94.6%

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

   6. Applied egg-rr 94.6%

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

      1. clear-num 94.5%

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

      2. associate-/r* 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+l+ 99.7%

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

      5. frac-times 99.7%

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

      6. metadata-eval 99.7%

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

      7. times-frac 99.7%

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

      8. *-un-lft-identity 99.7%

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

      9. *-un-lft-identity 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in y around 0 84.7%

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

       1. +-commutative 84.7%

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

   11. Simplified 84.7%

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

   if 8e20 < y < 4.2000000000000003e149

   1. Initial program 69.0%

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

      1. +-commutative 69.0%

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

      2. +-commutative 69.0%

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

      3. +-commutative 69.0%

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

      4. *-commutative 69.0%

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

      5. distribute-rgt1-in 60.2%

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

      6. fma-define 69.0%

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

      7. +-commutative 69.0%

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

      8. +-commutative 69.0%

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

      9. cube-unmult 69.0%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 69.0%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 69.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 69.0%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 60.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 60.2%

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

      4. distribute-rgt1-in 69.0%

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

      5. *-commutative 69.0%

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

      6. associate-*l* 69.0%

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

      7. times-frac 94.3%

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

      8. associate-+r+ 94.3%

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

   6. Applied egg-rr 94.3%

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

   7. Taylor expanded in y around inf 82.9%

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

   if 4.2000000000000003e149 < y

   1. Initial program 44.7%

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

      1. +-commutative 44.7%

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

      2. +-commutative 44.7%

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

      3. +-commutative 44.7%

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

      4. *-commutative 44.7%

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

      5. distribute-rgt1-in 44.7%

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

      6. fma-define 44.7%

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

      7. +-commutative 44.7%

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

      8. +-commutative 44.7%

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

      9. cube-unmult 44.7%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 44.7%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 44.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 44.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 44.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 44.7%

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

      4. distribute-rgt1-in 44.7%

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

      5. *-commutative 44.7%

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

      6. associate-*l* 44.7%

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

      7. times-frac 78.9%

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

      8. associate-+r+ 78.9%

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

   6. Applied egg-rr 78.9%

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

      1. clear-num 78.9%

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

      2. associate-/r* 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l+ 99.9%

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

      5. frac-times 99.9%

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

      6. metadata-eval 99.9%

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

      7. times-frac 99.9%

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

      8. *-un-lft-identity 99.9%

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

      9. *-un-lft-identity 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

      12. +-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in y around inf 92.3%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(x + 1\right) \cdot \frac{x + y}{y}}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(y + \left(x + 1\right)\right) \cdot \frac{y}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+169}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -26000000000:\\ \;\;\;\;\frac{y}{x \cdot \left(x + y\right)}\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -8e+169)
   (* (/ y x) (/ 1.0 x))
   (if (<= x -26000000000.0)
     (/ y (* x (+ x y)))
     (if (<= x -2.5e-54) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -8e+169) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -26000000000.0) {
		tmp = y / (x * (x + y));
	} else if (x <= -2.5e-54) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-8d+169)) then
        tmp = (y / x) * (1.0d0 / x)
    else if (x <= (-26000000000.0d0)) then
        tmp = y / (x * (x + y))
    else if (x <= (-2.5d-54)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -8e+169) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -26000000000.0) {
		tmp = y / (x * (x + y));
	} else if (x <= -2.5e-54) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -8e+169:
		tmp = (y / x) * (1.0 / x)
	elif x <= -26000000000.0:
		tmp = y / (x * (x + y))
	elif x <= -2.5e-54:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -8e+169)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	elseif (x <= -26000000000.0)
		tmp = Float64(y / Float64(x * Float64(x + y)));
	elseif (x <= -2.5e-54)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8e+169)
		tmp = (y / x) * (1.0 / x);
	elseif (x <= -26000000000.0)
		tmp = y / (x * (x + y));
	elseif (x <= -2.5e-54)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -8e+169], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -26000000000.0], N[(y / N[(x * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.5e-54], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.99999999999999947e169

   1. Initial program 50.2%

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

      1. +-commutative 50.2%

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

      2. +-commutative 50.2%

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

      3. +-commutative 50.2%

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

      4. *-commutative 50.2%

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

      5. distribute-rgt1-in 0.0%

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

      6. fma-define 50.2%

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

      7. +-commutative 50.2%

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

      8. +-commutative 50.2%

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

      9. cube-unmult 50.2%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 50.2%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 50.2%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 50.2%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 0.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 0.0%

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

      4. distribute-rgt1-in 50.2%

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

      5. *-commutative 50.2%

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

      6. associate-*l* 50.2%

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

      7. times-frac 79.1%

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

      8. associate-+r+ 79.1%

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

   6. Applied egg-rr 79.1%

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

   7. Taylor expanded in x around inf 81.7%

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

   8. Taylor expanded in y around 0 81.4%

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

   if -7.99999999999999947e169 < x < -2.6e10

   1. Initial program 68.0%

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

      1. +-commutative 68.0%

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

      2. +-commutative 68.0%

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

      3. +-commutative 68.0%

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

      4. *-commutative 68.0%

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

      5. distribute-rgt1-in 51.9%

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

      6. fma-define 68.0%

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

      7. +-commutative 68.0%

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

      8. +-commutative 68.0%

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

      9. cube-unmult 68.0%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 68.0%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 68.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 68.0%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 51.9%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 51.9%

