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

Percentage Accurate: 69.8% → 99.3%

Time: 17.7s

Alternatives: 17

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: 69.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{\frac{y}{y + x}}{\left(x + \mathsf{fma}\left(2, y, \frac{y}{\frac{x}{y + 1}}\right)\right) + 1} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (/ (/ y (+ y x)) (+ (+ x (fma 2.0 y (/ y (/ x (+ y 1.0))))) 1.0)))

C

assert(x < y);
double code(double x, double y) {
	return (y / (y + x)) / ((x + fma(2.0, y, (y / (x / (y + 1.0))))) + 1.0);
}

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(Float64(y / Float64(y + x)) / Float64(Float64(x + fma(2.0, y, Float64(y / Float64(x / Float64(y + 1.0))))) + 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[(2.0 * y + N[(y / N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.0%

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

   1. times-frac 86.7%

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

   2. +-commutative 86.7%

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

   3. +-commutative 86.7%

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

   4. +-commutative 86.7%

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

   5. times-frac 66.0%

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

   6. associate-*l/ 80.0%

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

   7. *-commutative 80.0%

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

   8. *-commutative 80.0%

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

   9. distribute-rgt1-in 61.2%

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

   10. fma-def 80.0%

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

   11. +-commutative 80.0%

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

   12. +-commutative 80.0%

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

   13. cube-unmult 80.0%

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

   14. +-commutative 80.0%

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

3. Simplified 80.0%

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

   1. associate-*r/ 66.0%

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

   2. fma-udef 51.3%

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

   3. cube-mult 51.3%

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

   4. distribute-rgt1-in 66.0%

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

   5. associate-+r+ 66.0%

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

   6. *-commutative 66.0%

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

   7. frac-times 86.7%

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

   8. *-commutative 86.7%

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

   9. associate-/r* 99.8%

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

   10. clear-num 99.7%

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

   11. frac-times 99.0%

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

   12. *-un-lft-identity 99.0%

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

   13. +-commutative 99.0%

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

   14. +-commutative 99.0%

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

   15. associate-+l+ 99.0%

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

   16. +-commutative 99.0%

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in x around 0 93.1%

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

   1. fma-def 93.1%

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

   2. associate-/l* 99.1%

      \[\leadsto \frac{\frac{y}{y + x}}{1 + \left(x + \mathsf{fma}\left(2, y, \color{blue}{\frac{y}{\frac{x}{1 + y}}}\right)\right)} \]

   3. +-commutative 99.1%

      \[\leadsto \frac{\frac{y}{y + x}}{1 + \left(x + \mathsf{fma}\left(2, y, \frac{y}{\frac{x}{\color{blue}{y + 1}}}\right)\right)} \]

9. Simplified 99.1%

   \[\leadsto \frac{\frac{y}{y + x}}{\color{blue}{1 + \left(x + \mathsf{fma}\left(2, y, \frac{y}{\frac{x}{y + 1}}\right)\right)}} \]

10. Final simplification 99.1%

    \[\leadsto \frac{\frac{y}{y + x}}{\left(x + \mathsf{fma}\left(2, y, \frac{y}{\frac{x}{y + 1}}\right)\right) + 1} \]
11. Add Preprocessing

Alternative 2: 95.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{y + x}\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{+118}:\\ \;\;\;\;\frac{t_0}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-16}:\\ \;\;\;\;y \cdot \frac{x}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot \frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y x))))
   (if (<= x -1.3e+118)
     (/ t_0 (+ x (+ y (+ y 1.0))))
     (if (<= x -1.4e-16)
       (* y (/ x (* (+ y x) (* (+ y x) (+ y (+ x 1.0))))))
       (/ (* t_0 (/ x (+ y 1.0))) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (x <= -1.3e+118) {
		tmp = t_0 / (x + (y + (y + 1.0)));
	} else if (x <= -1.4e-16) {
		tmp = y * (x / ((y + x) * ((y + x) * (y + (x + 1.0)))));
	} else {
		tmp = (t_0 * (x / (y + 1.0))) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (y + x)
    if (x <= (-1.3d+118)) then
        tmp = t_0 / (x + (y + (y + 1.0d0)))
    else if (x <= (-1.4d-16)) then
        tmp = y * (x / ((y + x) * ((y + x) * (y + (x + 1.0d0)))))
    else
        tmp = (t_0 * (x / (y + 1.0d0))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (x <= -1.3e+118) {
		tmp = t_0 / (x + (y + (y + 1.0)));
	} else if (x <= -1.4e-16) {
		tmp = y * (x / ((y + x) * ((y + x) * (y + (x + 1.0)))));
	} else {
		tmp = (t_0 * (x / (y + 1.0))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (y + x)
	tmp = 0
	if x <= -1.3e+118:
		tmp = t_0 / (x + (y + (y + 1.0)))
	elif x <= -1.4e-16:
		tmp = y * (x / ((y + x) * ((y + x) * (y + (x + 1.0)))))
	else:
		tmp = (t_0 * (x / (y + 1.0))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(y + x))
	tmp = 0.0
	if (x <= -1.3e+118)
		tmp = Float64(t_0 / Float64(x + Float64(y + Float64(y + 1.0))));
	elseif (x <= -1.4e-16)
		tmp = Float64(y * Float64(x / Float64(Float64(y + x) * Float64(Float64(y + x) * Float64(y + Float64(x + 1.0))))));
	else
		tmp = Float64(Float64(t_0 * Float64(x / Float64(y + 1.0))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (y + x);
	tmp = 0.0;
	if (x <= -1.3e+118)
		tmp = t_0 / (x + (y + (y + 1.0)));
	elseif (x <= -1.4e-16)
		tmp = y * (x / ((y + x) * ((y + x) * (y + (x + 1.0)))));
	else
		tmp = (t_0 * (x / (y + 1.0))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.3e+118], N[(t$95$0 / N[(x + N[(y + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.4e-16], N[(y * N[(x / N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000008e118

