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

Percentage Accurate: 69.4% → 99.8%

Time: 15.6s

Alternatives: 21

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.8%

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

   1. +-commutative 61.8%

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

   2. +-commutative 61.8%

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

   3. +-commutative 61.8%

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

   4. *-commutative 61.8%

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

   5. distribute-rgt1-in 50.4%

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

   6. fma-define 61.8%

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

   7. +-commutative 61.8%

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

   8. +-commutative 61.8%

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

   9. cube-unmult 61.9%

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

   10. +-commutative 61.9%

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

3. Simplified 61.9%

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

   1. *-commutative 61.9%

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

   2. fma-define 50.4%

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

   3. cube-mult 50.4%

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

   4. distribute-rgt1-in 61.8%

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

   5. *-commutative 61.8%

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

   6. associate-*l* 61.9%

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

   7. times-frac 90.3%

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

   8. associate-+r+ 90.3%

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

6. Applied egg-rr 90.3%

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

   1. clear-num 90.2%

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

   2. associate-/r* 99.7%

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

   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 99.4%

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

   6. metadata-eval 99.4%

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

   7. times-frac 99.4%

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

   8. *-un-lft-identity 99.4%

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

   9. *-un-lft-identity 99.4%

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

   10. +-commutative 99.4%

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

   11. +-commutative 99.4%

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

   12. +-commutative 99.4%

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

8. Applied egg-rr 99.4%

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

   1. *-un-lft-identity 99.4%

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

   2. times-frac 99.7%

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

   3. +-commutative 99.7%

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

   4. clear-num 99.8%

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

   5. +-commutative 99.8%

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

   6. associate-+r+ 99.8%

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

   7. +-commutative 99.8%

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

   8. associate-+l+ 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 94.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{y + x}\\ t_1 := y + \left(x + 1\right)\\ \mathbf{if}\;y \leq -4.6 \cdot 10^{+25}:\\ \;\;\;\;\frac{1}{\frac{y + x}{y} \cdot t\_1}\\ \mathbf{elif}\;y \leq 10^{-19}:\\ \;\;\;\;t\_0 \cdot \frac{x}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+160}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y x))) (t_1 (+ y (+ x 1.0))))
   (if (<= y -4.6e+25)
     (/ 1.0 (* (/ (+ y x) y) t_1))
     (if (<= y 1e-19)
       (* t_0 (/ x (* (+ y x) (+ x 1.0))))
       (if (<= y 1.02e+160)
         (/ x (* (+ y x) t_1))
         (* t_0 (/ (/ x y) (+ x (+ y 1.0)))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (y + x);
	double t_1 = y + (x + 1.0);
	double tmp;
	if (y <= -4.6e+25) {
		tmp = 1.0 / (((y + x) / y) * t_1);
	} else if (y <= 1e-19) {
		tmp = t_0 * (x / ((y + x) * (x + 1.0)));
	} else if (y <= 1.02e+160) {
		tmp = x / ((y + x) * t_1);
	} else {
		tmp = t_0 * ((x / y) / (x + (y + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y / (y + x)
    t_1 = y + (x + 1.0d0)
    if (y <= (-4.6d+25)) then
        tmp = 1.0d0 / (((y + x) / y) * t_1)
    else if (y <= 1d-19) then
        tmp = t_0 * (x / ((y + x) * (x + 1.0d0)))
    else if (y <= 1.02d+160) then
        tmp = x / ((y + x) * t_1)
    else
        tmp = t_0 * ((x / y) / (x + (y + 1.0d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (y + x);
	double t_1 = y + (x + 1.0);
	double tmp;
	if (y <= -4.6e+25) {
		tmp = 1.0 / (((y + x) / y) * t_1);
	} else if (y <= 1e-19) {
		tmp = t_0 * (x / ((y + x) * (x + 1.0)));
	} else if (y <= 1.02e+160) {
		tmp = x / ((y + x) * t_1);
	} else {
		tmp = t_0 * ((x / y) / (x + (y + 1.0)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (y + x)
	t_1 = y + (x + 1.0)
	tmp = 0
	if y <= -4.6e+25:
		tmp = 1.0 / (((y + x) / y) * t_1)
	elif y <= 1e-19:
		tmp = t_0 * (x / ((y + x) * (x + 1.0)))
	elif y <= 1.02e+160:
		tmp = x / ((y + x) * t_1)
	else:
		tmp = t_0 * ((x / y) / (x + (y + 1.0)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(y + x))
	t_1 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (y <= -4.6e+25)
		tmp = Float64(1.0 / Float64(Float64(Float64(y + x) / y) * t_1));
	elseif (y <= 1e-19)
		tmp = Float64(t_0 * Float64(x / Float64(Float64(y + x) * Float64(x + 1.0))));
	elseif (y <= 1.02e+160)
		tmp = Float64(x / Float64(Float64(y + x) * t_1));
	else
		tmp = Float64(t_0 * Float64(Float64(x / y) / Float64(x + Float64(y + 1.0))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (y + x);
	t_1 = y + (x + 1.0);
	tmp = 0.0;
	if (y <= -4.6e+25)
		tmp = 1.0 / (((y + x) / y) * t_1);
	elseif (y <= 1e-19)
		tmp = t_0 * (x / ((y + x) * (x + 1.0)));
	elseif (y <= 1.02e+160)
		tmp = x / ((y + x) * t_1);
	else
		tmp = t_0 * ((x / y) / (x + (y + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.6e+25], N[(1.0 / N[(N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e-19], N[(t$95$0 * N[(x / N[(N[(y + x), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.02e+160], N[(x / N[(N[(y + x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(x / y), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -4.5999999999999996e25

   1. Initial program 51.3%

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

      1. +-commutative 51.3%

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

      2. +-commutative 51.3%

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

      3. +-commutative 51.3%

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

      4. *-commutative 51.3%

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

      5. distribute-rgt1-in 21.1%

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

      6. fma-define 51.3%

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

      7. +-commutative 51.3%

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

      8. +-commutative 51.3%

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

      9. cube-unmult 51.3%

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

      10. +-commutative 51.3%

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

   3. Simplified 51.3%

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

      1. *-commutative 51.3%

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

      2. fma-define 21.1%

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

      3. cube-mult 21.1%

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

      4. distribute-rgt1-in 51.3%

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

      5. *-commutative 51.3%

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

      6. associate-*l* 51.3%

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

      7. times-frac 82.2%

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

      8. associate-+r+ 82.2%

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

   6. Applied egg-rr 82.2%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   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* 99.8%