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

      4. distribute-rgt1-in 68.0%

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

      5. *-commutative 68.0%

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

      6. associate-*l* 68.0%

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

      7. times-frac 91.1%

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

      8. associate-+r+ 91.1%

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

   6. Applied egg-rr 91.1%

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

   7. Taylor expanded in x around inf 57.8%

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

      1. frac-times 73.3%

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

      2. *-rgt-identity 73.3%

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

   9. Applied egg-rr 73.3%

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

   if -2.6e10 < x < -2.50000000000000008e-54

   1. Initial program 99.4%

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

      1. associate-/l* 99.6%

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

      2. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 39.3%

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

      1. +-commutative 39.3%

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

   7. Simplified 39.3%

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

   if -2.50000000000000008e-54 < x

   1. Initial program 68.4%

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

      1. associate-/l* 80.0%

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

      2. associate-+l+ 80.0%

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

   3. Simplified 80.0%

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

   5. Taylor expanded in x around 0 53.2%

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

      1. associate-/r* 55.5%

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

      2. +-commutative 55.5%

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

   7. Simplified 55.5%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+169}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -26000000000:\\ \;\;\;\;\frac{y}{x \cdot \left(x + y\right)}\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+171}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -26000000000:\\ \;\;\;\;\frac{y}{x \cdot \left(x + y\right)}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-55}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.75e+171)
   (* (/ y x) (/ 1.0 x))
   (if (<= x -26000000000.0)
     (/ y (* x (+ x y)))
     (if (<= x -8.2e-55) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.75e+171) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -26000000000.0) {
		tmp = y / (x * (x + y));
	} else if (x <= -8.2e-55) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.75d+171)) then
        tmp = (y / x) * (1.0d0 / x)
    else if (x <= (-26000000000.0d0)) then
        tmp = y / (x * (x + y))
    else if (x <= (-8.2d-55)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.75e+171) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -26000000000.0) {
		tmp = y / (x * (x + y));
	} else if (x <= -8.2e-55) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.75e+171:
		tmp = (y / x) * (1.0 / x)
	elif x <= -26000000000.0:
		tmp = y / (x * (x + y))
	elif x <= -8.2e-55:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.75e+171)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	elseif (x <= -26000000000.0)
		tmp = Float64(y / Float64(x * Float64(x + y)));
	elseif (x <= -8.2e-55)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.75e+171)
		tmp = (y / x) * (1.0 / x);
	elseif (x <= -26000000000.0)
		tmp = y / (x * (x + y));
	elseif (x <= -8.2e-55)
		tmp = y / (x * (x + 1.0));
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.75e+171], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -26000000000.0], N[(y / N[(x * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8.2e-55], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.75e171

   1. Initial program 50.2%

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

      1. +-commutative 50.2%

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

      2. +-commutative 50.2%

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

      3. +-commutative 50.2%

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

      4. *-commutative 50.2%

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

      5. distribute-rgt1-in 0.0%

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

      6. fma-define 50.2%

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

      7. +-commutative 50.2%

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

      8. +-commutative 50.2%

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

      9. cube-unmult 50.2%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 50.2%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 50.2%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 50.2%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 0.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 0.0%

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

      4. distribute-rgt1-in 50.2%

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

      5. *-commutative 50.2%

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

      6. associate-*l* 50.2%

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

      7. times-frac 79.1%

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

      8. associate-+r+ 79.1%

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

   6. Applied egg-rr 79.1%

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

   7. Taylor expanded in x around inf 81.7%

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

   8. Taylor expanded in y around 0 81.4%

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

   if -1.75e171 < x < -2.6e10

   1. Initial program 68.0%

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

      1. +-commutative 68.0%

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

      2. +-commutative 68.0%

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

      3. +-commutative 68.0%

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

      4. *-commutative 68.0%

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

      5. distribute-rgt1-in 51.9%

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

      6. fma-define 68.0%

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

      7. +-commutative 68.0%

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

      8. +-commutative 68.0%

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

      9. cube-unmult 68.0%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 68.0%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 68.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 68.0%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 51.9%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 51.9%

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

      4. distribute-rgt1-in 68.0%

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

      5. *-commutative 68.0%

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

      6. associate-*l* 68.0%

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

      7. times-frac 91.1%

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

      8. associate-+r+ 91.1%

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

   6. Applied egg-rr 91.1%

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

   7. Taylor expanded in x around inf 57.8%

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

      1. frac-times 73.3%

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

      2. *-rgt-identity 73.3%

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

   9. Applied egg-rr 73.3%

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

   if -2.6e10 < x < -8.1999999999999996e-55

   1. Initial program 99.4%

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

      1. associate-/l* 99.6%

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

      2. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 39.3%

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

      1. +-commutative 39.3%

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

   7. Simplified 39.3%

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

   if -8.1999999999999996e-55 < x

   1. Initial program 68.4%

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

      1. associate-/l* 80.0%

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

      2. associate-+l+ 80.0%

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

   3. Simplified 80.0%

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

   5. Taylor expanded in x around 0 53.2%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+171}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -26000000000:\\ \;\;\;\;\frac{y}{x \cdot \left(x + y\right)}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-55}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ t_1 := \frac{x + y}{y}\\ \mathbf{if}\;y \leq 0.215:\\ \;\;\;\;\frac{t\_0}{\left(x + 1\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{\frac{1}{x + y}}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))) (t_1 (/ (+ x y) y)))
   (if (<= y 0.215)
     (/ t_0 (* (+ x 1.0) t_1))
     (* t_0 (/ (/ 1.0 (+ x y)) t_1)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = (x + y) / y;
	double tmp;
	if (y <= 0.215) {
		tmp = t_0 / ((x + 1.0) * t_1);
	} else {
		tmp = t_0 * ((1.0 / (x + y)) / t_1);
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = x / (x + y)
    t_1 = (x + y) / y
    if (y <= 0.215d0) then
        tmp = t_0 / ((x + 1.0d0) * t_1)
    else
        tmp = t_0 * ((1.0d0 / (x + y)) / t_1)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = (x + y) / y;
	double tmp;
	if (y <= 0.215) {
		tmp = t_0 / ((x + 1.0) * t_1);
	} else {
		tmp = t_0 * ((1.0 / (x + y)) / t_1);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (x + y)
	t_1 = (x + y) / y
	tmp = 0
	if y <= 0.215:
		tmp = t_0 / ((x + 1.0) * t_1)
	else:
		tmp = t_0 * ((1.0 / (x + y)) / t_1)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	t_1 = Float64(Float64(x + y) / y)
	tmp = 0.0
	if (y <= 0.215)
		tmp = Float64(t_0 / Float64(Float64(x + 1.0) * t_1));
	else
		tmp = Float64(t_0 * Float64(Float64(1.0 / Float64(x + y)) / t_1));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	t_1 = (x + y) / y;
	tmp = 0.0;
	if (y <= 0.215)
		tmp = t_0 / ((x + 1.0) * t_1);
	else
		tmp = t_0 * ((1.0 / (x + y)) / t_1);
	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[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, 0.215], N[(t$95$0 / N[(N[(x + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < 0.214999999999999997