   1. Initial program 50.5%

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

      1. times-frac 79.1%

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

      2. +-commutative 79.1%

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

      3. +-commutative 79.1%

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

      4. +-commutative 79.1%

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

      5. times-frac 50.5%

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

      6. associate-*l/ 76.3%

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

      7. *-commutative 76.3%

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

      8. *-commutative 76.3%

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

      9. distribute-rgt1-in 6.4%

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

      10. fma-def 76.3%

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

      11. +-commutative 76.3%

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

      12. +-commutative 76.3%

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

      13. cube-unmult 76.3%

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

      14. +-commutative 76.3%

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

   3. Simplified 76.3%

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

      1. associate-*r/ 50.5%

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

      2. fma-udef 0.1%

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

      3. cube-mult 0.1%

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

      4. distribute-rgt1-in 50.5%

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

      5. associate-+r+ 50.5%

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

      6. *-commutative 50.5%

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

      7. frac-times 79.1%

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

      8. *-commutative 79.1%

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

      9. associate-/r* 99.8%

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

      10. clear-num 99.8%

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

      11. frac-times 99.9%

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

      12. *-un-lft-identity 99.9%

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

      13. +-commutative 99.9%

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

      14. +-commutative 99.9%

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

      15. associate-+l+ 99.9%

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

      16. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around -inf 85.5%

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

      1. mul-1-neg 85.5%

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

      2. unsub-neg 85.5%

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

      3. neg-mul-1 85.5%

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

      4. +-commutative 85.5%

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

      5. unsub-neg 85.5%

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

      6. distribute-lft-in 85.5%

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

      7. metadata-eval 85.5%

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

      8. neg-mul-1 85.5%

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

      9. unsub-neg 85.5%

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

   9. Simplified 85.5%

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

   if -1.30000000000000008e118 < x < -1.4000000000000001e-16

   1. Initial program 83.6%

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

      1. times-frac 92.4%

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

      2. +-commutative 92.4%

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

      3. +-commutative 92.4%

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

      4. +-commutative 92.4%

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

      5. times-frac 83.6%

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

      6. associate-*l/ 92.4%

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

      7. *-commutative 92.4%

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

      8. *-commutative 92.4%

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

      9. distribute-rgt1-in 75.1%

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

      10. fma-def 92.4%

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

      11. +-commutative 92.4%

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

      12. +-commutative 92.4%

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

      13. cube-unmult 92.3%

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

      14. +-commutative 92.3%

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

   3. Simplified 92.3%

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

      1. fma-udef 75.0%

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

      2. cube-mult 75.1%

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

      3. distribute-rgt1-in 92.4%

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

      4. associate-+r+ 92.4%

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

      5. *-commutative 92.4%

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

      6. associate-*l* 92.3%

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

      7. +-commutative 92.3%

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

      8. +-commutative 92.3%

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

      9. +-commutative 92.3%

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

      10. associate-+l+ 92.3%

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

   6. Applied egg-rr 92.3%

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

   if -1.4000000000000001e-16 < x

   1. Initial program 66.5%

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

   3. Taylor expanded in x around 0 59.7%

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

      1. +-commutative 59.7%

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

   5. Simplified 59.7%

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

      1. *-un-lft-identity 59.7%

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

      2. associate-*l* 59.7%

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

      3. +-commutative 59.7%

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

      4. times-frac 58.9%

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

      5. *-commutative 58.9%

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

      6. +-commutative 58.9%

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

   7. Applied egg-rr 58.9%

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

      1. associate-*l/ 58.9%

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

      2. *-lft-identity 58.9%

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

      3. times-frac 82.5%

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

   9. Simplified 82.5%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-16}:\\ \;\;\;\;y \cdot \frac{x}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{y + x}\\ \mathbf{if}\;x \leq -2.8 \cdot 10^{+131}:\\ \;\;\;\;\frac{t_0}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot \frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y x))))
   (if (<= x -2.8e+131)
     (/ t_0 (+ x (+ y (+ y 1.0))))
     (if (<= x -9.5e-17)
       (* (/ y (* (+ y x) (+ y x))) (/ x (+ x (+ y 1.0))))
       (/ (* t_0 (/ x (+ y 1.0))) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (x <= -2.8e+131) {
		tmp = t_0 / (x + (y + (y + 1.0)));
	} else if (x <= -9.5e-17) {
		tmp = (y / ((y + x) * (y + x))) * (x / (x + (y + 1.0)));
	} else {
		tmp = (t_0 * (x / (y + 1.0))) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (y + x)
    if (x <= (-2.8d+131)) then
        tmp = t_0 / (x + (y + (y + 1.0d0)))
    else if (x <= (-9.5d-17)) then
        tmp = (y / ((y + x) * (y + x))) * (x / (x + (y + 1.0d0)))
    else
        tmp = (t_0 * (x / (y + 1.0d0))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (x <= -2.8e+131) {
		tmp = t_0 / (x + (y + (y + 1.0)));
	} else if (x <= -9.5e-17) {
		tmp = (y / ((y + x) * (y + x))) * (x / (x + (y + 1.0)));
	} else {
		tmp = (t_0 * (x / (y + 1.0))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (y + x)
	tmp = 0
	if x <= -2.8e+131:
		tmp = t_0 / (x + (y + (y + 1.0)))
	elif x <= -9.5e-17:
		tmp = (y / ((y + x) * (y + x))) * (x / (x + (y + 1.0)))
	else:
		tmp = (t_0 * (x / (y + 1.0))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(y + x))
	tmp = 0.0
	if (x <= -2.8e+131)
		tmp = Float64(t_0 / Float64(x + Float64(y + Float64(y + 1.0))));
	elseif (x <= -9.5e-17)
		tmp = Float64(Float64(y / Float64(Float64(y + x) * Float64(y + x))) * Float64(x / Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(t_0 * Float64(x / Float64(y + 1.0))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (y + x);
	tmp = 0.0;
	if (x <= -2.8e+131)
		tmp = t_0 / (x + (y + (y + 1.0)));
	elseif (x <= -9.5e-17)
		tmp = (y / ((y + x) * (y + x))) * (x / (x + (y + 1.0)));
	else
		tmp = (t_0 * (x / (y + 1.0))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.8e+131], N[(t$95$0 / N[(x + N[(y + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -9.5e-17], N[(N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.8000000000000001e131