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

      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 x around inf 33.7%

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

   if -4.5999999999999996e25 < y < 9.9999999999999998e-20

   1. Initial program 68.8%

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

   3. Taylor expanded in x around inf 67.2%

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

      1. *-commutative 67.2%

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

      2. associate-*l* 67.2%

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

      3. times-frac 98.3%

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

      4. +-commutative 98.3%

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

      5. +-commutative 98.3%

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

   5. Applied egg-rr 98.3%

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

   if 9.9999999999999998e-20 < y < 1.01999999999999993e160

   1. Initial program 78.7%

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

      1. +-commutative 78.7%

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

      2. +-commutative 78.7%

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

      3. +-commutative 78.7%

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

      4. *-commutative 78.7%

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

      5. distribute-rgt1-in 76.2%

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

      6. fma-define 78.8%

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

      7. +-commutative 78.8%

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

      8. +-commutative 78.8%

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

      9. cube-unmult 78.8%

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

      10. +-commutative 78.8%

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

   3. Simplified 78.8%

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

      1. *-commutative 78.8%

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

      2. fma-define 76.2%

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

      3. cube-mult 76.2%

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

      4. distribute-rgt1-in 78.7%

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

      5. *-commutative 78.7%

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

      6. associate-*l* 78.8%

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

      7. times-frac 95.2%

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

      8. associate-+r+ 95.2%

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

   6. Applied egg-rr 95.2%

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

      1. associate-*r/ 95.2%

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

      2. +-commutative 95.2%

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

      3. +-commutative 95.2%

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

      4. +-commutative 95.2%

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

      5. associate-+l+ 95.2%

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

      6. +-commutative 95.2%

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

   8. Applied egg-rr 95.2%

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

   9. Taylor expanded in y around inf 91.7%

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

   if 1.01999999999999993e160 < y

   1. Initial program 38.3%

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

      1. +-commutative 38.3%

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

      2. +-commutative 38.3%

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

      3. +-commutative 38.3%

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

      4. *-commutative 38.3%

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

      5. distribute-rgt1-in 38.3%

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

      6. fma-define 38.3%

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

      7. +-commutative 38.3%

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

      8. +-commutative 38.3%

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

      9. cube-unmult 38.3%

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

      10. +-commutative 38.3%

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

   3. Simplified 38.3%

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

      1. *-commutative 38.3%

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

      2. fma-define 38.3%

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

      3. cube-mult 38.3%

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

      4. distribute-rgt1-in 38.3%

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

      5. *-commutative 38.3%

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

      6. associate-*l* 38.3%

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

      7. times-frac 67.5%

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

      8. associate-+r+ 67.5%

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

   6. Applied egg-rr 67.5%

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

      1. clear-num 67.5%

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

      2. associate-/r* 99.7%

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

      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 99.7%

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

      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. *-un-lft-identity 99.7%

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

      2. times-frac 99.7%

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

      3. +-commutative 99.7%

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

      4. clear-num 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in x around 0 88.8%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+25}:\\ \;\;\;\;\frac{1}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{elif}\;y \leq 10^{-19}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+160}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{y + x}\\ t_1 := x + \left(y + 1\right)\\ \mathbf{if}\;y \leq -1.5 \cdot 10^{-220}:\\ \;\;\;\;t\_0 \cdot \frac{1 - \frac{y}{x}}{t\_1}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+159}:\\ \;\;\;\;t\_0 \cdot \frac{x}{t\_1 \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y x))) (t_1 (+ x (+ y 1.0))))
   (if (<= y -1.5e-220)
     (* t_0 (/ (- 1.0 (/ y x)) t_1))
     (if (<= y 2.6e+159)
       (* t_0 (/ x (* t_1 (+ y x))))
       (/ (/ x y) (* (/ (+ y x) y) (+ y (+ x 1.0))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (y + x);
	double t_1 = x + (y + 1.0);
	double tmp;
	if (y <= -1.5e-220) {
		tmp = t_0 * ((1.0 - (y / x)) / t_1);
	} else if (y <= 2.6e+159) {
		tmp = t_0 * (x / (t_1 * (y + x)));
	} else {
		tmp = (x / y) / (((y + x) / y) * (y + (x + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y / (y + x)
    t_1 = x + (y + 1.0d0)
    if (y <= (-1.5d-220)) then
        tmp = t_0 * ((1.0d0 - (y / x)) / t_1)
    else if (y <= 2.6d+159) then
        tmp = t_0 * (x / (t_1 * (y + x)))
    else
        tmp = (x / y) / (((y + x) / y) * (y + (x + 1.0d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (y + x);
	double t_1 = x + (y + 1.0);
	double tmp;
	if (y <= -1.5e-220) {
		tmp = t_0 * ((1.0 - (y / x)) / t_1);
	} else if (y <= 2.6e+159) {
		tmp = t_0 * (x / (t_1 * (y + x)));
	} else {
		tmp = (x / y) / (((y + x) / y) * (y + (x + 1.0)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (y + x)
	t_1 = x + (y + 1.0)
	tmp = 0
	if y <= -1.5e-220:
		tmp = t_0 * ((1.0 - (y / x)) / t_1)
	elif y <= 2.6e+159:
		tmp = t_0 * (x / (t_1 * (y + x)))
	else:
		tmp = (x / y) / (((y + x) / y) * (y + (x + 1.0)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(y + x))
	t_1 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (y <= -1.5e-220)
		tmp = Float64(t_0 * Float64(Float64(1.0 - Float64(y / x)) / t_1));
	elseif (y <= 2.6e+159)
		tmp = Float64(t_0 * Float64(x / Float64(t_1 * Float64(y + x))));
	else
		tmp = Float64(Float64(x / y) / Float64(Float64(Float64(y + x) / y) * Float64(y + Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (y + x);
	t_1 = x + (y + 1.0);
	tmp = 0.0;
	if (y <= -1.5e-220)
		tmp = t_0 * ((1.0 - (y / x)) / t_1);
	elseif (y <= 2.6e+159)
		tmp = t_0 * (x / (t_1 * (y + x)));
	else
		tmp = (x / y) / (((y + x) / y) * (y + (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.5e-220], N[(t$95$0 * N[(N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e+159], N[(t$95$0 * N[(x / N[(t$95$1 * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.50000000000000009e-220

   1. Initial program 59.2%

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

      1. +-commutative 59.2%

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

      2. +-commutative 59.2%

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

      3. +-commutative 59.2%

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

      4. *-commutative 59.2%

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

      5. distribute-rgt1-in 36.7%

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

      6. fma-define 59.2%

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

      7. +-commutative 59.2%

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

      8. +-commutative 59.2%

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

      9. cube-unmult 59.3%

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

      10. +-commutative 59.3%

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

   3. Simplified 59.3%

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

      1. *-commutative 59.3%

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

      2. fma-define 36.8%

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

      3. cube-mult 36.7%

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

      4. distribute-rgt1-in 59.2%

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

      5. *-commutative 59.2%

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

      6. associate-*l* 59.2%

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

      7. times-frac 89.7%

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

      8. associate-+r+ 89.7%

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

   6. Applied egg-rr 89.7%

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

      1. clear-num 89.7%

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

      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 99.5%

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

      6. metadata-eval 99.5%

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

      7. times-frac 99.5%

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

      8. *-un-lft-identity 99.5%

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

      9. *-un-lft-identity 99.5%

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

      10. +-commutative 99.5%

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

      11. +-commutative 99.5%

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

      12. +-commutative 99.5%

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

   8. Applied egg-rr 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. times-frac 99.7%

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

      3. +-commutative 99.7%

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

      4. clear-num 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in x around inf 45.9%

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

       1. neg-mul-1 45.9%

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

       2. unsub-neg 45.9%

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

   13. Simplified 45.9%

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

   if -1.50000000000000009e-220 < y < 2.6e159

   1. Initial program 71.8%

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

      1. +-commutative 71.8%

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

      2. +-commutative 71.8%

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

      3. +-commutative 71.8%

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

      4. *-commutative 71.8%

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

      5. distribute-rgt1-in 65.9%

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

      6. fma-define 71.9%

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

      7. +-commutative 71.9%

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

      8. +-commutative 71.9%

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

      9. cube-unmult 71.9%

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

      10. +-commutative 71.9%

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

   3. Simplified 71.9%

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

      1. *-commutative 71.9%

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

      2. fma-define 65.9%

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

      3. cube-mult 65.9%

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

      4. distribute-rgt1-in 71.8%

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

      5. *-commutative 71.8%

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

      6. associate-*l* 71.9%

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

      7. times-frac 98.3%

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

      8. associate-+r+ 98.3%

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

   6. Applied egg-rr 98.3%

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

   if 2.6e159 < y

   1. Initial program 38.3%

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

      1. +-commutative 38.3%

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

      2. +-commutative 38.3%

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

      3. +-commutative 38.3%

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

      4. *-commutative 38.3%

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

      5. distribute-rgt1-in 38.3%

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

      6. fma-define 38.3%

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

      7. +-commutative 38.3%

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

      8. +-commutative 38.3%

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

      9. cube-unmult 38.3%

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

      10. +-commutative 38.3%

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

   3. Simplified 38.3%

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

      1. *-commutative 38.3%

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

      2. fma-define 38.3%

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

      3. cube-mult 38.3%

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

      4. distribute-rgt1-in 38.3%

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

      5. *-commutative 38.3%

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

      6. associate-*l* 38.3%

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

      7. times-frac 67.5%

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

      8. associate-+r+ 67.5%

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

   6. Applied egg-rr 67.5%

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

      1. clear-num 67.5%

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

      2. associate-/r* 99.7%

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

      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 99.7%

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

      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 x around 0 93.4%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-220}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1 - \frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+159}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(x + \left(y + 1\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;y \leq -6.8 \cdot 10^{+40}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{\frac{x}{y} \cdot t\_0}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+159}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(x + \left(y + 1\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y + x}{y} \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= y -6.8e+40)
     (/ (/ x (+ y x)) (* (/ x y) t_0))
     (if (<= y 2.6e+159)
       (* (/ y (+ y x)) (/ x (* (+ x (+ y 1.0)) (+ y x))))
       (/ (/ x y) (* (/ (+ y x) y) t_0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= -6.8e+40) {
		tmp = (x / (y + x)) / ((x / y) * t_0);
	} else if (y <= 2.6e+159) {
		tmp = (y / (y + x)) * (x / ((x + (y + 1.0)) * (y + x)));
	} else {
		tmp = (x / y) / (((y + x) / y) * t_0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (y <= (-6.8d+40)) then
        tmp = (x / (y + x)) / ((x / y) * t_0)
    else if (y <= 2.6d+159) then
        tmp = (y / (y + x)) * (x / ((x + (y + 1.0d0)) * (y + x)))
    else
        tmp = (x / y) / (((y + x) / y) * t_0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= -6.8e+40) {
		tmp = (x / (y + x)) / ((x / y) * t_0);
	} else if (y <= 2.6e+159) {
		tmp = (y / (y + x)) * (x / ((x + (y + 1.0)) * (y + x)));
	} else {
		tmp = (x / y) / (((y + x) / y) * t_0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if y <= -6.8e+40:
		tmp = (x / (y + x)) / ((x / y) * t_0)
	elif y <= 2.6e+159:
		tmp = (y / (y + x)) * (x / ((x + (y + 1.0)) * (y + x)))
	else:
		tmp = (x / y) / (((y + x) / y) * t_0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (y <= -6.8e+40)
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(Float64(x / y) * t_0));
	elseif (y <= 2.6e+159)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(Float64(x + Float64(y + 1.0)) * Float64(y + x))));
	else
		tmp = Float64(Float64(x / y) / Float64(Float64(Float64(y + x) / y) * t_0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (y <= -6.8e+40)
		tmp = (x / (y + x)) / ((x / y) * t_0);
	elseif (y <= 2.6e+159)
		tmp = (y / (y + x)) * (x / ((x + (y + 1.0)) * (y + x)));
	else
		tmp = (x / y) / (((y + x) / y) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.8e+40], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(N[(x / y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e+159], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.79999999999999977e40