   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. +-commutative 72.4%

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

      2. +-commutative 72.4%

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

      3. +-commutative 72.4%

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

      4. *-commutative 72.4%

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

      5. distribute-rgt1-in 52.7%

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

      6. fma-define 72.4%

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

      7. +-commutative 72.4%

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

      8. +-commutative 72.4%

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

      9. cube-unmult 72.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 72.4%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 72.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 72.4%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 52.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 52.7%

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

      4. distribute-rgt1-in 72.4%

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

      5. *-commutative 72.4%

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

      6. associate-*l* 72.4%

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

      7. times-frac 94.5%

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

      8. associate-+r+ 94.5%

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

   6. Applied egg-rr 94.5%

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

      1. clear-num 94.4%

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

      2. associate-/r* 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+l+ 99.7%

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

      5. frac-times 99.7%

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

      6. metadata-eval 99.7%

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

      7. times-frac 99.7%

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

      8. *-un-lft-identity 99.7%

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

      9. *-un-lft-identity 99.7%

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

      10. +-commutative 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in y around 0 85.0%

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

       1. +-commutative 85.0%

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

   11. Simplified 85.0%

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

   if 0.214999999999999997 < y

   1. Initial program 57.5%

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

      1. +-commutative 57.5%

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

      2. +-commutative 57.5%

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

      3. +-commutative 57.5%

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

      4. *-commutative 57.5%

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

      5. distribute-rgt1-in 53.4%

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

      6. fma-define 57.5%

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

      7. +-commutative 57.5%

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

      8. +-commutative 57.5%

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

      9. cube-unmult 57.5%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 57.5%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 57.5%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 57.5%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 53.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 53.4%

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

      4. distribute-rgt1-in 57.5%

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

      5. *-commutative 57.5%

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

      6. associate-*l* 57.5%

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

      7. times-frac 86.6%

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

      8. associate-+r+ 86.6%

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

   6. Applied egg-rr 86.6%

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

      1. clear-num 86.6%

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

      2. associate-/r* 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l+ 99.9%

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

      5. frac-times 99.8%

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

      6. metadata-eval 99.8%

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

      7. times-frac 99.8%

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

      8. *-un-lft-identity 99.8%

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

      9. *-un-lft-identity 99.8%

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

      10. +-commutative 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. div-inv 99.8%

         \[\leadsto \frac{\color{blue}{x \cdot \frac{1}{y + x}}}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)} \]

      2. *-commutative 99.8%

         \[\leadsto \frac{x \cdot \frac{1}{y + x}}{\color{blue}{\left(y + \left(x + 1\right)\right) \cdot \frac{y + x}{y}}} \]

      3. times-frac 99.8%

         \[\leadsto \color{blue}{\frac{x}{y + \left(x + 1\right)} \cdot \frac{\frac{1}{y + x}}{\frac{y + x}{y}}} \]

      4. +-commutative 99.8%

         \[\leadsto \frac{x}{y + \left(x + 1\right)} \cdot \frac{\frac{1}{\color{blue}{x + y}}}{\frac{y + x}{y}} \]

      5. +-commutative 99.8%

         \[\leadsto \frac{x}{y + \left(x + 1\right)} \cdot \frac{\frac{1}{x + y}}{\frac{\color{blue}{x + y}}{y}} \]

   10. Applied egg-rr 99.8%

       \[\leadsto \color{blue}{\frac{x}{y + \left(x + 1\right)} \cdot \frac{\frac{1}{x + y}}{\frac{x + y}{y}}} \]

   11. Taylor expanded in x around inf 98.9%

       \[\leadsto \frac{x}{y + \color{blue}{x}} \cdot \frac{\frac{1}{x + y}}{\frac{x + y}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.215:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(x + 1\right) \cdot \frac{x + y}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{\frac{1}{x + y}}{\frac{x + y}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 86.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-117}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.4e-117)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 4.2e+149)
     (/ x (* (+ x y) (+ x (+ y 1.0))))
     (/ (/ x (+ x y)) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.4e-117) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 4.2e+149) {
		tmp = x / ((x + y) * (x + (y + 1.0)));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.4d-117) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 4.2d+149) then
        tmp = x / ((x + y) * (x + (y + 1.0d0)))
    else
        tmp = (x / (x + y)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.4e-117) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 4.2e+149) {
		tmp = x / ((x + y) * (x + (y + 1.0)));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.4e-117:
		tmp = (y / x) / (x + 1.0)
	elif y <= 4.2e+149:
		tmp = x / ((x + y) * (x + (y + 1.0)))
	else:
		tmp = (x / (x + y)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.4e-117)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 4.2e+149)
		tmp = Float64(x / Float64(Float64(x + y) * Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.4e-117)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 4.2e+149)
		tmp = x / ((x + y) * (x + (y + 1.0)));
	else
		tmp = (x / (x + y)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.4e-117], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.2e+149], N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.40000000000000035e-117

   1. Initial program 67.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 67.8%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 67.8%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 67.8%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 67.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 46.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 67.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 67.8%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 67.8%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 67.8%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 67.8%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 67.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 67.8%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 46.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 46.4%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 67.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 67.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 67.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 93.3%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 93.3%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 93.3%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   7. Step-by-step derivation