   1. Initial program 50.5%

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

      1. times-frac 79.1%

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

      2. +-commutative 79.1%

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

      3. +-commutative 79.1%

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

      4. +-commutative 79.1%

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

      5. times-frac 50.5%

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

      6. associate-*l/ 76.3%

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

      7. *-commutative 76.3%

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

      8. *-commutative 76.3%

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

      9. distribute-rgt1-in 6.4%

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

      10. fma-def 76.3%

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

      11. +-commutative 76.3%

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

      12. +-commutative 76.3%

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

      13. cube-unmult 76.3%

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

      14. +-commutative 76.3%

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

   3. Simplified 76.3%

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

      1. associate-*r/ 50.5%

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

      2. fma-udef 0.1%

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

      3. cube-mult 0.1%

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

      4. distribute-rgt1-in 50.5%

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

      5. associate-+r+ 50.5%

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

      6. *-commutative 50.5%

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

      7. frac-times 79.1%

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

      8. *-commutative 79.1%

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

      9. associate-/r* 99.8%

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

      10. clear-num 99.8%

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

      11. frac-times 99.9%

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

      12. *-un-lft-identity 99.9%

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

      13. +-commutative 99.9%

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

      14. +-commutative 99.9%

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

      15. associate-+l+ 99.9%

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

      16. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around -inf 85.5%

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

      1. mul-1-neg 85.5%

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

      2. unsub-neg 85.5%

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

      3. neg-mul-1 85.5%

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

      4. +-commutative 85.5%

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

      5. unsub-neg 85.5%

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

      6. distribute-lft-in 85.5%

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

      7. metadata-eval 85.5%

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

      8. neg-mul-1 85.5%

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

      9. unsub-neg 85.5%

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

   9. Simplified 85.5%

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

   if -2.8000000000000001e131 < x < -9.50000000000000029e-17

   1. Initial program 83.6%

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

      1. associate-/r* 83.6%

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

      2. *-commutative 83.6%

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

      3. +-commutative 83.6%

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

      4. +-commutative 83.6%

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

      5. associate-*l/ 92.3%

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

      6. +-commutative 92.3%

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

      7. associate-*r/ 92.3%

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

      8. remove-double-neg 92.3%

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

      9. +-commutative 92.3%

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

      10. +-commutative 92.3%

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

      11. remove-double-neg 92.3%

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

      12. +-commutative 92.3%

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

      13. associate-+l+ 92.3%

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

   3. Simplified 92.3%

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

   if -9.50000000000000029e-17 < x

   1. Initial program 66.5%

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

   3. Taylor expanded in x around 0 59.7%

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

      1. +-commutative 59.7%

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

   5. Simplified 59.7%

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

      1. *-un-lft-identity 59.7%

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

      2. associate-*l* 59.7%

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

      3. +-commutative 59.7%

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

      4. times-frac 58.9%

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

      5. *-commutative 58.9%

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

      6. +-commutative 58.9%

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

   7. Applied egg-rr 58.9%

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

      1. associate-*l/ 58.9%

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

      2. *-lft-identity 58.9%

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

      3. times-frac 82.5%

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

   9. Simplified 82.5%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+131}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{y + x}\\ \mathbf{if}\;x \leq -2.8 \cdot 10^{+131}:\\ \;\;\;\;\frac{t_0}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-79}:\\ \;\;\;\;\frac{x \cdot t_0}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot \frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y x))))
   (if (<= x -2.8e+131)
     (/ t_0 (+ x (+ y (+ y 1.0))))
     (if (<= x -2e-79)
       (/ (* x t_0) (* (+ y x) (+ y (+ x 1.0))))
       (/ (* t_0 (/ x (+ y 1.0))) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (x <= -2.8e+131) {
		tmp = t_0 / (x + (y + (y + 1.0)));
	} else if (x <= -2e-79) {
		tmp = (x * t_0) / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = (t_0 * (x / (y + 1.0))) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (y + x)
    if (x <= (-2.8d+131)) then
        tmp = t_0 / (x + (y + (y + 1.0d0)))
    else if (x <= (-2d-79)) then
        tmp = (x * t_0) / ((y + x) * (y + (x + 1.0d0)))
    else
        tmp = (t_0 * (x / (y + 1.0d0))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (x <= -2.8e+131) {
		tmp = t_0 / (x + (y + (y + 1.0)));
	} else if (x <= -2e-79) {
		tmp = (x * t_0) / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = (t_0 * (x / (y + 1.0))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (y + x)
	tmp = 0
	if x <= -2.8e+131:
		tmp = t_0 / (x + (y + (y + 1.0)))
	elif x <= -2e-79:
		tmp = (x * t_0) / ((y + x) * (y + (x + 1.0)))
	else:
		tmp = (t_0 * (x / (y + 1.0))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(y + x))
	tmp = 0.0
	if (x <= -2.8e+131)
		tmp = Float64(t_0 / Float64(x + Float64(y + Float64(y + 1.0))));
	elseif (x <= -2e-79)
		tmp = Float64(Float64(x * t_0) / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(t_0 * Float64(x / Float64(y + 1.0))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (y + x);
	tmp = 0.0;
	if (x <= -2.8e+131)
		tmp = t_0 / (x + (y + (y + 1.0)));
	elseif (x <= -2e-79)
		tmp = (x * t_0) / ((y + x) * (y + (x + 1.0)));
	else
		tmp = (t_0 * (x / (y + 1.0))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.8e+131], N[(t$95$0 / N[(x + N[(y + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2e-79], N[(N[(x * t$95$0), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.8000000000000001e131

   1. Initial program 50.5%

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

      1. times-frac 79.1%

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

      2. +-commutative 79.1%

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

      3. +-commutative 79.1%

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

      4. +-commutative 79.1%

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

      5. times-frac 50.5%

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

      6. associate-*l/ 76.3%

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

      7. *-commutative 76.3%

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

      8. *-commutative 76.3%

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

      9. distribute-rgt1-in 6.4%

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

      10. fma-def 76.3%

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

      11. +-commutative 76.3%

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

      12. +-commutative 76.3%

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

      13. cube-unmult 76.3%

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

      14. +-commutative 76.3%

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

   3. Simplified 76.3%

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

      1. associate-*r/ 50.5%

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

      2. fma-udef 0.1%

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

      3. cube-mult 0.1%

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

      4. distribute-rgt1-in 50.5%

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

      5. associate-+r+ 50.5%

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

      6. *-commutative 50.5%

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

      7. frac-times 79.1%

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

      8. *-commutative 79.1%

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

      9. associate-/r* 99.8%

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

      10. clear-num 99.8%

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

      11. frac-times 99.9%

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

      12. *-un-lft-identity 99.9%

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

      13. +-commutative 99.9%

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

      14. +-commutative 99.9%

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

      15. associate-+l+ 99.9%

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

      16. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around -inf 85.5%

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

      1. mul-1-neg 85.5%

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

      2. unsub-neg 85.5%

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

      3. neg-mul-1 85.5%

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

      4. +-commutative 85.5%

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

      5. unsub-neg 85.5%

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

      6. distribute-lft-in 85.5%

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

      7. metadata-eval 85.5%

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

      8. neg-mul-1 85.5%

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

      9. unsub-neg 85.5%

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

   9. Simplified 85.5%

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

   if -2.8000000000000001e131 < x < -2e-79

   1. Initial program 84.1%

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

      1. times-frac 94.9%

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

      2. +-commutative 94.9%

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

      3. +-commutative 94.9%

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

      4. +-commutative 94.9%

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

      5. times-frac 84.1%

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

      6. associate-*l/ 90.0%

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

      7. *-commutative 90.0%

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

      8. *-commutative 90.0%

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

      9. distribute-rgt1-in 78.8%

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

      10. fma-def 90.0%

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

      11. +-commutative 90.0%

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

      12. +-commutative 90.0%

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

      13. cube-unmult 90.0%

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

      14. +-commutative 90.0%

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

   3. Simplified 90.0%

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

      1. associate-*r/ 84.2%

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

      2. fma-udef 78.4%

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

      3. cube-mult 78.3%

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

      4. distribute-rgt1-in 84.1%

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

      5. associate-+r+ 84.1%

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

      6. *-commutative 84.1%

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

      7. frac-times 94.8%

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

      8. associate-/r* 99.7%

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

      9. frac-times 94.9%

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

      10. +-commutative 94.9%

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

      11. +-commutative 94.9%

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

      12. +-commutative 94.9%

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

      13. associate-+l+ 94.9%

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

   6. Applied egg-rr 94.9%

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

   if -2e-79 < x

   1. Initial program 65.2%

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

   3. Taylor expanded in x around 0 57.9%

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

      1. +-commutative 57.9%

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

   5. Simplified 57.9%

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

      1. *-un-lft-identity 57.9%

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

      2. associate-*l* 57.9%

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

      3. +-commutative 57.9%

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

      4. times-frac 56.0%

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

      5. *-commutative 56.0%

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

      6. +-commutative 56.0%

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

   7. Applied egg-rr 56.0%

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

      1. associate-*l/ 56.0%

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

      2. *-lft-identity 56.0%

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

      3. times-frac 81.3%

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

   9. Simplified 81.3%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+131}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-79}:\\ \;\;\;\;\frac{x \cdot \frac{y}{y + x}}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-117}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.9e+94)
   (/ (/ y (+ y x)) (+ x (+ y (+ y 1.0))))
   (if (<= x -1.55e-117)
     (* (/ y (* (+ y x) (+ y x))) (/ x (+ x 1.0)))
     (/ (/ x (+ y 1.0)) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.9e+94) {
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)));
	} else if (x <= -1.55e-117) {
		tmp = (y / ((y + x) * (y + x))) * (x / (x + 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 (x <= (-2.9d+94)) then
        tmp = (y / (y + x)) / (x + (y + (y + 1.0d0)))
    else if (x <= (-1.55d-117)) then
        tmp = (y / ((y + x) * (y + x))) * (x / (x + 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 (x <= -2.9e+94) {
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)));
	} else if (x <= -1.55e-117) {
		tmp = (y / ((y + x) * (y + x))) * (x / (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.9e+94:
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)))
	elif x <= -1.55e-117:
		tmp = (y / ((y + x) * (y + x))) * (x / (x + 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 (x <= -2.9e+94)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + Float64(y + Float64(y + 1.0))));
	elseif (x <= -1.55e-117)
		tmp = Float64(Float64(y / Float64(Float64(y + x) * Float64(y + x))) * Float64(x / Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.9e+94)
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)));
	elseif (x <= -1.55e-117)
		tmp = (y / ((y + x) * (y + x))) * (x / (x + 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[x, -2.9e+94], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.55e-117], N[(N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.8999999999999998e94