   1. Initial program 49.5%

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

      1. +-commutative 49.5%

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

      2. +-commutative 49.5%

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

      3. +-commutative 49.5%

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

      4. *-commutative 49.5%

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

      5. distribute-rgt1-in 18.2%

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

      6. fma-define 49.5%

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

      7. +-commutative 49.5%

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

      8. +-commutative 49.5%

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

      9. cube-unmult 49.5%

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

      10. +-commutative 49.5%

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

   3. Simplified 49.5%

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

      1. *-commutative 49.5%

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

      2. fma-define 18.2%

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

      3. cube-mult 18.2%

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

      4. distribute-rgt1-in 49.5%

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

      5. *-commutative 49.5%

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

      6. associate-*l* 49.5%

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

      7. times-frac 81.6%

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

      8. associate-+r+ 81.6%

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

   6. Applied egg-rr 81.6%

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

      1. clear-num 81.6%

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

      2. associate-/r* 99.7%

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

      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 y around 0 31.8%

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

   if -6.79999999999999977e40 < y < 2.6e159

   1. Initial program 71.7%

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

      1. +-commutative 71.7%

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

      2. +-commutative 71.7%

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

      3. +-commutative 71.7%

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

      4. *-commutative 71.7%

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

      5. distribute-rgt1-in 64.2%

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

      6. fma-define 71.7%

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

      7. +-commutative 71.7%

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

      8. +-commutative 71.7%

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

      9. cube-unmult 71.7%

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

      10. +-commutative 71.7%

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

   3. Simplified 71.7%

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

      1. *-commutative 71.7%

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

      2. fma-define 64.3%

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

      3. cube-mult 64.2%

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

      4. distribute-rgt1-in 71.7%

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

      5. *-commutative 71.7%

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

      6. associate-*l* 71.7%

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

      7. times-frac 98.7%

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

      8. associate-+r+ 98.7%

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

   6. Applied egg-rr 98.7%

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

   if 2.6e159 < y

   1. Initial program 38.3%

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

      1. +-commutative 38.3%

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

      2. +-commutative 38.3%

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

      3. +-commutative 38.3%

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

      4. *-commutative 38.3%

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

      5. distribute-rgt1-in 38.3%

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

      6. fma-define 38.3%

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

      7. +-commutative 38.3%

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

      8. +-commutative 38.3%

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

      9. cube-unmult 38.3%

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

      10. +-commutative 38.3%

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

   3. Simplified 38.3%

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

      1. *-commutative 38.3%

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

      2. fma-define 38.3%

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

      3. cube-mult 38.3%

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

      4. distribute-rgt1-in 38.3%

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

      5. *-commutative 38.3%

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

      6. associate-*l* 38.3%

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

      7. times-frac 67.5%

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

      8. associate-+r+ 67.5%

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

   6. Applied egg-rr 67.5%

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

      1. clear-num 67.5%

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

      2. associate-/r* 99.7%

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

      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 99.7%

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

      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 x around 0 93.4%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+40}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{\frac{x}{y} \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+159}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(x + \left(y + 1\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;y \leq -2.1 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{\frac{x}{y} \cdot t\_0}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y + x}{y} \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= y -2.1e+20)
     (/ (/ x (+ y x)) (* (/ x y) t_0))
     (if (<= y 1.8e-17)
       (* (/ y (+ y x)) (/ x (* (+ y x) (+ x 1.0))))
       (/ (/ x y) (* (/ (+ y x) y) t_0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= -2.1e+20) {
		tmp = (x / (y + x)) / ((x / y) * t_0);
	} else if (y <= 1.8e-17) {
		tmp = (y / (y + x)) * (x / ((y + x) * (x + 1.0)));
	} else {
		tmp = (x / y) / (((y + x) / y) * t_0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (y <= (-2.1d+20)) then
        tmp = (x / (y + x)) / ((x / y) * t_0)
    else if (y <= 1.8d-17) then
        tmp = (y / (y + x)) * (x / ((y + x) * (x + 1.0d0)))
    else
        tmp = (x / y) / (((y + x) / y) * t_0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= -2.1e+20) {
		tmp = (x / (y + x)) / ((x / y) * t_0);
	} else if (y <= 1.8e-17) {
		tmp = (y / (y + x)) * (x / ((y + x) * (x + 1.0)));
	} else {
		tmp = (x / y) / (((y + x) / y) * t_0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if y <= -2.1e+20:
		tmp = (x / (y + x)) / ((x / y) * t_0)
	elif y <= 1.8e-17:
		tmp = (y / (y + x)) * (x / ((y + x) * (x + 1.0)))
	else:
		tmp = (x / y) / (((y + x) / y) * t_0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (y <= -2.1e+20)
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(Float64(x / y) * t_0));
	elseif (y <= 1.8e-17)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(Float64(y + x) * Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / y) / Float64(Float64(Float64(y + x) / y) * t_0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (y <= -2.1e+20)
		tmp = (x / (y + x)) / ((x / y) * t_0);
	elseif (y <= 1.8e-17)
		tmp = (y / (y + x)) * (x / ((y + x) * (x + 1.0)));
	else
		tmp = (x / y) / (((y + x) / y) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.1e+20], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(N[(x / y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e-17], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(y + x), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.1e20

   1. Initial program 51.3%

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

      1. +-commutative 51.3%

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

      2. +-commutative 51.3%

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

      3. +-commutative 51.3%

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

      4. *-commutative 51.3%

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

      5. distribute-rgt1-in 21.1%

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

      6. fma-define 51.3%

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

      7. +-commutative 51.3%

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

      8. +-commutative 51.3%

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

      9. cube-unmult 51.3%

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

      10. +-commutative 51.3%

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

   3. Simplified 51.3%

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

      1. *-commutative 51.3%

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

      2. fma-define 21.1%

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

      3. cube-mult 21.1%

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

      4. distribute-rgt1-in 51.3%

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

      5. *-commutative 51.3%

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

      6. associate-*l* 51.3%

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

      7. times-frac 82.2%

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

      8. associate-+r+ 82.2%

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

   6. Applied egg-rr 82.2%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   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* 99.8%

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

      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 32.5%

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

   if -2.1e20 < y < 1.79999999999999997e-17

   1. Initial program 68.8%

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

   3. Taylor expanded in x around inf 67.2%

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

      1. *-commutative 67.2%

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

      2. associate-*l* 67.2%

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

      3. times-frac 98.3%

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

      4. +-commutative 98.3%

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

      5. +-commutative 98.3%

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

   5. Applied egg-rr 98.3%

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

   if 1.79999999999999997e-17 < y

   1. Initial program 59.0%

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

      1. +-commutative 59.0%

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

      2. +-commutative 59.0%

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

      3. +-commutative 59.0%

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

      4. *-commutative 59.0%

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

      5. distribute-rgt1-in 57.7%

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

      6. fma-define 59.0%

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

      7. +-commutative 59.0%

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

      8. +-commutative 59.0%

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

      9. cube-unmult 59.0%

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

      10. +-commutative 59.0%

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

   3. Simplified 59.0%

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

      1. *-commutative 59.0%

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

      2. fma-define 57.7%

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

      3. cube-mult 57.7%

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

      4. distribute-rgt1-in 59.0%

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

      5. *-commutative 59.0%

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

      6. associate-*l* 59.0%

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

      7. times-frac 81.7%

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

      8. associate-+r+ 81.7%

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

   6. Applied egg-rr 81.7%

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

      1. clear-num 81.7%

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

      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 92.5%

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

Alternative 6: 95.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)\\ \mathbf{if}\;y \leq -5.7 \cdot 10^{+35}:\\ \;\;\;\;\frac{1}{t\_0}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ (+ y x) y) (+ y (+ x 1.0)))))
   (if (<= y -5.7e+35)
     (/ 1.0 t_0)
     (if (<= y 1.8e-17)
       (* (/ y (+ y x)) (/ x (* (+ y x) (+ x 1.0))))
       (/ (/ x y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = ((y + x) / y) * (y + (x + 1.0));
	double tmp;
	if (y <= -5.7e+35) {
		tmp = 1.0 / t_0;
	} else if (y <= 1.8e-17) {
		tmp = (y / (y + x)) * (x / ((y + x) * (x + 1.0)));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y + x) / y) * (y + (x + 1.0d0))
    if (y <= (-5.7d+35)) then
        tmp = 1.0d0 / t_0
    else if (y <= 1.8d-17) then
        tmp = (y / (y + x)) * (x / ((y + x) * (x + 1.0d0)))
    else
        tmp = (x / y) / t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = ((y + x) / y) * (y + (x + 1.0));
	double tmp;
	if (y <= -5.7e+35) {
		tmp = 1.0 / t_0;
	} else if (y <= 1.8e-17) {
		tmp = (y / (y + x)) * (x / ((y + x) * (x + 1.0)));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = ((y + x) / y) * (y + (x + 1.0))
	tmp = 0
	if y <= -5.7e+35:
		tmp = 1.0 / t_0
	elif y <= 1.8e-17:
		tmp = (y / (y + x)) * (x / ((y + x) * (x + 1.0)))
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(Float64(y + x) / y) * Float64(y + Float64(x + 1.0)))
	tmp = 0.0
	if (y <= -5.7e+35)
		tmp = Float64(1.0 / t_0);
	elseif (y <= 1.8e-17)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(Float64(y + x) * Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = ((y + x) / y) * (y + (x + 1.0));
	tmp = 0.0;
	if (y <= -5.7e+35)
		tmp = 1.0 / t_0;
	elseif (y <= 1.8e-17)
		tmp = (y / (y + x)) * (x / ((y + x) * (x + 1.0)));
	else
		tmp = (x / y) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.7e+35], N[(1.0 / t$95$0), $MachinePrecision], If[LessEqual[y, 1.8e-17], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(y + x), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.69999999999999993e35