      1. clear-num 93.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + y}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)} \]

      2. associate-/r* 99.8%

         \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + \left(y + 1\right)}} \]

      3. +-commutative 99.8%

         \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \frac{\frac{x}{x + y}}{x + \color{blue}{\left(1 + y\right)}} \]

      4. associate-+l+ 99.8%

         \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \frac{\frac{x}{x + y}}{\color{blue}{\left(x + 1\right) + y}} \]

      5. frac-times 99.8%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)}} \]

      6. metadata-eval 99.8%

         \[\leadsto \frac{\color{blue}{\frac{1}{1}} \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      7. times-frac 99.8%

         \[\leadsto \frac{\color{blue}{\frac{1 \cdot x}{1 \cdot \left(x + y\right)}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      8. *-un-lft-identity 99.8%

         \[\leadsto \frac{\frac{\color{blue}{x}}{1 \cdot \left(x + y\right)}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      9. *-un-lft-identity 99.8%

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      10. +-commutative 99.8%

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      11. +-commutative 99.8%

          \[\leadsto \frac{\frac{x}{y + x}}{\frac{\color{blue}{y + x}}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      12. +-commutative 99.8%

          \[\leadsto \frac{\frac{x}{y + x}}{\frac{y + x}{y} \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

   8. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)}} \]

   9. Taylor expanded in y around 0 59.8%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   10. Step-by-step derivation

       1. associate-/r* 62.3%

          \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

       2. +-commutative 62.3%

          \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   11. Simplified 62.3%

       \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if 3.40000000000000035e-117 < y < 4.2000000000000003e149

   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. +-commutative 81.5%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 81.5%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 81.5%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 81.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 71.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 81.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 81.6%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 81.6%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 81.6%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 81.6%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 81.6%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 81.6%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 71.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 71.4%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 81.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 81.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 81.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 97.1%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 97.1%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 97.1%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

   7. Taylor expanded in y around inf 72.9%

      \[\leadsto \color{blue}{1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)} \]

   if 4.2000000000000003e149 < y

   1. Initial program 44.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 44.7%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 44.7%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 44.7%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 44.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 44.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 44.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 44.7%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 44.7%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 44.7%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 44.7%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 44.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 44.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 44.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 44.7%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 44.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 44.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 44.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 78.9%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 78.9%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 78.9%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   7. Step-by-step derivation

      1. clear-num 78.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + y}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)} \]

      2. associate-/r* 99.9%

         \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + \left(y + 1\right)}} \]

      3. +-commutative 99.9%

         \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \frac{\frac{x}{x + y}}{x + \color{blue}{\left(1 + y\right)}} \]

      4. associate-+l+ 99.9%

         \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \frac{\frac{x}{x + y}}{\color{blue}{\left(x + 1\right) + y}} \]

      5. frac-times 99.9%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)}} \]

      6. metadata-eval 99.9%

         \[\leadsto \frac{\color{blue}{\frac{1}{1}} \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      7. times-frac 99.9%

         \[\leadsto \frac{\color{blue}{\frac{1 \cdot x}{1 \cdot \left(x + y\right)}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      8. *-un-lft-identity 99.9%

         \[\leadsto \frac{\frac{\color{blue}{x}}{1 \cdot \left(x + y\right)}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      9. *-un-lft-identity 99.9%

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      10. +-commutative 99.9%

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      11. +-commutative 99.9%

          \[\leadsto \frac{\frac{x}{y + x}}{\frac{\color{blue}{y + x}}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      12. +-commutative 99.9%

          \[\leadsto \frac{\frac{x}{y + x}}{\frac{y + x}{y} \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

   8. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)}} \]

   9. Taylor expanded in x around 0 92.1%

      \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{1 + y}} \]
   10. Step-by-step derivation

       1. +-commutative 92.1%

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + 1}} \]

   11. Simplified 92.1%

       \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + 1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-117}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 72.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1250000000:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -7.4 \cdot 10^{-163} \lor \neg \left(x \leq 9 \cdot 10^{-169}\right):\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1250000000.0)
   (* (/ y x) (/ 1.0 x))
   (if (or (<= x -7.4e-163) (not (<= x 9e-169)))
     (/ x (* y y))
     (/ 1.0 (/ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1250000000.0) {
		tmp = (y / x) * (1.0 / x);
	} else if ((x <= -7.4e-163) || !(x <= 9e-169)) {
		tmp = x / (y * y);
	} else {
		tmp = 1.0 / (y / x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1250000000.0d0)) then
        tmp = (y / x) * (1.0d0 / x)
    else if ((x <= (-7.4d-163)) .or. (.not. (x <= 9d-169))) then
        tmp = x / (y * y)
    else
        tmp = 1.0d0 / (y / x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1250000000.0) {
		tmp = (y / x) * (1.0 / x);
	} else if ((x <= -7.4e-163) || !(x <= 9e-169)) {
		tmp = x / (y * y);
	} else {
		tmp = 1.0 / (y / x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1250000000.0:
		tmp = (y / x) * (1.0 / x)
	elif (x <= -7.4e-163) or not (x <= 9e-169):
		tmp = x / (y * y)
	else:
		tmp = 1.0 / (y / x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1250000000.0)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	elseif ((x <= -7.4e-163) || !(x <= 9e-169))
		tmp = Float64(x / Float64(y * y));
	else
		tmp = Float64(1.0 / Float64(y / x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1250000000.0)
		tmp = (y / x) * (1.0 / x);
	elseif ((x <= -7.4e-163) || ~((x <= 9e-169)))
		tmp = x / (y * y);
	else
		tmp = 1.0 / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1250000000.0], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -7.4e-163], N[Not[LessEqual[x, 9e-169]], $MachinePrecision]], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25e9