   1. Initial program 48.2%

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

      1. times-frac 76.7%

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

      2. +-commutative 76.7%

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

      3. +-commutative 76.7%

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

      4. +-commutative 76.7%

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

      5. times-frac 48.2%

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

      6. associate-*l/ 74.2%

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

      7. *-commutative 74.2%

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

      8. *-commutative 74.2%

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

      9. distribute-rgt1-in 8.6%

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

      10. fma-def 74.2%

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

      11. +-commutative 74.2%

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

      12. +-commutative 74.2%

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

      13. cube-unmult 74.2%

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

      14. +-commutative 74.2%

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

   3. Simplified 74.2%

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

      1. associate-*r/ 48.2%

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

      2. fma-udef 2.8%

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

      3. cube-mult 2.8%

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

      4. distribute-rgt1-in 48.2%

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

      5. associate-+r+ 48.2%

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

      6. *-commutative 48.2%

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

      7. frac-times 76.7%

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

      8. *-commutative 76.7%

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

      9. associate-/r* 99.7%

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

      10. clear-num 99.7%

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

      11. frac-times 99.9%

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

      12. *-un-lft-identity 99.9%

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

      13. +-commutative 99.9%

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

      14. +-commutative 99.9%

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

      15. associate-+l+ 99.9%

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

      16. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around -inf 79.6%

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

      1. mul-1-neg 79.6%

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

      2. unsub-neg 79.6%

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

      3. neg-mul-1 79.6%

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

      4. +-commutative 79.6%

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

      5. unsub-neg 79.6%

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

      6. distribute-lft-in 79.6%

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

      7. metadata-eval 79.6%

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

      8. neg-mul-1 79.6%

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

      9. unsub-neg 79.6%

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

   9. Simplified 79.6%

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

   if -2.8999999999999998e94 < x < -1.55000000000000005e-117

   1. Initial program 91.5%

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

      1. associate-/r* 95.6%

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

      2. *-commutative 95.6%

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

      3. +-commutative 95.6%

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

      4. +-commutative 95.6%

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

      5. associate-*l/ 99.6%

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

      6. +-commutative 99.6%

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

      7. associate-*r/ 99.6%

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

      8. remove-double-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. remove-double-neg 99.6%

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

      12. +-commutative 99.6%

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

      13. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 76.0%

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

      1. +-commutative 76.0%

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

   7. Simplified 76.0%

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

   if -1.55000000000000005e-117 < x

   1. Initial program 63.3%

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

      1. associate-/r* 69.2%

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

      2. *-commutative 69.2%

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

      3. +-commutative 69.2%

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

      4. +-commutative 69.2%

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

      5. associate-*l/ 85.5%

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

      6. +-commutative 85.5%

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

      7. associate-*r/ 85.5%

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

      8. remove-double-neg 85.5%

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

      9. +-commutative 85.5%

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

      10. +-commutative 85.5%

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

      11. remove-double-neg 85.5%

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

      12. +-commutative 85.5%

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

      13. associate-+l+ 85.5%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in x around 0 52.8%

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

      1. +-commutative 52.8%

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

   7. Simplified 52.8%

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

      1. *-un-lft-identity 52.8%

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

      2. times-frac 56.1%

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

   9. Applied egg-rr 56.1%

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

       1. associate-*l/ 56.2%

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

       2. +-commutative 56.2%

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

       3. *-lft-identity 56.2%

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

       4. +-commutative 56.2%

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

   11. Simplified 56.2%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-117}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0042:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-268}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \frac{y + 1}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -0.0042)
   (/ (/ y (+ y x)) (+ x (+ y (+ y 1.0))))
   (if (<= x -2.2e-268)
     (/ y (* (+ y x) (* (+ y x) (/ (+ y 1.0) x))))
     (/ (/ x (+ y 1.0)) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -0.0042) {
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)));
	} else if (x <= -2.2e-268) {
		tmp = y / ((y + x) * ((y + x) * ((y + 1.0) / x)));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.0042d0)) then
        tmp = (y / (y + x)) / (x + (y + (y + 1.0d0)))
    else if (x <= (-2.2d-268)) then
        tmp = y / ((y + x) * ((y + x) * ((y + 1.0d0) / x)))
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -0.0042) {
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)));
	} else if (x <= -2.2e-268) {
		tmp = y / ((y + x) * ((y + x) * ((y + 1.0) / x)));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -0.0042:
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)))
	elif x <= -2.2e-268:
		tmp = y / ((y + x) * ((y + x) * ((y + 1.0) / x)))
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -0.0042)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + Float64(y + Float64(y + 1.0))));
	elseif (x <= -2.2e-268)
		tmp = Float64(y / Float64(Float64(y + x) * Float64(Float64(y + x) * Float64(Float64(y + 1.0) / x))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.0042)
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)));
	elseif (x <= -2.2e-268)
		tmp = y / ((y + x) * ((y + x) * ((y + 1.0) / x)));
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -0.0042], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.2e-268], N[(y / N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00419999999999999974

   1. Initial program 62.3%

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

      1. times-frac 83.8%

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

      2. +-commutative 83.8%

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

      3. +-commutative 83.8%

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

      4. +-commutative 83.8%

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

      5. times-frac 62.3%

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

      6. associate-*l/ 82.0%

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

      7. *-commutative 82.0%

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

      8. *-commutative 82.0%

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

      9. distribute-rgt1-in 31.2%

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

      10. fma-def 82.0%

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

      11. +-commutative 82.0%

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

      12. +-commutative 82.0%

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

      13. cube-unmult 82.0%

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

      14. +-commutative 82.0%

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

   3. Simplified 82.0%

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

      1. associate-*r/ 62.3%

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

      2. fma-udef 27.3%

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

      3. cube-mult 27.2%

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

      4. distribute-rgt1-in 62.3%

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

      5. associate-+r+ 62.3%

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

      6. *-commutative 62.3%

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

      7. frac-times 83.7%

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

      8. *-commutative 83.7%

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

      9. associate-/r* 99.7%

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

      10. clear-num 99.7%

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

      11. frac-times 99.7%

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

      12. *-un-lft-identity 99.7%

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

      13. +-commutative 99.7%

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

      14. +-commutative 99.7%

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

      15. associate-+l+ 99.7%

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

      16. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around -inf 75.5%

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

      1. mul-1-neg 75.5%

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

      2. unsub-neg 75.5%

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

      3. neg-mul-1 75.5%

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

      4. +-commutative 75.5%

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

      5. unsub-neg 75.5%

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

      6. distribute-lft-in 75.5%

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

      7. metadata-eval 75.5%

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

      8. neg-mul-1 75.5%

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

      9. unsub-neg 75.5%

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

   9. Simplified 75.5%

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

   if -0.00419999999999999974 < x < -2.20000000000000004e-268

   1. Initial program 73.4%

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

      1. times-frac 87.2%

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

      2. +-commutative 87.2%

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

      3. +-commutative 87.2%

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

      4. +-commutative 87.2%

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

      5. times-frac 73.4%

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

      6. associate-*l/ 80.2%

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

      7. *-commutative 80.2%

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

      8. *-commutative 80.2%

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

      9. distribute-rgt1-in 67.3%

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

      10. fma-def 80.2%

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

      11. +-commutative 80.2%

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

      12. +-commutative 80.2%

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

      13. cube-unmult 80.2%

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

      14. +-commutative 80.2%

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

   3. Simplified 80.2%

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

      1. associate-*r/ 73.5%

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

      2. fma-udef 60.6%

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

      3. cube-mult 60.5%

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

      4. distribute-rgt1-in 73.4%

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

      5. associate-+r+ 73.4%

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

      6. *-commutative 73.4%

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

      7. frac-times 87.2%

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

      8. *-commutative 87.2%

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

      9. associate-/r* 99.8%

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

      10. clear-num 99.7%

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

      11. frac-times 98.7%

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

      12. *-un-lft-identity 98.7%

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

      13. +-commutative 98.7%

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

      14. +-commutative 98.7%

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

      15. associate-+l+ 98.7%

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

      16. +-commutative 98.7%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in x around 0 97.9%