   1. Initial program 50.4%

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

      1. +-commutative 50.4%

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

      2. +-commutative 50.4%

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

      3. +-commutative 50.4%

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

      4. *-commutative 50.4%

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

      5. distribute-rgt1-in 19.7%

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

      6. fma-define 50.4%

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

      7. +-commutative 50.4%

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

      8. +-commutative 50.4%

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

      9. cube-unmult 50.4%

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

      10. +-commutative 50.4%

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

   3. Simplified 50.4%

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

      1. *-commutative 50.4%

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

      2. fma-define 19.7%

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

      3. cube-mult 19.7%

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

      4. distribute-rgt1-in 50.4%

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

      5. *-commutative 50.4%

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

      6. associate-*l* 50.4%

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

      7. times-frac 81.9%

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

      8. associate-+r+ 81.9%

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

   6. Applied egg-rr 81.9%

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

      1. clear-num 81.9%

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

      2. associate-/r* 99.7%

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

      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 inf 34.3%

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

   if -5.69999999999999993e35 < y < 1.79999999999999997e-17

   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. Add Preprocessing

   3. Taylor expanded in x around inf 66.7%

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

      1. *-commutative 66.7%

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

      2. associate-*l* 66.7%

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

      3. times-frac 97.5%

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

      4. +-commutative 97.5%

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

      5. +-commutative 97.5%

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

   5. Applied egg-rr 97.5%

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

   if 1.79999999999999997e-17 < y

   1. Initial program 59.0%

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

      1. +-commutative 59.0%

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

      2. +-commutative 59.0%

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

      3. +-commutative 59.0%

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

      4. *-commutative 59.0%

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

      5. distribute-rgt1-in 57.7%

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

      6. fma-define 59.0%

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

      7. +-commutative 59.0%

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

      8. +-commutative 59.0%

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

      9. cube-unmult 59.0%

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

      10. +-commutative 59.0%

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

   3. Simplified 59.0%

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

      1. *-commutative 59.0%

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

      2. fma-define 57.7%

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

      3. cube-mult 57.7%

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

      4. distribute-rgt1-in 59.0%

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

      5. *-commutative 59.0%

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

      6. associate-*l* 59.0%

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

      7. times-frac 81.7%

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

      8. associate-+r+ 81.7%

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

   6. Applied egg-rr 81.7%

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

      1. clear-num 81.7%

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

      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 92.5%

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

Alternative 7: 86.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-187}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{+159}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.4e-187)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 2.95e+159)
     (/ x (* (+ y x) (+ y (+ x 1.0))))
     (* (/ y (+ y x)) (/ (/ x y) (+ x (+ y 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.4e-187) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 2.95e+159) {
		tmp = x / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = (y / (y + x)) * ((x / y) / (x + (y + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.4d-187) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 2.95d+159) then
        tmp = x / ((y + x) * (y + (x + 1.0d0)))
    else
        tmp = (y / (y + x)) * ((x / y) / (x + (y + 1.0d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.4e-187) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 2.95e+159) {
		tmp = x / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = (y / (y + x)) * ((x / y) / (x + (y + 1.0)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.4e-187:
		tmp = (y / x) / (x + 1.0)
	elif y <= 2.95e+159:
		tmp = x / ((y + x) * (y + (x + 1.0)))
	else:
		tmp = (y / (y + x)) * ((x / y) / (x + (y + 1.0)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.4e-187)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 2.95e+159)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(Float64(x / y) / Float64(x + Float64(y + 1.0))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.4e-187)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 2.95e+159)
		tmp = x / ((y + x) * (y + (x + 1.0)));
	else
		tmp = (y / (y + x)) * ((x / y) / (x + (y + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.4e-187], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.95e+159], N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(N[(x / y), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.4000000000000001e-187