   1. Initial program 60.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 60.2%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 60.2%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 60.2%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 60.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 27.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 60.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 60.2%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 60.2%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 60.2%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 60.2%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 60.2%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 60.2%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 27.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 27.8%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 60.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 60.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 60.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 85.4%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 85.4%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 85.4%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

   7. Taylor expanded in x around inf 69.8%

      \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{x}} \]

   8. Taylor expanded in y around 0 69.4%

      \[\leadsto \color{blue}{\frac{y}{x}} \cdot \frac{1}{x} \]

   if -1.25e9 < x < -7.3999999999999998e-163 or 8.9999999999999997e-169 < x

   1. Initial program 75.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 84.9%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 84.9%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 84.9%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 41.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   6. Taylor expanded in y around inf 39.2%

      \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

   if -7.3999999999999998e-163 < x < 8.9999999999999997e-169

   1. Initial program 58.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 72.0%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 72.0%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 72.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 88.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   6. Taylor expanded in y around 0 79.4%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
   7. Step-by-step derivation

      1. clear-num 79.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]

      2. inv-pow 79.1%

         \[\leadsto \color{blue}{{\left(\frac{y}{x}\right)}^{-1}} \]

   8. Applied egg-rr 79.1%

      \[\leadsto \color{blue}{{\left(\frac{y}{x}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. unpow-1 79.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]

   10. Simplified 79.1%

       \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1250000000:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -7.4 \cdot 10^{-163} \lor \neg \left(x \leq 9 \cdot 10^{-169}\right):\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 82.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{\left(x + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.7e+170)
   (/ (/ y x) (+ x y))
   (if (<= x -7.2e-54)
     (/ y (* (+ x 1.0) (+ x y)))
     (/ (/ x (+ x y)) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.7e+170) {
		tmp = (y / x) / (x + y);
	} else if (x <= -7.2e-54) {
		tmp = y / ((x + 1.0) * (x + y));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.7d+170)) then
        tmp = (y / x) / (x + y)
    else if (x <= (-7.2d-54)) then
        tmp = y / ((x + 1.0d0) * (x + y))
    else
        tmp = (x / (x + y)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.7e+170) {
		tmp = (y / x) / (x + y);
	} else if (x <= -7.2e-54) {
		tmp = y / ((x + 1.0) * (x + y));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.7e+170:
		tmp = (y / x) / (x + y)
	elif x <= -7.2e-54:
		tmp = y / ((x + 1.0) * (x + y))
	else:
		tmp = (x / (x + y)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.7e+170)
		tmp = Float64(Float64(y / x) / Float64(x + y));
	elseif (x <= -7.2e-54)
		tmp = Float64(y / Float64(Float64(x + 1.0) * Float64(x + y)));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.7e+170)
		tmp = (y / x) / (x + y);
	elseif (x <= -7.2e-54)
		tmp = y / ((x + 1.0) * (x + y));
	else
		tmp = (x / (x + y)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.7e+170], N[(N[(y / x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7.2e-54], N[(y / N[(N[(x + 1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7000000000000001e170

   1. Initial program 50.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 50.2%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 50.2%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 50.2%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 50.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 0.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 50.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 50.2%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 50.2%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 50.2%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 50.2%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 50.2%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 50.2%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 0.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 0.0%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 50.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 50.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 50.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 79.1%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 79.1%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 79.1%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

   7. Taylor expanded in x around inf 81.7%

      \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{x}} \]
   8. Step-by-step derivation

      1. frac-times 79.1%

         \[\leadsto \color{blue}{\frac{y \cdot 1}{\left(x + y\right) \cdot x}} \]

      2. *-rgt-identity 79.1%

         \[\leadsto \frac{\color{blue}{y}}{\left(x + y\right) \cdot x} \]

   9. Applied egg-rr 79.1%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot x}} \]
   10. Step-by-step derivation

       1. associate-/l/ 81.8%

          \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + y}} \]

   11. Simplified 81.8%

       \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + y}} \]

   if -1.7000000000000001e170 < x < -7.19999999999999953e-54

   1. Initial program 79.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 67.0%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{x} + 1\right)} \]
   4. Step-by-step derivation

      1. *-commutative 67.0%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + 1\right)} \]

      2. associate-*l* 67.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + 1\right)\right)}} \]

      3. times-frac 64.0%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + 1\right)}} \]

      4. +-commutative 64.0%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + 1\right)} \]

      5. +-commutative 64.0%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + 1\right)} \]

   5. Applied egg-rr 64.0%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(x + 1\right)}} \]

   6. Taylor expanded in y around 0 51.5%

      \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{1 + x}} \]
   7. Step-by-step derivation

      1. +-commutative 51.5%

         \[\leadsto \frac{y}{y + x} \cdot \frac{1}{\color{blue}{x + 1}} \]

   8. Simplified 51.5%

      \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{x + 1}} \]
   9. Step-by-step derivation

      1. +-commutative 51.5%

         \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{1}{x + 1} \]

      2. clear-num 51.5%

         \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{\frac{x + 1}{1}}} \]

      3. frac-times 61.5%

         \[\leadsto \color{blue}{\frac{y \cdot 1}{\left(x + y\right) \cdot \frac{x + 1}{1}}} \]

      4. *-rgt-identity 61.5%

         \[\leadsto \frac{\color{blue}{y}}{\left(x + y\right) \cdot \frac{x + 1}{1}} \]

      5. /-rgt-identity 61.5%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + 1\right)}} \]

      6. +-commutative 61.5%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(1 + x\right)}} \]

   10. Applied egg-rr 61.5%

       \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(1 + x\right)}} \]

   if -7.19999999999999953e-54 < x

   1. Initial program 68.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 68.4%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 68.4%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 68.4%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 68.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 60.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 68.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 68.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 68.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 68.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 68.4%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 68.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 68.4%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 60.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 60.5%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 68.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 68.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 68.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 94.1%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 94.1%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 94.1%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   7. Step-by-step derivation

      1. clear-num 94.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + y}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)} \]