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

      1. +-commutative 97.9%

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

   9. Simplified 97.9%

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

       1. associate-/l/ 91.9%

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

       2. div-inv 91.9%

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

       3. *-commutative 91.9%

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

   11. Applied egg-rr 91.9%

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

       1. associate-*r/ 91.9%

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

       2. *-rgt-identity 91.9%

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

       3. *-commutative 91.9%

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

   13. Simplified 91.9%

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

   if -2.20000000000000004e-268 < x

   1. Initial program 64.2%

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

      1. times-frac 87.7%

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

      2. +-commutative 87.7%

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

      3. +-commutative 87.7%

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

      4. +-commutative 87.7%

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

      5. times-frac 64.2%

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

      6. associate-*l/ 79.2%

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

      7. *-commutative 79.2%

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

      8. *-commutative 79.2%

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

      9. distribute-rgt1-in 70.3%

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

      10. fma-def 79.2%

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

      11. +-commutative 79.2%

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

      12. +-commutative 79.2%

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

      13. cube-unmult 79.2%

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

      14. +-commutative 79.2%

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

   3. Simplified 79.2%

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

      1. associate-*r/ 64.2%

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

      2. fma-udef 56.7%

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

      3. cube-mult 56.8%

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

      4. distribute-rgt1-in 64.2%

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

      5. associate-+r+ 64.2%

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

      6. *-commutative 64.2%

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

      7. frac-times 87.6%

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

      8. associate-*l/ 79.3%

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

      9. associate-/r* 86.5%

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

      10. div-inv 86.4%

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

      11. div-inv 86.5%

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

      12. +-commutative 86.5%

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

      13. associate-+l+ 86.5%

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

      14. +-commutative 86.5%

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

      15. +-commutative 86.5%

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

   6. Applied egg-rr 86.5%

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

   7. Taylor expanded in x around 0 51.2%

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

      1. +-commutative 51.2%

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

   9. Simplified 51.2%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0042:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-268}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \frac{y + 1}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{\frac{y}{y + x}}{1 - \left(\left(\frac{y}{x} \cdot \left(-1 - y\right) - y \cdot 2\right) - x\right)} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (/ (/ y (+ y x)) (- 1.0 (- (- (* (/ y x) (- -1.0 y)) (* y 2.0)) x))))

C

assert(x < y);
double code(double x, double y) {
	return (y / (y + x)) / (1.0 - ((((y / x) * (-1.0 - y)) - (y * 2.0)) - x));
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (y / (y + x)) / (1.0d0 - ((((y / x) * ((-1.0d0) - y)) - (y * 2.0d0)) - x))
end function

Java

assert x < y;
public static double code(double x, double y) {
	return (y / (y + x)) / (1.0 - ((((y / x) * (-1.0 - y)) - (y * 2.0)) - x));
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return (y / (y + x)) / (1.0 - ((((y / x) * (-1.0 - y)) - (y * 2.0)) - x))

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(Float64(y / Float64(y + x)) / Float64(1.0 - Float64(Float64(Float64(Float64(y / x) * Float64(-1.0 - y)) - Float64(y * 2.0)) - x)))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = (y / (y + x)) / (1.0 - ((((y / x) * (-1.0 - y)) - (y * 2.0)) - x));
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[(1.0 - N[(N[(N[(N[(y / x), $MachinePrecision] * N[(-1.0 - y), $MachinePrecision]), $MachinePrecision] - N[(y * 2.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.0%

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

   1. times-frac 86.7%

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

   2. +-commutative 86.7%

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

   3. +-commutative 86.7%

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

   4. +-commutative 86.7%

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

   5. times-frac 66.0%

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

   6. associate-*l/ 80.0%

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

   7. *-commutative 80.0%

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

   8. *-commutative 80.0%

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

   9. distribute-rgt1-in 61.2%

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

   10. fma-def 80.0%

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

   11. +-commutative 80.0%

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

   12. +-commutative 80.0%

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

   13. cube-unmult 80.0%

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

   14. +-commutative 80.0%

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

3. Simplified 80.0%

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

   1. associate-*r/ 66.0%

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

   2. fma-udef 51.3%

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

   3. cube-mult 51.3%

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

   4. distribute-rgt1-in 66.0%

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

   5. associate-+r+ 66.0%

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

   6. *-commutative 66.0%

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

   7. frac-times 86.7%

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

   8. *-commutative 86.7%

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

   9. associate-/r* 99.8%

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

   10. clear-num 99.7%

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

   11. frac-times 99.0%

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

   12. *-un-lft-identity 99.0%

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

   13. +-commutative 99.0%

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

   14. +-commutative 99.0%

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

   15. associate-+l+ 99.0%

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

   16. +-commutative 99.0%

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in x around 0 93.1%

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

   1. fma-def 93.1%

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

   2. associate-/l* 99.1%

      \[\leadsto \frac{\frac{y}{y + x}}{1 + \left(x + \mathsf{fma}\left(2, y, \color{blue}{\frac{y}{\frac{x}{1 + y}}}\right)\right)} \]

   3. +-commutative 99.1%

      \[\leadsto \frac{\frac{y}{y + x}}{1 + \left(x + \mathsf{fma}\left(2, y, \frac{y}{\frac{x}{\color{blue}{y + 1}}}\right)\right)} \]

9. Simplified 99.1%

   \[\leadsto \frac{\frac{y}{y + x}}{\color{blue}{1 + \left(x + \mathsf{fma}\left(2, y, \frac{y}{\frac{x}{y + 1}}\right)\right)}} \]
10. Step-by-step derivation

    1. fma-udef 99.1%

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

    2. associate-/r/ 99.1%

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

11. Applied egg-rr 99.1%

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

12. Final simplification 99.1%

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

Alternative 8: 93.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{y + x}\\ \mathbf{if}\;x \leq -0.032:\\ \;\;\;\;\frac{t_0}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot \frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y x))))
   (if (<= x -0.032)
     (/ t_0 (+ x (+ y (+ y 1.0))))
     (/ (* t_0 (/ x (+ y 1.0))) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (x <= -0.032) {
		tmp = t_0 / (x + (y + (y + 1.0)));
	} else {
		tmp = (t_0 * (x / (y + 1.0))) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (y + x)
    if (x <= (-0.032d0)) then
        tmp = t_0 / (x + (y + (y + 1.0d0)))
    else
        tmp = (t_0 * (x / (y + 1.0d0))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (x <= -0.032) {
		tmp = t_0 / (x + (y + (y + 1.0)));
	} else {
		tmp = (t_0 * (x / (y + 1.0))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (y + x)
	tmp = 0
	if x <= -0.032:
		tmp = t_0 / (x + (y + (y + 1.0)))
	else:
		tmp = (t_0 * (x / (y + 1.0))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(y + x))
	tmp = 0.0
	if (x <= -0.032)
		tmp = Float64(t_0 / Float64(x + Float64(y + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(t_0 * Float64(x / Float64(y + 1.0))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (y + x);
	tmp = 0.0;
	if (x <= -0.032)
		tmp = t_0 / (x + (y + (y + 1.0)));
	else
		tmp = (t_0 * (x / (y + 1.0))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.032], N[(t$95$0 / N[(x + N[(y + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.032000000000000001