   1. Initial program 56.1%

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

      1. +-commutative 56.1%

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

      2. +-commutative 56.1%

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

      3. +-commutative 56.1%

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

      4. *-commutative 56.1%

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

      5. distribute-rgt1-in 38.0%

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

      6. fma-define 56.1%

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

      7. +-commutative 56.1%

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

      8. +-commutative 56.1%

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

      9. cube-unmult 56.1%

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

      10. +-commutative 56.1%

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

   3. Simplified 56.1%

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

      1. *-commutative 56.1%

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

      2. fma-define 38.0%

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

      3. cube-mult 38.0%

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

      4. distribute-rgt1-in 56.1%

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

      5. *-commutative 56.1%

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

      6. associate-*l* 56.1%

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

      7. times-frac 92.4%

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

      8. associate-+r+ 92.4%

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

   6. Applied egg-rr 92.4%

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

      1. clear-num 92.3%

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

      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 99.5%

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

      6. metadata-eval 99.5%

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

      7. times-frac 99.5%

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

      8. *-un-lft-identity 99.5%

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

      9. *-un-lft-identity 99.5%

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

      10. +-commutative 99.5%

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

      11. +-commutative 99.5%

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

      12. +-commutative 99.5%

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

   8. Applied egg-rr 99.5%

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

   9. Taylor expanded in y around 0 54.4%

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

       1. associate-/r* 56.6%

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

       2. +-commutative 56.6%

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

   11. Simplified 56.6%

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

   if 3.4000000000000001e-187 < y < 2.94999999999999996e159

   1. Initial program 82.2%

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

      1. +-commutative 82.2%

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

      2. +-commutative 82.2%

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

      3. +-commutative 82.2%

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

      4. *-commutative 82.2%

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

      5. distribute-rgt1-in 76.2%

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

      6. fma-define 82.2%

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

      7. +-commutative 82.2%

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

      8. +-commutative 82.2%

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

      9. cube-unmult 82.2%

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

      10. +-commutative 82.2%

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

   3. Simplified 82.2%

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

      1. *-commutative 82.2%

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

      2. fma-define 76.2%

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

      3. cube-mult 76.2%

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

      4. distribute-rgt1-in 82.2%

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

      5. *-commutative 82.2%

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

      6. associate-*l* 82.2%

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

      7. times-frac 97.6%

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

      8. associate-+r+ 97.6%

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

   6. Applied egg-rr 97.6%

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

      1. associate-*r/ 97.5%

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

      2. +-commutative 97.5%

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

      3. +-commutative 97.5%

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

      4. +-commutative 97.5%

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

      5. associate-+l+ 97.5%

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

      6. +-commutative 97.5%

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

   8. Applied egg-rr 97.5%

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

   9. Taylor expanded in y around inf 80.6%

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

   if 2.94999999999999996e159 < y

   1. Initial program 38.3%

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

      1. +-commutative 38.3%

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

      2. +-commutative 38.3%

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

      3. +-commutative 38.3%

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

      4. *-commutative 38.3%

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

      5. distribute-rgt1-in 38.3%

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

      6. fma-define 38.3%

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

      7. +-commutative 38.3%

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

      8. +-commutative 38.3%

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

      9. cube-unmult 38.3%

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

      10. +-commutative 38.3%

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

   3. Simplified 38.3%

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

      1. *-commutative 38.3%

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

      2. fma-define 38.3%

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

      3. cube-mult 38.3%

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

      4. distribute-rgt1-in 38.3%

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

      5. *-commutative 38.3%

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

      6. associate-*l* 38.3%

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

      7. times-frac 67.5%

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

      8. associate-+r+ 67.5%

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

   6. Applied egg-rr 67.5%

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

      1. clear-num 67.5%

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

      2. associate-/r* 99.7%

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

      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 99.7%

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

      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. *-un-lft-identity 99.7%

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

      2. times-frac 99.7%

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

      3. +-commutative 99.7%

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

      4. clear-num 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in x around 0 88.8%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-187}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{+159}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+24}:\\ \;\;\;\;\frac{x}{y \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-182}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{elif}\;y \leq 3.15 \cdot 10^{+159}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -1e+24)
   (/ x (* y (+ x 1.0)))
   (if (<= y 3.8e-182)
     (/ y (+ y x))
     (if (<= y 3.15e+159) (/ x (* y (+ y 1.0))) (* (/ x y) (/ 1.0 y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -1e+24) {
		tmp = x / (y * (x + 1.0));
	} else if (y <= 3.8e-182) {
		tmp = y / (y + x);
	} else if (y <= 3.15e+159) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / 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 <= (-1d+24)) then
        tmp = x / (y * (x + 1.0d0))
    else if (y <= 3.8d-182) then
        tmp = y / (y + x)
    else if (y <= 3.15d+159) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= -1e+24) {
		tmp = x / (y * (x + 1.0));
	} else if (y <= 3.8e-182) {
		tmp = y / (y + x);
	} else if (y <= 3.15e+159) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -1e+24:
		tmp = x / (y * (x + 1.0))
	elif y <= 3.8e-182:
		tmp = y / (y + x)
	elif y <= 3.15e+159:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -1e+24)
		tmp = Float64(x / Float64(y * Float64(x + 1.0)));
	elseif (y <= 3.8e-182)
		tmp = Float64(y / Float64(y + x));
	elseif (y <= 3.15e+159)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1e+24)
		tmp = x / (y * (x + 1.0));
	elseif (y <= 3.8e-182)
		tmp = y / (y + x);
	elseif (y <= 3.15e+159)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) * (1.0 / 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, -1e+24], N[(x / N[(y * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e-182], N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.15e+159], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -9.9999999999999998e23

   1. Initial program 51.3%

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

   3. Taylor expanded in x around inf 38.6%

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

   4. Taylor expanded in y around inf 42.7%

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

   if -9.9999999999999998e23 < y < 3.8000000000000003e-182

   1. Initial program 61.6%

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

      1. +-commutative 61.6%

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

      2. +-commutative 61.6%

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

      3. +-commutative 61.6%

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

      4. *-commutative 61.6%

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

      5. distribute-rgt1-in 51.7%

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

      6. fma-define 61.6%

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

      7. +-commutative 61.6%

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

      8. +-commutative 61.6%

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

      9. cube-unmult 61.6%

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

      10. +-commutative 61.6%

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

   3. Simplified 61.6%

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

      1. *-commutative 61.6%

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

      2. fma-define 51.8%

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

      3. cube-mult 51.7%

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

      4. distribute-rgt1-in 61.6%

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

      5. *-commutative 61.6%

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

      6. associate-*l* 61.6%

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

      7. times-frac 99.9%

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

      8. associate-+r+ 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around 0 76.6%

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

      1. +-commutative 76.6%

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

   9. Simplified 76.6%

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

       1. associate-*l/ 76.6%

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

       2. un-div-inv 76.6%

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

   11. Applied egg-rr 76.6%

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

   12. Taylor expanded in x around 0 53.0%

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

   if 3.8000000000000003e-182 < y < 3.15000000000000003e159

   1. Initial program 81.3%

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

      1. associate-/l* 85.6%

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

      2. associate-+l+ 85.6%

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

   3. Simplified 85.6%

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

   5. Taylor expanded in x around 0 64.2%

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

   if 3.15000000000000003e159 < y

   1. Initial program 38.3%

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

      1. associate-/l* 67.5%

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

      2. associate-+l+ 67.5%

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

   3. Simplified 67.5%

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

   5. Taylor expanded in x around 0 67.5%

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

      1. associate-/r* 88.1%

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

      2. +-commutative 88.1%

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

      3. div-inv 88.1%

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

   7. Applied egg-rr 88.1%

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

   8. Taylor expanded in y around inf 88.1%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+24}:\\ \;\;\;\;\frac{x}{y \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-182}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{elif}\;y \leq 3.15 \cdot 10^{+159}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y \cdot \left(x + 1\right)}\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-182}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{elif}\;y \leq 1.62:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y (+ x 1.0)))))
   (if (<= y -6.2e+19)
     t_0
     (if (<= y 3.8e-182) (/ y (+ y x)) (if (<= y 1.62) t_0 (/ (/ x y) y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y * (x + 1.0));
	double tmp;
	if (y <= -6.2e+19) {
		tmp = t_0;
	} else if (y <= 3.8e-182) {
		tmp = y / (y + x);
	} else if (y <= 1.62) {
		tmp = t_0;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y * (x + 1.0d0))
    if (y <= (-6.2d+19)) then
        tmp = t_0
    else if (y <= 3.8d-182) then
        tmp = y / (y + x)
    else if (y <= 1.62d0) then
        tmp = t_0
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y * (x + 1.0));
	double tmp;
	if (y <= -6.2e+19) {
		tmp = t_0;
	} else if (y <= 3.8e-182) {
		tmp = y / (y + x);
	} else if (y <= 1.62) {
		tmp = t_0;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y * (x + 1.0))
	tmp = 0
	if y <= -6.2e+19:
		tmp = t_0
	elif y <= 3.8e-182:
		tmp = y / (y + x)
	elif y <= 1.62:
		tmp = t_0
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y * Float64(x + 1.0)))
	tmp = 0.0
	if (y <= -6.2e+19)
		tmp = t_0;
	elseif (y <= 3.8e-182)
		tmp = Float64(y / Float64(y + x));
	elseif (y <= 1.62)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y * (x + 1.0));
	tmp = 0.0;
	if (y <= -6.2e+19)
		tmp = t_0;
	elseif (y <= 3.8e-182)
		tmp = y / (y + x);
	elseif (y <= 1.62)
		tmp = t_0;
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.2e+19], t$95$0, If[LessEqual[y, 3.8e-182], N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.62], t$95$0, N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.2e19 or 3.8000000000000003e-182 < y < 1.6200000000000001

   1. Initial program 65.5%

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

   3. Taylor expanded in x around inf 57.5%

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

   4. Taylor expanded in y around inf 46.9%

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

   if -6.2e19 < y < 3.8000000000000003e-182

   1. Initial program 61.6%

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

      1. +-commutative 61.6%

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

      2. +-commutative 61.6%

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

      3. +-commutative 61.6%

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

      4. *-commutative 61.6%

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

      5. distribute-rgt1-in 51.7%

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

      6. fma-define 61.6%

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

      7. +-commutative 61.6%

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

      8. +-commutative 61.6%

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

      9. cube-unmult 61.6%

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

      10. +-commutative 61.6%

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

   3. Simplified 61.6%

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

      1. *-commutative 61.6%

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

      2. fma-define 51.8%

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

      3. cube-mult 51.7%

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

      4. distribute-rgt1-in 61.6%

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

      5. *-commutative 61.6%

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

      6. associate-*l* 61.6%

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

      7. times-frac 99.9%

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

      8. associate-+r+ 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around 0 76.6%

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

      1. +-commutative 76.6%

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

   9. Simplified 76.6%

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

       1. associate-*l/ 76.6%

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

       2. un-div-inv 76.6%

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

   11. Applied egg-rr 76.6%

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

   12. Taylor expanded in x around 0 53.0%

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

   if 1.6200000000000001 < y

   1. Initial program 57.4%

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

      1. +-commutative 57.4%

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

      2. +-commutative 57.4%

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

      3. +-commutative 57.4%

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

      4. *-commutative 57.4%

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

      5. distribute-rgt1-in 56.1%

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

      6. fma-define 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 56.1%

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

      3. cube-mult 56.1%

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

      4. distribute-rgt1-in 57.4%

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

      5. *-commutative 57.4%

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

      6. associate-*l* 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 81.0%

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

      8. associate-+r+ 81.0%

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

   6. Applied egg-rr 81.0%

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

      1. clear-num 81.0%

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

      2. associate-/r* 99.7%

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

      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 y around inf 81.2%

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

   10. Taylor expanded in x around 0 80.8%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{x}{y \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-182}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{elif}\;y \leq 1.62:\\ \;\;\;\;\frac{x}{y \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-187}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+159}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{x + \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.4e-187)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 7e+159)
     (/ x (* (+ y x) (+ y (+ x 1.0))))
     (/ (/ x (+ y x)) (+ x (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.4e-187) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 7e+159) {
		tmp = x / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = (x / (y + x)) / (x + (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.4d-187) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 7d+159) then
        tmp = x / ((y + x) * (y + (x + 1.0d0)))
    else
        tmp = (x / (y + x)) / (x + (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.4e-187) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 7e+159) {
		tmp = x / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = (x / (y + x)) / (x + (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.4e-187:
		tmp = (y / x) / (x + 1.0)
	elif y <= 7e+159:
		tmp = x / ((y + x) * (y + (x + 1.0)))
	else:
		tmp = (x / (y + x)) / (x + (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.4e-187)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 7e+159)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(x + Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.4e-187)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 7e+159)
		tmp = x / ((y + x) * (y + (x + 1.0)));
	else
		tmp = (x / (y + x)) / (x + (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.4e-187], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7e+159], N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.4000000000000001e-187