      2. associate-/r* 99.7%

         \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + \left(y + 1\right)}} \]

      3. +-commutative 99.7%

         \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \frac{\frac{x}{x + y}}{x + \color{blue}{\left(1 + y\right)}} \]

      4. associate-+l+ 99.7%

         \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \frac{\frac{x}{x + y}}{\color{blue}{\left(x + 1\right) + y}} \]

      5. frac-times 99.8%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)}} \]

      6. metadata-eval 99.8%

         \[\leadsto \frac{\color{blue}{\frac{1}{1}} \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      7. times-frac 99.8%

         \[\leadsto \frac{\color{blue}{\frac{1 \cdot x}{1 \cdot \left(x + y\right)}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      8. *-un-lft-identity 99.8%

         \[\leadsto \frac{\frac{\color{blue}{x}}{1 \cdot \left(x + y\right)}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      9. *-un-lft-identity 99.8%

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      10. +-commutative 99.8%

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      11. +-commutative 99.8%

          \[\leadsto \frac{\frac{x}{y + x}}{\frac{\color{blue}{y + x}}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      12. +-commutative 99.8%

          \[\leadsto \frac{\frac{x}{y + x}}{\frac{y + x}{y} \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

   8. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)}} \]

   9. Taylor expanded in x around 0 56.0%

      \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{1 + y}} \]
   10. Step-by-step derivation

       1. +-commutative 56.0%

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + 1}} \]

   11. Simplified 56.0%

       \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + 1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{\left(x + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 82.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-55}:\\ \;\;\;\;\frac{y}{\left(x + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.45e+170)
   (/ (/ y x) (+ x y))
   (if (<= x -3.3e-55) (/ y (* (+ x 1.0) (+ x y))) (/ (/ x y) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.45e+170) {
		tmp = (y / x) / (x + y);
	} else if (x <= -3.3e-55) {
		tmp = y / ((x + 1.0) * (x + y));
	} 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 <= (-1.45d+170)) then
        tmp = (y / x) / (x + y)
    else if (x <= (-3.3d-55)) then
        tmp = y / ((x + 1.0d0) * (x + y))
    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 <= -1.45e+170) {
		tmp = (y / x) / (x + y);
	} else if (x <= -3.3e-55) {
		tmp = y / ((x + 1.0) * (x + y));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.45e+170:
		tmp = (y / x) / (x + y)
	elif x <= -3.3e-55:
		tmp = y / ((x + 1.0) * (x + y))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.45e+170)
		tmp = Float64(Float64(y / x) / Float64(x + y));
	elseif (x <= -3.3e-55)
		tmp = Float64(y / Float64(Float64(x + 1.0) * Float64(x + y)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.45e+170)
		tmp = (y / x) / (x + y);
	elseif (x <= -3.3e-55)
		tmp = y / ((x + 1.0) * (x + y));
	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, -1.45e+170], N[(N[(y / x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.3e-55], N[(y / N[(N[(x + 1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.45e170

   1. Initial program 50.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 50.2%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 50.2%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 50.2%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 50.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 0.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 50.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 50.2%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 50.2%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 50.2%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 50.2%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 50.2%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 50.2%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 0.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 0.0%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 50.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 50.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 50.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 79.1%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 79.1%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 79.1%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

   7. Taylor expanded in x around inf 81.7%

      \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{x}} \]
   8. Step-by-step derivation

      1. frac-times 79.1%

         \[\leadsto \color{blue}{\frac{y \cdot 1}{\left(x + y\right) \cdot x}} \]

      2. *-rgt-identity 79.1%

         \[\leadsto \frac{\color{blue}{y}}{\left(x + y\right) \cdot x} \]

   9. Applied egg-rr 79.1%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot x}} \]
   10. Step-by-step derivation

       1. associate-/l/ 81.8%

          \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + y}} \]

   11. Simplified 81.8%

       \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + y}} \]

   if -1.45e170 < x < -3.2999999999999999e-55

   1. Initial program 79.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 67.0%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{x} + 1\right)} \]
   4. Step-by-step derivation

      1. *-commutative 67.0%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + 1\right)} \]

      2. associate-*l* 67.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + 1\right)\right)}} \]

      3. times-frac 64.0%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + 1\right)}} \]

      4. +-commutative 64.0%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + 1\right)} \]

      5. +-commutative 64.0%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + 1\right)} \]

   5. Applied egg-rr 64.0%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(x + 1\right)}} \]

   6. Taylor expanded in y around 0 51.5%

      \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{1 + x}} \]
   7. Step-by-step derivation

      1. +-commutative 51.5%

         \[\leadsto \frac{y}{y + x} \cdot \frac{1}{\color{blue}{x + 1}} \]

   8. Simplified 51.5%

      \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{1}{x + 1}} \]
   9. Step-by-step derivation

      1. +-commutative 51.5%

         \[\leadsto \frac{y}{\color{blue}{x + y}} \cdot \frac{1}{x + 1} \]

      2. clear-num 51.5%

         \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{\frac{x + 1}{1}}} \]

      3. frac-times 61.5%

         \[\leadsto \color{blue}{\frac{y \cdot 1}{\left(x + y\right) \cdot \frac{x + 1}{1}}} \]

      4. *-rgt-identity 61.5%

         \[\leadsto \frac{\color{blue}{y}}{\left(x + y\right) \cdot \frac{x + 1}{1}} \]

      5. /-rgt-identity 61.5%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + 1\right)}} \]

      6. +-commutative 61.5%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(1 + x\right)}} \]

   10. Applied egg-rr 61.5%

       \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(1 + x\right)}} \]

   if -3.2999999999999999e-55 < x

   1. Initial program 68.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 80.0%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 80.0%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 80.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 53.2%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 55.5%

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

      2. +-commutative 55.5%

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y + 1}} \]