   1. Initial program 62.3%

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

      1. times-frac 83.8%

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

      2. +-commutative 83.8%

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

      3. +-commutative 83.8%

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

      4. +-commutative 83.8%

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

      5. times-frac 62.3%

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

      6. associate-*l/ 82.0%

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

      7. *-commutative 82.0%

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

      8. *-commutative 82.0%

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

      9. distribute-rgt1-in 31.2%

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

      10. fma-def 82.0%

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

      11. +-commutative 82.0%

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

      12. +-commutative 82.0%

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

      13. cube-unmult 82.0%

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

      14. +-commutative 82.0%

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

   3. Simplified 82.0%

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

      1. associate-*r/ 62.3%

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

      2. fma-udef 27.3%

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

      3. cube-mult 27.2%

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

      4. distribute-rgt1-in 62.3%

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

      5. associate-+r+ 62.3%

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

      6. *-commutative 62.3%

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

      7. frac-times 83.7%

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

      8. *-commutative 83.7%

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

      9. associate-/r* 99.7%

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

      10. clear-num 99.7%

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

      11. frac-times 99.7%

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

      12. *-un-lft-identity 99.7%

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

      13. +-commutative 99.7%

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

      14. +-commutative 99.7%

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

      15. associate-+l+ 99.7%

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

      16. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around -inf 75.5%

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

      1. mul-1-neg 75.5%

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

      2. unsub-neg 75.5%

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

      3. neg-mul-1 75.5%

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

      4. +-commutative 75.5%

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

      5. unsub-neg 75.5%

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

      6. distribute-lft-in 75.5%

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

      7. metadata-eval 75.5%

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

      8. neg-mul-1 75.5%

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

      9. unsub-neg 75.5%

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

   9. Simplified 75.5%

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

   if -0.032000000000000001 < x

   1. Initial program 67.0%

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

   3. Taylor expanded in x around 0 60.1%

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

      1. +-commutative 60.1%

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

   5. Simplified 60.1%

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

      1. *-un-lft-identity 60.1%

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

      2. associate-*l* 60.1%

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

      3. +-commutative 60.1%

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

      4. times-frac 59.2%

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

      5. *-commutative 59.2%

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

      6. +-commutative 59.2%

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

   7. Applied egg-rr 59.2%

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

      1. associate-*l/ 59.3%

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

      2. *-lft-identity 59.3%

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

      3. times-frac 82.5%

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

   9. Simplified 82.5%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.032:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 66.0%

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

   1. times-frac 86.7%

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

   2. +-commutative 86.7%

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

   3. +-commutative 86.7%

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

   4. +-commutative 86.7%

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

   5. times-frac 66.0%

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

   6. associate-*l/ 80.0%

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

   7. *-commutative 80.0%

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

   8. *-commutative 80.0%

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

   9. distribute-rgt1-in 61.2%

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

   10. fma-def 80.0%

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

   11. +-commutative 80.0%

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

   12. +-commutative 80.0%

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

   13. cube-unmult 80.0%

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

   14. +-commutative 80.0%

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

3. Simplified 80.0%

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

   1. associate-*r/ 66.0%

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

   2. fma-udef 51.3%

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

   3. cube-mult 51.3%

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

   4. distribute-rgt1-in 66.0%

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

   5. associate-+r+ 66.0%

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

   6. *-commutative 66.0%

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

   7. frac-times 86.7%

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

   8. *-commutative 86.7%

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

   9. associate-/r* 99.8%

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

   10. clear-num 99.7%

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

   11. frac-times 99.0%

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

   12. *-un-lft-identity 99.0%

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

   13. +-commutative 99.0%

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

   14. +-commutative 99.0%

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

   15. associate-+l+ 99.0%

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

   16. +-commutative 99.0%

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

6. Applied egg-rr 99.0%

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

7. Final simplification 99.0%

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

Alternative 10: 85.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.95e-158)
   (/ (/ y (+ y x)) (+ x 1.0))
   (/ x (* (+ y x) (+ y (+ x 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.95e-158) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = x / ((y + x) * (y + (x + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.95d-158) then
        tmp = (y / (y + x)) / (x + 1.0d0)
    else
        tmp = x / ((y + x) * (y + (x + 1.0d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.95e-158) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = x / ((y + x) * (y + (x + 1.0)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.95e-158:
		tmp = (y / (y + x)) / (x + 1.0)
	else:
		tmp = x / ((y + x) * (y + (x + 1.0)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.95e-158)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + 1.0));
	else
		tmp = Float64(x / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.95e-158)
		tmp = (y / (y + x)) / (x + 1.0);
	else
		tmp = x / ((y + x) * (y + (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.95e-158], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.9499999999999998e-158

   1. Initial program 70.3%

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

      1. times-frac 86.3%

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

      2. +-commutative 86.3%

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

      3. +-commutative 86.3%

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

      4. +-commutative 86.3%

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

      5. times-frac 70.3%

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

      6. associate-*l/ 81.9%

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

      7. *-commutative 81.9%

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

      8. *-commutative 81.9%

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

      9. distribute-rgt1-in 56.4%

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

      10. fma-def 81.9%

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

      11. +-commutative 81.9%

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

      12. +-commutative 81.9%

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

      13. cube-unmult 81.9%

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

      14. +-commutative 81.9%

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

   3. Simplified 81.9%

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

      1. associate-*r/ 70.4%

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

      2. fma-udef 48.7%

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

      3. cube-mult 48.7%

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

      4. distribute-rgt1-in 70.3%

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

      5. associate-+r+ 70.3%

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

      6. *-commutative 70.3%

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

      7. frac-times 86.3%

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

      8. *-commutative 86.3%

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

      9. associate-/r* 99.8%

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

      10. clear-num 99.7%

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

      11. frac-times 99.3%

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

      12. *-un-lft-identity 99.3%

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

      13. +-commutative 99.3%

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

      14. +-commutative 99.3%

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

      15. associate-+l+ 99.3%

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

      16. +-commutative 99.3%

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

   6. Applied egg-rr 99.3%

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

   7. Taylor expanded in y around 0 55.4%

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

      1. +-commutative 55.4%

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

   9. Simplified 55.4%

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

   if 1.9499999999999998e-158 < y

   1. Initial program 58.9%

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

      1. times-frac 87.3%

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

      2. +-commutative 87.3%

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

      3. +-commutative 87.3%

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

      4. +-commutative 87.3%

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

      5. times-frac 58.9%

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

      6. associate-*l/ 77.0%

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

      7. *-commutative 77.0%

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

      8. *-commutative 77.0%

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

      9. distribute-rgt1-in 69.1%

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

      10. fma-def 77.0%

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

      11. +-commutative 77.0%

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

      12. +-commutative 77.0%

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

      13. cube-unmult 77.0%

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

      14. +-commutative 77.0%

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

   3. Simplified 77.0%

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

      1. associate-*r/ 58.9%

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

      2. fma-udef 55.7%

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

      3. cube-mult 55.7%

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

      4. distribute-rgt1-in 58.9%

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

      5. associate-+r+ 58.9%

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

      6. *-commutative 58.9%

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

      7. frac-times 87.3%

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

      8. associate-/r* 99.7%

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

      9. frac-times 87.4%

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

      10. +-commutative 87.4%

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

      11. +-commutative 87.4%

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

      12. +-commutative 87.4%

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

      13. associate-+l+ 87.4%

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

   6. Applied egg-rr 87.4%

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

   7. Taylor expanded in y around inf 71.7%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-144}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4.5e-144) (/ y (* x (+ x 1.0))) (/ (/ x (+ y 1.0)) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.5e-144) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.5d-144) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 4.5e-144) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 4.5e-144:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4.5e-144)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.5e-144)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 4.5e-144], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.4999999999999998e-144