   1. Initial program 56.1%

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

      1. +-commutative 56.1%

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

      2. +-commutative 56.1%

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

      3. +-commutative 56.1%

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

      4. *-commutative 56.1%

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

      5. distribute-rgt1-in 38.0%

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

      6. fma-define 56.1%

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

      7. +-commutative 56.1%

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

      8. +-commutative 56.1%

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

      9. cube-unmult 56.1%

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

      10. +-commutative 56.1%

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

   3. Simplified 56.1%

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

      1. *-commutative 56.1%

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

      2. fma-define 38.0%

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

      3. cube-mult 38.0%

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

      4. distribute-rgt1-in 56.1%

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

      5. *-commutative 56.1%

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

      6. associate-*l* 56.1%

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

      7. times-frac 92.4%

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

      8. associate-+r+ 92.4%

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

   6. Applied egg-rr 92.4%

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

      1. clear-num 92.3%

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

      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 99.5%

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

      6. metadata-eval 99.5%

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

      7. times-frac 99.5%

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

      8. *-un-lft-identity 99.5%

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

      9. *-un-lft-identity 99.5%

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

      10. +-commutative 99.5%

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

      11. +-commutative 99.5%

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

      12. +-commutative 99.5%

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

   8. Applied egg-rr 99.5%

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

   9. Taylor expanded in y around 0 54.4%

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

       1. associate-/r* 56.6%

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

       2. +-commutative 56.6%

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

   11. Simplified 56.6%

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

   if 3.4000000000000001e-187 < y < 6.9999999999999999e159

   1. Initial program 82.2%

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

      1. +-commutative 82.2%

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

      2. +-commutative 82.2%

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

      3. +-commutative 82.2%

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

      4. *-commutative 82.2%

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

      5. distribute-rgt1-in 76.2%

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

      6. fma-define 82.2%

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

      7. +-commutative 82.2%

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

      8. +-commutative 82.2%

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

      9. cube-unmult 82.2%

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

      10. +-commutative 82.2%

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

   3. Simplified 82.2%

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

      1. *-commutative 82.2%

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

      2. fma-define 76.2%

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

      3. cube-mult 76.2%

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

      4. distribute-rgt1-in 82.2%

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

      5. *-commutative 82.2%

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

      6. associate-*l* 82.2%

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

      7. times-frac 97.6%

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

      8. associate-+r+ 97.6%

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

   6. Applied egg-rr 97.6%

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

      1. associate-*r/ 97.5%

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

      2. +-commutative 97.5%

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

      3. +-commutative 97.5%

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

      4. +-commutative 97.5%

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

      5. associate-+l+ 97.5%

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

      6. +-commutative 97.5%

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

   8. Applied egg-rr 97.5%

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

   9. Taylor expanded in y around inf 80.6%

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

   if 6.9999999999999999e159 < y

   1. Initial program 38.3%

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

      1. +-commutative 38.3%

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

      2. +-commutative 38.3%

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

      3. +-commutative 38.3%

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

      4. *-commutative 38.3%

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

      5. distribute-rgt1-in 38.3%

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

      6. fma-define 38.3%

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

      7. +-commutative 38.3%

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

      8. +-commutative 38.3%

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

      9. cube-unmult 38.3%

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

      10. +-commutative 38.3%

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

   3. Simplified 38.3%

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

      1. *-commutative 38.3%

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

      2. fma-define 38.3%

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

      3. cube-mult 38.3%

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

      4. distribute-rgt1-in 38.3%

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

      5. *-commutative 38.3%

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

      6. associate-*l* 38.3%

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

      7. times-frac 67.5%

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

      8. associate-+r+ 67.5%

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

   6. Applied egg-rr 67.5%

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

      1. clear-num 67.5%

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

      2. associate-/r* 99.7%

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

      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 99.7%

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

      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. *-un-lft-identity 99.7%

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

      2. times-frac 99.7%

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

      3. +-commutative 99.7%

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

      4. clear-num 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in y around inf 88.8%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-187}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+159}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{x + \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-187}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.4e-187)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 2.65e+149)
     (/ x (* (+ y x) (+ y (+ x 1.0))))
     (* (/ x (+ y x)) (/ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.4e-187) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 2.65e+149) {
		tmp = x / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = (x / (y + x)) * (1.0 / 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 <= 3.4d-187) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 2.65d+149) then
        tmp = x / ((y + x) * (y + (x + 1.0d0)))
    else
        tmp = (x / (y + x)) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.4e-187) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 2.65e+149) {
		tmp = x / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = (x / (y + x)) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.4e-187:
		tmp = (y / x) / (x + 1.0)
	elif y <= 2.65e+149:
		tmp = x / ((y + x) * (y + (x + 1.0)))
	else:
		tmp = (x / (y + x)) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.4e-187)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 2.65e+149)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.4e-187)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 2.65e+149)
		tmp = x / ((y + x) * (y + (x + 1.0)));
	else
		tmp = (x / (y + x)) * (1.0 / 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, 3.4e-187], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.65e+149], N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.4000000000000001e-187

   1. Initial program 56.1%

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

      1. +-commutative 56.1%

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

      2. +-commutative 56.1%

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

      3. +-commutative 56.1%

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

      4. *-commutative 56.1%

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

      5. distribute-rgt1-in 38.0%

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

      6. fma-define 56.1%

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

      7. +-commutative 56.1%

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

      8. +-commutative 56.1%

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

      9. cube-unmult 56.1%

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

      10. +-commutative 56.1%

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

   3. Simplified 56.1%

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

      1. *-commutative 56.1%

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

      2. fma-define 38.0%

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

      3. cube-mult 38.0%

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

      4. distribute-rgt1-in 56.1%

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

      5. *-commutative 56.1%

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

      6. associate-*l* 56.1%

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

      7. times-frac 92.4%

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

      8. associate-+r+ 92.4%

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

   6. Applied egg-rr 92.4%

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

      1. clear-num 92.3%

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

      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 99.5%

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

      6. metadata-eval 99.5%

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

      7. times-frac 99.5%

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

      8. *-un-lft-identity 99.5%

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

      9. *-un-lft-identity 99.5%

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

      10. +-commutative 99.5%

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

      11. +-commutative 99.5%

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

      12. +-commutative 99.5%

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

   8. Applied egg-rr 99.5%

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

   9. Taylor expanded in y around 0 54.4%

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

       1. associate-/r* 56.6%

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

       2. +-commutative 56.6%

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

   11. Simplified 56.6%

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

   if 3.4000000000000001e-187 < y < 2.65000000000000016e149

   1. Initial program 83.0%

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

      1. +-commutative 83.0%

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

      2. +-commutative 83.0%

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

      3. +-commutative 83.0%

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

      4. *-commutative 83.0%

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

      5. distribute-rgt1-in 76.8%

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

      6. fma-define 83.0%

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

      7. +-commutative 83.0%

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

      8. +-commutative 83.0%

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

      9. cube-unmult 83.0%

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

      10. +-commutative 83.0%

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

   3. Simplified 83.0%

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

      1. *-commutative 83.0%

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

      2. fma-define 76.8%

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

      3. cube-mult 76.8%

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

      4. distribute-rgt1-in 83.0%

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

      5. *-commutative 83.0%

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

      6. associate-*l* 83.0%

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

      7. times-frac 98.7%

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

      8. associate-+r+ 98.7%

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

   6. Applied egg-rr 98.7%

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

      1. associate-*r/ 98.6%

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

      2. +-commutative 98.6%

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

      3. +-commutative 98.6%

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

      4. +-commutative 98.6%

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

      5. associate-+l+ 98.6%

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

      6. +-commutative 98.6%

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

   8. Applied egg-rr 98.6%

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

   9. Taylor expanded in y around inf 81.3%

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

   if 2.65000000000000016e149 < y

   1. Initial program 38.9%

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

      1. +-commutative 38.9%

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

      2. +-commutative 38.9%

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

      3. +-commutative 38.9%

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

      4. *-commutative 38.9%

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

      5. distribute-rgt1-in 38.8%

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

      6. fma-define 38.9%

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

      7. +-commutative 38.9%

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

      8. +-commutative 38.9%

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

      9. cube-unmult 38.9%

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

      10. +-commutative 38.9%

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

   3. Simplified 38.9%

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

      1. *-commutative 38.9%

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

      2. fma-define 38.8%

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

      3. cube-mult 38.8%

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

      4. distribute-rgt1-in 38.9%

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

      5. *-commutative 38.9%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 38.9%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 66.8%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 66.8%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 66.8%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   7. Step-by-step derivation

      1. clear-num 66.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + y}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)} \]

      2. associate-/r* 99.7%

         \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + \left(y + 1\right)}} \]

      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 99.7%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)}} \]

      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 inf 86.7%

      \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y}} \]
   10. Step-by-step derivation

       1. div-inv 86.7%

          \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{1}{y}} \]

       2. +-commutative 86.7%

          \[\leadsto \frac{x}{\color{blue}{x + y}} \cdot \frac{1}{y} \]