   7. Simplified 55.5%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-55}:\\ \;\;\;\;\frac{y}{\left(x + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+170}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.7e+170)
   (* (/ y x) (/ 1.0 x))
   (if (<= x -7.2e-54) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.7e+170) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -7.2e-54) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.7d+170)) then
        tmp = (y / x) * (1.0d0 / x)
    else if (x <= (-7.2d-54)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.7e+170) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -7.2e-54) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.7e+170:
		tmp = (y / x) * (1.0 / x)
	elif x <= -7.2e-54:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.7e+170)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	elseif (x <= -7.2e-54)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.7e+170)
		tmp = (y / x) * (1.0 / x);
	elseif (x <= -7.2e-54)
		tmp = y / (x * (x + 1.0));
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.7e+170], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7.2e-54], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7000000000000001e170

   1. Initial program 50.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 50.2%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 50.2%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 50.2%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 50.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 0.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 50.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 50.2%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 50.2%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 50.2%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 50.2%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 50.2%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 50.2%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 0.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 0.0%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 50.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 50.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 50.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 79.1%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 79.1%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 79.1%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

   7. Taylor expanded in x around inf 81.7%

      \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{x}} \]

   8. Taylor expanded in y around 0 81.4%

      \[\leadsto \color{blue}{\frac{y}{x}} \cdot \frac{1}{x} \]

   if -1.7000000000000001e170 < x < -7.19999999999999953e-54

   1. Initial program 79.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 89.6%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 89.6%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 89.6%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 52.8%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 52.8%

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   7. Simplified 52.8%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]

   if -7.19999999999999953e-54 < x

   1. Initial program 68.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 80.0%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 80.0%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 80.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 53.2%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+170}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{\frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(1 + \left(x + y\right)\right)} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (/ (/ x (+ x y)) (* (/ (+ x y) y) (+ 1.0 (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	return (x / (x + y)) / (((x + y) / y) * (1.0 + (x + y)));
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x / (x + y)) / (((x + y) / y) * (1.0d0 + (x + y)))
end function

Java

assert x < y;
public static double code(double x, double y) {
	return (x / (x + y)) / (((x + y) / y) * (1.0 + (x + y)));
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return (x / (x + y)) / (((x + y) / y) * (1.0 + (x + y)))

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(Float64(x / Float64(x + y)) / Float64(Float64(Float64(x + y) / y) * Float64(1.0 + Float64(x + y))))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = (x / (x + y)) / (((x + y) / y) * (1.0 + (x + y)));
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision] * N[(1.0 + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.2%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. +-commutative 68.2%

      \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. +-commutative 68.2%

      \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   3. +-commutative 68.2%

      \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

   4. *-commutative 68.2%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   5. distribute-rgt1-in 52.9%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   6. fma-define 68.2%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

   7. +-commutative 68.2%

      \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   8. +-commutative 68.2%

      \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   9. cube-unmult 68.2%

      \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

   10. +-commutative 68.2%

       \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

3. Simplified 68.2%

   \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative 68.2%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

   2. fma-define 52.9%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   3. cube-mult 52.9%

      \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   4. distribute-rgt1-in 68.2%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   5. *-commutative 68.2%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   6. associate-*l* 68.1%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

   7. times-frac 92.3%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   8. associate-+r+ 92.3%

      \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

6. Applied egg-rr 92.3%

   \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
7. Step-by-step derivation

   1. clear-num 92.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{x + y}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)} \]

   2. associate-/r* 99.8%

      \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + \left(y + 1\right)}} \]

   3. +-commutative 99.8%

      \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \frac{\frac{x}{x + y}}{x + \color{blue}{\left(1 + y\right)}} \]

   4. associate-+l+ 99.8%

      \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \frac{\frac{x}{x + y}}{\color{blue}{\left(x + 1\right) + y}} \]

   5. frac-times 99.7%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)}} \]

   6. metadata-eval 99.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{1}} \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

   7. times-frac 99.7%

      \[\leadsto \frac{\color{blue}{\frac{1 \cdot x}{1 \cdot \left(x + y\right)}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

   8. *-un-lft-identity 99.7%

      \[\leadsto \frac{\frac{\color{blue}{x}}{1 \cdot \left(x + y\right)}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

   9. *-un-lft-identity 99.7%

      \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

   10. +-commutative 99.7%

       \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

   11. +-commutative 99.7%

       \[\leadsto \frac{\frac{x}{y + x}}{\frac{\color{blue}{y + x}}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

   12. +-commutative 99.7%

       \[\leadsto \frac{\frac{x}{y + x}}{\frac{y + x}{y} \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

8. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)}} \]

9. Taylor expanded in y around 0 99.7%

   \[\leadsto \frac{\frac{x}{y + x}}{\frac{y + x}{y} \cdot \color{blue}{\left(1 + \left(x + y\right)\right)}} \]

10. Final simplification 99.7%

    \[\leadsto \frac{\frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(1 + \left(x + y\right)\right)} \]
11. Add Preprocessing

Alternative 18: 82.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-117}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.3e-117) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.3e-117) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.3d-117) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.3e-117) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.3e-117:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.3e-117)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.3e-117)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.3e-117], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.30000000000000015e-117

   1. Initial program 67.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 67.8%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 67.8%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 67.8%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 67.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 46.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 67.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 67.8%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 67.8%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 67.8%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 67.8%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 67.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 67.8%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 46.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 46.4%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 67.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 67.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 67.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 93.3%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 93.3%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 93.3%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   7. Step-by-step derivation

      1. clear-num 93.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + y}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)} \]

      2. associate-/r* 99.8%

         \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + \left(y + 1\right)}} \]

      3. +-commutative 99.8%

         \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \frac{\frac{x}{x + y}}{x + \color{blue}{\left(1 + y\right)}} \]

      4. associate-+l+ 99.8%

         \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \frac{\frac{x}{x + y}}{\color{blue}{\left(x + 1\right) + y}} \]