   1. Initial program 70.5%

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

      1. associate-/r* 74.0%

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

      2. *-commutative 74.0%

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

      3. +-commutative 74.0%

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

      4. +-commutative 74.0%

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

      5. associate-*l/ 86.4%

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

      6. +-commutative 86.4%

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

      7. associate-*r/ 86.4%

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

      8. remove-double-neg 86.4%

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

      9. +-commutative 86.4%

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

      10. +-commutative 86.4%

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

      11. remove-double-neg 86.4%

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

      12. +-commutative 86.4%

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

      13. associate-+l+ 86.4%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in y around 0 54.5%

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

      1. +-commutative 54.5%

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

   7. Simplified 54.5%

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

   if 4.4999999999999998e-144 < y

   1. Initial program 58.5%

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

      1. times-frac 87.2%

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

      2. +-commutative 87.2%

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

      3. +-commutative 87.2%

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

      4. +-commutative 87.2%

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

      5. times-frac 58.5%

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

      6. associate-*l/ 76.8%

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

      7. *-commutative 76.8%

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

      8. *-commutative 76.8%

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

      9. distribute-rgt1-in 68.8%

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

      10. fma-def 76.8%

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

      11. +-commutative 76.8%

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

      12. +-commutative 76.8%

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

      13. cube-unmult 76.8%

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

      14. +-commutative 76.8%

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

   3. Simplified 76.8%

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

      1. associate-*r/ 58.5%

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

      2. fma-udef 55.2%

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

      3. cube-mult 55.2%

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

      4. distribute-rgt1-in 58.5%

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

      5. associate-+r+ 58.5%

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

      6. *-commutative 58.5%

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

      7. frac-times 87.2%

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

      8. associate-*l/ 81.4%

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

      9. associate-/r* 93.9%

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

      10. div-inv 93.9%

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

      11. div-inv 93.9%

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

      12. +-commutative 93.9%

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

      13. associate-+l+ 93.9%

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

      14. +-commutative 93.9%

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

      15. +-commutative 93.9%

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

   6. Applied egg-rr 93.9%

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

   7. Taylor expanded in x around 0 62.5%

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

      1. +-commutative 62.5%

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

   9. Simplified 62.5%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-144}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 82.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4.5e-144) (/ (/ y (+ y x)) (+ x 1.0)) (/ (/ x (+ y 1.0)) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.5e-144) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.5d-144) then
        tmp = (y / (y + x)) / (x + 1.0d0)
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 4.5e-144) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 4.5e-144:
		tmp = (y / (y + x)) / (x + 1.0)
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4.5e-144)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.5e-144)
		tmp = (y / (y + x)) / (x + 1.0);
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 4.5e-144], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.4999999999999998e-144

   1. Initial program 70.5%

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

      1. times-frac 86.4%

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

      2. +-commutative 86.4%

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

      3. +-commutative 86.4%

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

      4. +-commutative 86.4%

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

      5. times-frac 70.5%

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

      6. associate-*l/ 82.0%

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

      7. *-commutative 82.0%

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

      8. *-commutative 82.0%

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

      9. distribute-rgt1-in 56.7%

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

      10. fma-def 82.0%

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

      11. +-commutative 82.0%

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

      12. +-commutative 82.0%

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

      13. cube-unmult 82.0%

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

      14. +-commutative 82.0%

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

   3. Simplified 82.0%

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

      1. associate-*r/ 70.5%

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

      2. fma-udef 49.0%

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

      3. cube-mult 49.0%

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

      4. distribute-rgt1-in 70.5%

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

      5. associate-+r+ 70.5%

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

      6. *-commutative 70.5%

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

      7. frac-times 86.4%

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

      8. *-commutative 86.4%

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

      9. associate-/r* 99.8%

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

      10. clear-num 99.7%

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

      11. frac-times 99.3%

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

      12. *-un-lft-identity 99.3%

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

      13. +-commutative 99.3%

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

      14. +-commutative 99.3%

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

      15. associate-+l+ 99.3%

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

      16. +-commutative 99.3%

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

   6. Applied egg-rr 99.3%

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

   7. Taylor expanded in y around 0 55.7%

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

      1. +-commutative 55.7%

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

   9. Simplified 55.7%

      \[\leadsto \frac{\frac{y}{y + x}}{\color{blue}{x + 1}} \]

   if 4.4999999999999998e-144 < y

   1. Initial program 58.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 87.2%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 87.2%

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1} \]

      3. +-commutative 87.2%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      4. +-commutative 87.2%

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      5. times-frac 58.5%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. associate-*l/ 76.8%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

      7. *-commutative 76.8%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 76.8%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 68.8%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 76.8%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 76.8%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 76.8%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 76.8%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 76.8%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 76.8%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 58.5%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 55.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 55.2%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 58.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. associate-+r+ 58.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 58.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. frac-times 87.2%

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

      8. associate-*l/ 81.4%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      9. associate-/r* 93.9%

         \[\leadsto \color{blue}{\frac{\frac{y \cdot \frac{x}{x + \left(y + 1\right)}}{x + y}}{x + y}} \]

      10. div-inv 93.9%

          \[\leadsto \frac{\frac{y \cdot \color{blue}{\left(x \cdot \frac{1}{x + \left(y + 1\right)}\right)}}{x + y}}{x + y} \]

      11. div-inv 93.9%

          \[\leadsto \frac{\frac{y \cdot \color{blue}{\frac{x}{x + \left(y + 1\right)}}}{x + y}}{x + y} \]

      12. +-commutative 93.9%

          \[\leadsto \frac{\frac{y \cdot \frac{x}{\color{blue}{\left(y + 1\right) + x}}}{x + y}}{x + y} \]

      13. associate-+l+ 93.9%

          \[\leadsto \frac{\frac{y \cdot \frac{x}{\color{blue}{y + \left(1 + x\right)}}}{x + y}}{x + y} \]

      14. +-commutative 93.9%

          \[\leadsto \frac{\frac{y \cdot \frac{x}{y + \left(1 + x\right)}}{\color{blue}{y + x}}}{x + y} \]

      15. +-commutative 93.9%

          \[\leadsto \frac{\frac{y \cdot \frac{x}{y + \left(1 + x\right)}}{y + x}}{\color{blue}{y + x}} \]

   6. Applied egg-rr 93.9%

      \[\leadsto \color{blue}{\frac{\frac{y \cdot \frac{x}{y + \left(1 + x\right)}}{y + x}}{y + x}} \]

   7. Taylor expanded in x around 0 62.5%

      \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{y + x} \]
   8. Step-by-step derivation

      1. +-commutative 62.5%

         \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{y + x} \]

   9. Simplified 62.5%

      \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{y + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 78.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-144}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4.5e-144) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.5e-144) {
		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 <= 4.5d-144) 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 <= 4.5e-144) {
		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 <= 4.5e-144:
		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 <= 4.5e-144)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.5e-144)
		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, 4.5e-144], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.4999999999999998e-144

   1. Initial program 70.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 74.0%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. *-commutative 74.0%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 74.0%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      4. +-commutative 74.0%

         \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      5. associate-*l/ 86.4%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 86.4%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 86.4%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 86.4%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 86.4%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 86.4%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 86.4%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 86.4%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 86.4%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 86.4%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 54.5%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 54.5%

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   7. Simplified 54.5%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]

   if 4.4999999999999998e-144 < y

   1. Initial program 58.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 66.2%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. *-commutative 66.2%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 66.2%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      4. +-commutative 66.2%