   11. Applied egg-rr 86.7%

       \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{1}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-187}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-182}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -2.3e+18)
   (/ x (* y y))
   (if (<= y 3.8e-182) (/ y (+ y x)) (if (<= y 1.0) (/ x y) (/ (/ x y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -2.3e+18) {
		tmp = x / (y * y);
	} else if (y <= 3.8e-182) {
		tmp = y / (y + x);
	} else 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 <= (-2.3d+18)) then
        tmp = x / (y * y)
    else if (y <= 3.8d-182) then
        tmp = y / (y + x)
    else 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 <= -2.3e+18) {
		tmp = x / (y * y);
	} else if (y <= 3.8e-182) {
		tmp = y / (y + x);
	} else 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 <= -2.3e+18:
		tmp = x / (y * y)
	elif y <= 3.8e-182:
		tmp = y / (y + x)
	elif 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 <= -2.3e+18)
		tmp = Float64(x / Float64(y * y));
	elseif (y <= 3.8e-182)
		tmp = Float64(y / Float64(y + x));
	elseif (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.3e+18)
		tmp = x / (y * y);
	elseif (y <= 3.8e-182)
		tmp = y / (y + x);
	elseif (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, -2.3e+18], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e-182], N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.3e18

   1. Initial program 52.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 70.6%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 70.6%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 70.6%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 69.3%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   6. Taylor expanded in y around inf 69.3%

      \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

   if -2.3e18 < y < 3.8000000000000003e-182

   1. Initial program 61.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 61.1%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 61.1%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 61.1%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 61.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 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 61.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 61.1%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 61.1%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 61.2%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 61.2%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 61.2%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 61.2%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 51.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 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 61.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 61.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 61.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 99.9%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 99.9%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

   7. Taylor expanded in y around 0 77.5%

      \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{1 + x}} \]
   8. Step-by-step derivation

      1. +-commutative 77.5%

         \[\leadsto \frac{y}{x + y} \cdot \frac{1}{\color{blue}{x + 1}} \]

   9. Simplified 77.5%

      \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{x + 1}} \]
   10. Step-by-step derivation

       1. associate-*l/ 77.5%

          \[\leadsto \color{blue}{\frac{y \cdot \frac{1}{x + 1}}{x + y}} \]

       2. un-div-inv 77.5%

          \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{x + y} \]

   11. Applied egg-rr 77.5%

       \[\leadsto \color{blue}{\frac{\frac{y}{x + 1}}{x + y}} \]

   12. Taylor expanded in x around 0 53.6%

       \[\leadsto \frac{\color{blue}{y}}{x + y} \]

   if 3.8000000000000003e-182 < y < 1

   1. Initial program 85.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 91.0%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 91.0%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 91.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 54.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   6. Taylor expanded in y around 0 52.5%

      \[\leadsto \color{blue}{\frac{x}{y}} \]

   if 1 < y

   1. Initial program 57.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 57.4%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 57.4%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 57.4%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 57.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 56.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 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 56.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 56.1%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 57.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 57.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 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 81.0%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 81.0%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 81.0%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   7. Step-by-step derivation

      1. clear-num 81.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + y}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)} \]

      2. associate-/r* 99.7%

         \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + \left(y + 1\right)}} \]

      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 y around inf 81.2%

      \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y}} \]

   10. Taylor expanded in x around 0 80.8%

       \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-182}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{+18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-182}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y y))))
   (if (<= y -2.3e+18)
     t_0
     (if (<= y 3.7e-182) (/ y (+ y x)) (if (<= y 1.0) (/ x y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (y <= -2.3e+18) {
		tmp = t_0;
	} else if (y <= 3.7e-182) {
		tmp = y / (y + x);
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y * y)
    if (y <= (-2.3d+18)) then
        tmp = t_0
    else if (y <= 3.7d-182) then
        tmp = y / (y + x)
    else if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (y <= -2.3e+18) {
		tmp = t_0;
	} else if (y <= 3.7e-182) {
		tmp = y / (y + x);
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y * y)
	tmp = 0
	if y <= -2.3e+18:
		tmp = t_0
	elif y <= 3.7e-182:
		tmp = y / (y + x)
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y * y))
	tmp = 0.0
	if (y <= -2.3e+18)
		tmp = t_0;
	elseif (y <= 3.7e-182)
		tmp = Float64(y / Float64(y + x));
	elseif (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y * y);
	tmp = 0.0;
	if (y <= -2.3e+18)
		tmp = t_0;
	elseif (y <= 3.7e-182)
		tmp = y / (y + x);
	elseif (y <= 1.0)
		tmp = x / y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.3e+18], t$95$0, If[LessEqual[y, 3.7e-182], N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.3e18 or 1 < y

   1. Initial program 55.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 72.3%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 72.3%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 72.3%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 70.4%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   6. Taylor expanded in y around inf 69.9%

      \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

   if -2.3e18 < y < 3.69999999999999971e-182

   1. Initial program 61.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 61.1%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 61.1%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 61.1%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 61.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 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 61.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 61.1%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 61.1%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 61.2%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 61.2%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 61.2%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 61.2%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 51.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 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 61.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 61.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 61.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 99.9%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 99.9%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

   7. Taylor expanded in y around 0 77.5%

      \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{1 + x}} \]
   8. Step-by-step derivation

      1. +-commutative 77.5%

         \[\leadsto \frac{y}{x + y} \cdot \frac{1}{\color{blue}{x + 1}} \]

   9. Simplified 77.5%

      \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{x + 1}} \]
   10. Step-by-step derivation

       1. associate-*l/ 77.5%

          \[\leadsto \color{blue}{\frac{y \cdot \frac{1}{x + 1}}{x + y}} \]

       2. un-div-inv 77.5%

          \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{x + y} \]

   11. Applied egg-rr 77.5%

       \[\leadsto \color{blue}{\frac{\frac{y}{x + 1}}{x + y}} \]

   12. Taylor expanded in x around 0 53.6%

       \[\leadsto \frac{\color{blue}{y}}{x + y} \]

   if 3.69999999999999971e-182 < y < 1

   1. Initial program 85.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 91.0%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 91.0%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 91.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 54.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   6. Taylor expanded in y around 0 52.5%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-182}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{-158}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+160}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 8.2e-158)
   (/ y (* x (+ x 1.0)))
   (if (<= y 1.9e+160) (/ x (* y (+ y 1.0))) (* (/ x y) (/ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 8.2e-158) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 1.9e+160) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / 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 <= 8.2d-158) then
        tmp = y / (x * (x + 1.0d0))
    else if (y <= 1.9d+160) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 8.2e-158) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 1.9e+160) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 8.2e-158:
		tmp = y / (x * (x + 1.0))
	elif y <= 1.9e+160:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 8.2e-158)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (y <= 1.9e+160)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8.2e-158)
		tmp = y / (x * (x + 1.0));
	elseif (y <= 1.9e+160)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) * (1.0 / 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, 8.2e-158], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.9e+160], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 8.20000000000000008e-158

   1. Initial program 57.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 71.9%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 71.9%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 71.9%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 55.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 55.9%

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   7. Simplified 55.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]

   if 8.20000000000000008e-158 < y < 1.90000000000000006e160

   1. Initial program 82.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 87.3%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 87.3%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 87.3%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 65.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   if 1.90000000000000006e160 < y

   1. Initial program 38.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 67.5%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 67.5%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 67.5%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 67.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 88.1%

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

      2. +-commutative 88.1%

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y + 1}} \]

      3. div-inv 88.1%

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{1}{y + 1}} \]

   7. Applied egg-rr 88.1%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{1}{y + 1}} \]

   8. Taylor expanded in y around inf 88.1%

      \[\leadsto \frac{x}{y} \cdot \color{blue}{\frac{1}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{-158}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+160}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 48.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-61} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.7e-61) (not (<= y 1.0))) (/ x (* y y)) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if ((y <= -1.7e-61) || !(y <= 1.0)) {
		tmp = x / (y * y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.7d-61)) .or. (.not. (y <= 1.0d0))) then
        tmp = x / (y * y)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.7e-61) || !(y <= 1.0)) {
		tmp = x / (y * y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if (y <= -1.7e-61) or not (y <= 1.0):
		tmp = x / (y * y)
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if ((y <= -1.7e-61) || !(y <= 1.0))
		tmp = Float64(x / Float64(y * y));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.7e-61) || ~((y <= 1.0)))
		tmp = x / (y * y);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[Or[LessEqual[y, -1.7e-61], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6999999999999999e-61 or 1 < y

   1. Initial program 57.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 73.5%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 73.5%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 73.5%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 65.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   6. Taylor expanded in y around inf 64.3%

      \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]

   if -1.6999999999999999e-61 < y < 1

   1. Initial program 67.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 78.6%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 78.6%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 78.6%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 35.9%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   6. Taylor expanded in y around 0 35.3%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-61} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 81.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.35 \cdot 10^{-166}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.35e-166) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ y x)) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.35e-166) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + x)) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.35d-166) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (y + x)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.35e-166) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + x)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.35e-166:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (y + x)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.35e-166)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.35e-166)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (y + x)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 2.35e-166], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.35000000000000007e-166