      5. frac-times 99.8%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)}} \]

      6. metadata-eval 99.8%

         \[\leadsto \frac{\color{blue}{\frac{1}{1}} \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      7. times-frac 99.8%

         \[\leadsto \frac{\color{blue}{\frac{1 \cdot x}{1 \cdot \left(x + y\right)}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      8. *-un-lft-identity 99.8%

         \[\leadsto \frac{\frac{\color{blue}{x}}{1 \cdot \left(x + y\right)}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      9. *-un-lft-identity 99.8%

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      10. +-commutative 99.8%

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      11. +-commutative 99.8%

          \[\leadsto \frac{\frac{x}{y + x}}{\frac{\color{blue}{y + x}}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      12. +-commutative 99.8%

          \[\leadsto \frac{\frac{x}{y + x}}{\frac{y + x}{y} \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

   8. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)}} \]

   9. Taylor expanded in y around 0 59.8%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   10. Step-by-step derivation

       1. associate-/r* 62.3%

          \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

       2. +-commutative 62.3%

          \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   11. Simplified 62.3%

       \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if 3.30000000000000015e-117 < y

   1. Initial program 68.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 81.7%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 81.7%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 81.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 60.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 65.5%

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

      2. +-commutative 65.5%

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y + 1}} \]

   7. Simplified 65.5%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 78.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -250000000:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -250000000.0) (* (/ y x) (/ 1.0 x)) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -250000000.0) {
		tmp = (y / x) * (1.0 / x);
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-250000000.0d0)) then
        tmp = (y / x) * (1.0d0 / x)
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -250000000.0) {
		tmp = (y / x) * (1.0 / x);
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -250000000.0:
		tmp = (y / x) * (1.0 / x)
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -250000000.0)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -250000000.0)
		tmp = (y / x) * (1.0 / x);
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -250000000.0], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5e8

   1. Initial program 60.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 60.2%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 60.2%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 60.2%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 60.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 27.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 60.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 60.2%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 60.2%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 60.2%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 60.2%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 60.2%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 60.2%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 27.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 27.8%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 60.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 60.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 60.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 85.4%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 85.4%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 85.4%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

   7. Taylor expanded in x around inf 69.8%

      \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{x}} \]

   8. Taylor expanded in y around 0 69.4%

      \[\leadsto \color{blue}{\frac{y}{x}} \cdot \frac{1}{x} \]

   if -2.5e8 < x

   1. Initial program 70.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 81.6%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 81.6%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 81.6%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 53.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -250000000:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 47.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 1.0) (/ x y) (/ x (* y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.0) {
		tmp = x / y;
	} 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 <= 1.0d0) then
        tmp = x / y
    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 <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.0:
		tmp = x / y
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.0)
		tmp = Float64(x / y);
	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 <= 1.0)
		tmp = x / y;
	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, 1.0], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 72.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 83.5%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 83.5%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 83.5%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 36.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   6. Taylor expanded in y around 0 23.3%

      \[\leadsto \color{blue}{\frac{x}{y}} \]

   if 1 < y

   1. Initial program 56.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 77.2%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 77.2%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 77.2%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 74.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   6. Taylor expanded in y around inf 73.1%

      \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 26.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{1}{\frac{y}{x}} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ 1.0 (/ y x)))

C

assert(x < y);
double code(double x, double y) {
	return 1.0 / (y / x);
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / (y / x)
end function

Java

assert x < y;
public static double code(double x, double y) {
	return 1.0 / (y / x);
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return 1.0 / (y / x)

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(1.0 / Float64(y / x))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 1.0 / (y / x);
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.2%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l* 81.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   2. associate-+l+ 81.7%

      \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

3. Simplified 81.7%

   \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 47.2%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

6. Taylor expanded in y around 0 20.5%

   \[\leadsto \color{blue}{\frac{x}{y}} \]
7. Step-by-step derivation

   1. clear-num 20.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]

   2. inv-pow 20.8%

      \[\leadsto \color{blue}{{\left(\frac{y}{x}\right)}^{-1}} \]

8. Applied egg-rr 20.8%

   \[\leadsto \color{blue}{{\left(\frac{y}{x}\right)}^{-1}} \]
9. Step-by-step derivation

   1. unpow-1 20.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]

10. Simplified 20.8%

    \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]
11. Add Preprocessing

Alternative 22: 26.5% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{x}{y} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ x y))

C

assert(x < y);
double code(double x, double y) {
	return x / y;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / y
end function

Java

assert x < y;
public static double code(double x, double y) {
	return x / y;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return x / y

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x / y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x / y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 68.2%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l* 81.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   2. associate-+l+ 81.7%

      \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

3. Simplified 81.7%

   \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 47.2%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

6. Taylor expanded in y around 0 20.5%

   \[\leadsto \color{blue}{\frac{x}{y}} \]
7. Add Preprocessing

Alternative 23: 4.4% 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 68.2%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. +-commutative 68.2%

      \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. +-commutative 68.2%

      \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   3. +-commutative 68.2%

      \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

   4. *-commutative 68.2%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   5. distribute-rgt1-in 52.9%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   6. fma-define 68.2%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

   7. +-commutative 68.2%

      \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   8. +-commutative 68.2%

      \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   9. cube-unmult 68.2%

      \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

   10. +-commutative 68.2%

       \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

3. Simplified 68.2%

   \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative 68.2%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

   2. fma-define 52.9%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   3. cube-mult 52.9%

      \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   4. distribute-rgt1-in 68.2%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   5. *-commutative 68.2%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   6. associate-*l* 68.1%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

   7. times-frac 92.3%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   8. associate-+r+ 92.3%

      \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

6. Applied egg-rr 92.3%

   \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

7. Taylor expanded in x around inf 41.2%

   \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{x}} \]

8. Taylor expanded in y around inf 4.4%

   \[\leadsto \color{blue}{\frac{1}{x}} \]
9. 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 2024180 
(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))))