   1. Initial program 57.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 57.8%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 57.8%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 57.8%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 57.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 39.3%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 57.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 57.8%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 57.8%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 57.9%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 57.9%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 57.9%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 57.9%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 39.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 39.3%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 57.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 57.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 57.9%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 92.8%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 92.8%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 92.8%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   7. Step-by-step derivation

      1. clear-num 92.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + y}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)} \]

      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 99.6%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)}} \]

      6. metadata-eval 99.6%

         \[\leadsto \frac{\color{blue}{\frac{1}{1}} \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      7. times-frac 99.6%

         \[\leadsto \frac{\color{blue}{\frac{1 \cdot x}{1 \cdot \left(x + y\right)}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      8. *-un-lft-identity 99.6%

         \[\leadsto \frac{\frac{\color{blue}{x}}{1 \cdot \left(x + y\right)}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      9. *-un-lft-identity 99.6%

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      10. +-commutative 99.6%

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      11. +-commutative 99.6%

          \[\leadsto \frac{\frac{x}{y + x}}{\frac{\color{blue}{y + x}}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      12. +-commutative 99.6%

          \[\leadsto \frac{\frac{x}{y + x}}{\frac{y + x}{y} \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

   8. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)}} \]

   9. Taylor expanded in y around 0 56.3%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   10. Step-by-step derivation

       1. associate-/r* 58.4%

          \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

       2. +-commutative 58.4%

          \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   11. Simplified 58.4%

       \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if 2.35000000000000007e-166 < y

   1. Initial program 66.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 66.8%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 66.8%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 66.8%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 66.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 64.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 66.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 66.8%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 66.8%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 66.8%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 66.8%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 66.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 66.8%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 64.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 64.2%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 66.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 66.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 66.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 87.1%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 87.1%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 87.1%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   7. Step-by-step derivation

      1. clear-num 87.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + y}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)} \]

      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 99.2%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)}} \]

      6. metadata-eval 99.2%

         \[\leadsto \frac{\color{blue}{\frac{1}{1}} \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      7. times-frac 99.2%

         \[\leadsto \frac{\color{blue}{\frac{1 \cdot x}{1 \cdot \left(x + y\right)}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      8. *-un-lft-identity 99.2%

         \[\leadsto \frac{\frac{\color{blue}{x}}{1 \cdot \left(x + y\right)}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      9. *-un-lft-identity 99.2%

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      10. +-commutative 99.2%

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      11. +-commutative 99.2%

          \[\leadsto \frac{\frac{x}{y + x}}{\frac{\color{blue}{y + x}}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      12. +-commutative 99.2%

          \[\leadsto \frac{\frac{x}{y + x}}{\frac{y + x}{y} \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

   8. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)}} \]

   9. Taylor expanded in x around 0 74.1%

      \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{1 + y}} \]
   10. Step-by-step derivation

       1. +-commutative 74.1%

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + 1}} \]

   11. Simplified 74.1%

       \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + 1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 82.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.7e-151) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.7e-151) {
		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 <= 1.7d-151) 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 <= 1.7e-151) {
		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 <= 1.7e-151:
		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 <= 1.7e-151)
		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 <= 1.7e-151)
		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, 1.7e-151], 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 < 1.7000000000000001e-151

   1. Initial program 57.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 57.4%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 57.4%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 57.4%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 57.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 39.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 57.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 57.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 57.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 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 39.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 39.1%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 57.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 57.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 57.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 92.8%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 92.8%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 92.8%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   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 99.6%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)}} \]

      6. metadata-eval 99.6%

         \[\leadsto \frac{\color{blue}{\frac{1}{1}} \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      7. times-frac 99.6%

         \[\leadsto \frac{\color{blue}{\frac{1 \cdot x}{1 \cdot \left(x + y\right)}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      8. *-un-lft-identity 99.6%

         \[\leadsto \frac{\frac{\color{blue}{x}}{1 \cdot \left(x + y\right)}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      9. *-un-lft-identity 99.6%

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      10. +-commutative 99.6%

          \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      11. +-commutative 99.6%

          \[\leadsto \frac{\frac{x}{y + x}}{\frac{\color{blue}{y + x}}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      12. +-commutative 99.6%

          \[\leadsto \frac{\frac{x}{y + x}}{\frac{y + x}{y} \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

   8. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)}} \]

   9. Taylor expanded in y around 0 55.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   10. Step-by-step derivation

       1. associate-/r* 58.0%

          \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

       2. +-commutative 58.0%

          \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   11. Simplified 58.0%

       \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if 1.7000000000000001e-151 < y

   1. Initial program 67.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 80.5%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 80.5%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 80.5%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 66.4%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 73.5%

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

      2. +-commutative 73.5%

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y + 1}} \]

   7. Simplified 73.5%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 80.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{-151}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.7e-151) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.7e-151) {
		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 <= 1.7d-151) 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 <= 1.7e-151) {
		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 <= 1.7e-151:
		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 <= 1.7e-151)
		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 (y <= 1.7e-151)
		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, 1.7e-151], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.7000000000000001e-151

   1. Initial program 57.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 71.9%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 71.9%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 71.9%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 55.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 55.9%

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   7. Simplified 55.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]

   if 1.7000000000000001e-151 < y

   1. Initial program 67.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 80.5%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 80.5%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 80.5%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 66.4%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 73.5%

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

      2. +-commutative 73.5%

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y + 1}} \]

   7. Simplified 73.5%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 27.3% 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 61.8%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l* 75.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   2. associate-+l+ 75.7%

      \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

3. Simplified 75.7%

   \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 53.2%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

6. Taylor expanded in y around 0 29.7%

   \[\leadsto \color{blue}{\frac{x}{y}} \]
7. Step-by-step derivation

   1. clear-num 29.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]

   2. inv-pow 29.7%

      \[\leadsto \color{blue}{{\left(\frac{y}{x}\right)}^{-1}} \]

8. Applied egg-rr 29.7%

   \[\leadsto \color{blue}{{\left(\frac{y}{x}\right)}^{-1}} \]
9. Step-by-step derivation

   1. unpow-1 29.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]

10. Simplified 29.7%

    \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]
11. Add Preprocessing

Alternative 20: 26.9% 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 61.8%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l* 75.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   2. associate-+l+ 75.7%

      \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

3. Simplified 75.7%

   \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 53.2%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

6. Taylor expanded in y around 0 29.7%

   \[\leadsto \color{blue}{\frac{x}{y}} \]
7. Add Preprocessing

Alternative 21: 4.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{1}{y} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ 1.0 y))

C

assert(x < y);
double code(double x, double y) {
	return 1.0 / y;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / y
end function

Java

assert x < y;
public static double code(double x, double y) {
	return 1.0 / y;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return 1.0 / y

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(1.0 / y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 1.0 / y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(1.0 / y), $MachinePrecision]

Derivation

1. Initial program 61.8%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. +-commutative 61.8%

      \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. +-commutative 61.8%

      \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   3. +-commutative 61.8%

      \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

   4. *-commutative 61.8%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   5. distribute-rgt1-in 50.4%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   6. fma-define 61.8%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

   7. +-commutative 61.8%

      \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   8. +-commutative 61.8%

      \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   9. cube-unmult 61.9%

      \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

   10. +-commutative 61.9%

       \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

3. Simplified 61.9%

   \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative 61.9%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

   2. fma-define 50.4%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   3. cube-mult 50.4%

      \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   4. distribute-rgt1-in 61.8%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   5. *-commutative 61.8%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   6. associate-*l* 61.9%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

   7. times-frac 90.3%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   8. associate-+r+ 90.3%

      \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

6. Applied egg-rr 90.3%

   \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
7. Step-by-step derivation

   1. clear-num 90.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{x + y}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)} \]

   2. associate-/r* 99.7%

      \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + \left(y + 1\right)}} \]

   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 99.4%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)}} \]

   6. metadata-eval 99.4%

      \[\leadsto \frac{\color{blue}{\frac{1}{1}} \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

   7. times-frac 99.4%

      \[\leadsto \frac{\color{blue}{\frac{1 \cdot x}{1 \cdot \left(x + y\right)}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

   8. *-un-lft-identity 99.4%

      \[\leadsto \frac{\frac{\color{blue}{x}}{1 \cdot \left(x + y\right)}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

   9. *-un-lft-identity 99.4%

      \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

   10. +-commutative 99.4%

       \[\leadsto \frac{\frac{x}{\color{blue}{y + x}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

   11. +-commutative 99.4%

       \[\leadsto \frac{\frac{x}{y + x}}{\frac{\color{blue}{y + x}}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

   12. +-commutative 99.4%

       \[\leadsto \frac{\frac{x}{y + x}}{\frac{y + x}{y} \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

8. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)}} \]

9. Taylor expanded in y around inf 42.4%

   \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y}} \]

10. Taylor expanded in x around inf 4.4%

    \[\leadsto \color{blue}{\frac{1}{y}} \]
11. 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 2024141 
(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))))