         \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      5. associate-*l/ 87.2%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 87.2%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 87.2%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 87.2%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 87.2%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 87.2%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 87.2%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 87.2%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 87.2%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 87.2%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 61.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 61.8%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   7. Simplified 61.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-144}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 80.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-144}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4.3e-144) (/ y (* x (+ x 1.0))) (/ (/ x (+ y 1.0)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.3e-144) {
		tmp = y / (x * (x + 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 <= 4.3d-144) then
        tmp = y / (x * (x + 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 <= 4.3e-144) {
		tmp = y / (x * (x + 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 <= 4.3e-144:
		tmp = y / (x * (x + 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 <= 4.3e-144)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + 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 <= 4.3e-144)
		tmp = y / (x * (x + 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, 4.3e-144], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.2999999999999999e-144

   1. Initial program 70.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 74.0%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. *-commutative 74.0%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 74.0%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      4. +-commutative 74.0%

         \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      5. associate-*l/ 86.4%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 86.4%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 86.4%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 86.4%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 86.4%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 86.4%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 86.4%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 86.4%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 86.4%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 86.4%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 54.5%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 54.5%

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   7. Simplified 54.5%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]

   if 4.2999999999999999e-144 < y

   1. Initial program 58.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 66.2%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. *-commutative 66.2%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 66.2%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      4. +-commutative 66.2%

         \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      5. associate-*l/ 87.2%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 87.2%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 87.2%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 87.2%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 87.2%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 87.2%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 87.2%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 87.2%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 87.2%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 87.2%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 61.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 61.8%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   7. Simplified 61.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 61.8%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{y \cdot \left(y + 1\right)} \]

      2. times-frac 61.8%

         \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y + 1}} \]

   9. Applied egg-rr 61.8%

      \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y + 1}} \]
   10. Step-by-step derivation

       1. associate-*l/ 61.9%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y + 1}}{y}} \]

       2. +-commutative 61.9%

          \[\leadsto \frac{1 \cdot \frac{x}{\color{blue}{1 + y}}}{y} \]

       3. *-lft-identity 61.9%

          \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{y} \]

       4. +-commutative 61.9%

          \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{y} \]

   11. Simplified 61.9%

       \[\leadsto \color{blue}{\frac{\frac{x}{y + 1}}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-144}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 48.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{x}{y \cdot \left(y + 1\right)} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ x (* y (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	return x / (y * (y + 1.0));
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / (y * (y + 1.0d0))
end function

Java

assert x < y;
public static double code(double x, double y) {
	return x / (y * (y + 1.0));
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return x / (y * (y + 1.0))

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x / Float64(y * Float64(y + 1.0)))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x / (y * (y + 1.0));
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.0%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 71.1%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

   2. *-commutative 71.1%

      \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

   3. +-commutative 71.1%

      \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

   4. +-commutative 71.1%

      \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

   5. associate-*l/ 86.7%

      \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

   6. +-commutative 86.7%

      \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

   7. associate-*r/ 86.7%

      \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

   8. remove-double-neg 86.7%

      \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

   9. +-commutative 86.7%

      \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

   10. +-commutative 86.7%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

   11. remove-double-neg 86.7%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

   12. +-commutative 86.7%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

   13. associate-+l+ 86.7%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

3. Simplified 86.7%

   \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 49.2%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
6. Step-by-step derivation

   1. +-commutative 49.2%

      \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

7. Simplified 49.2%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

8. Final simplification 49.2%

   \[\leadsto \frac{x}{y \cdot \left(y + 1\right)} \]
9. Add Preprocessing

Alternative 16: 4.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{1}{x} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ 1.0 x))

C

assert(x < y);
double code(double x, double y) {
	return 1.0 / x;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / x
end function

Java

assert x < y;
public static double code(double x, double y) {
	return 1.0 / x;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return 1.0 / x

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(1.0 / x)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 1.0 / x;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(1.0 / x), $MachinePrecision]

Derivation

1. Initial program 66.0%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. times-frac 86.7%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. +-commutative 86.7%

      \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1} \]

   3. +-commutative 86.7%

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

   4. +-commutative 86.7%

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

   5. times-frac 66.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

   6. associate-*l/ 80.0%

      \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

   7. *-commutative 80.0%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

   8. *-commutative 80.0%

      \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   9. distribute-rgt1-in 61.2%

      \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   10. fma-def 80.0%

       \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

   11. +-commutative 80.0%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   12. +-commutative 80.0%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   13. cube-unmult 80.0%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

   14. +-commutative 80.0%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

3. Simplified 80.0%

   \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r/ 66.0%

      \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   2. fma-udef 51.3%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   3. cube-mult 51.3%

      \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   4. distribute-rgt1-in 66.0%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   5. associate-+r+ 66.0%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

   6. *-commutative 66.0%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

   7. frac-times 86.7%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]

   8. *-commutative 86.7%

      \[\leadsto \color{blue}{\frac{x}{x + \left(y + 1\right)} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   9. associate-/r* 99.8%

      \[\leadsto \frac{x}{x + \left(y + 1\right)} \cdot \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \]

   10. clear-num 99.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{x + \left(y + 1\right)}{x}}} \cdot \frac{\frac{y}{x + y}}{x + y} \]

   11. frac-times 99.0%

       \[\leadsto \color{blue}{\frac{1 \cdot \frac{y}{x + y}}{\frac{x + \left(y + 1\right)}{x} \cdot \left(x + y\right)}} \]

   12. *-un-lft-identity 99.0%

       \[\leadsto \frac{\color{blue}{\frac{y}{x + y}}}{\frac{x + \left(y + 1\right)}{x} \cdot \left(x + y\right)} \]

   13. +-commutative 99.0%

       \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{\frac{x + \left(y + 1\right)}{x} \cdot \left(x + y\right)} \]

   14. +-commutative 99.0%

       \[\leadsto \frac{\frac{y}{y + x}}{\frac{\color{blue}{\left(y + 1\right) + x}}{x} \cdot \left(x + y\right)} \]

   15. associate-+l+ 99.0%

       \[\leadsto \frac{\frac{y}{y + x}}{\frac{\color{blue}{y + \left(1 + x\right)}}{x} \cdot \left(x + y\right)} \]

   16. +-commutative 99.0%

       \[\leadsto \frac{\frac{y}{y + x}}{\frac{y + \left(1 + x\right)}{x} \cdot \color{blue}{\left(y + x\right)}} \]

6. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{\frac{y + \left(1 + x\right)}{x} \cdot \left(y + x\right)}} \]

7. Taylor expanded in x around inf 35.8%

   \[\leadsto \frac{\frac{y}{y + x}}{\color{blue}{x}} \]

8. Taylor expanded in y around inf 4.2%

   \[\leadsto \color{blue}{\frac{1}{x}} \]

9. Final simplification 4.2%

   \[\leadsto \frac{1}{x} \]
10. Add Preprocessing

Alternative 17: 25.8% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{x}{y} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ x y))

C

assert(x < y);
double code(double x, double y) {
	return x / y;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / y
end function

Java

assert x < y;
public static double code(double x, double y) {
	return x / y;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return x / y

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x / y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x / y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 66.0%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 71.1%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

   2. *-commutative 71.1%

      \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

   3. +-commutative 71.1%

      \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

   4. +-commutative 71.1%

      \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

   5. associate-*l/ 86.7%

      \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

   6. +-commutative 86.7%

      \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

   7. associate-*r/ 86.7%

      \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

   8. remove-double-neg 86.7%

      \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

   9. +-commutative 86.7%

      \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

   10. +-commutative 86.7%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

   11. remove-double-neg 86.7%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

   12. +-commutative 86.7%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

   13. associate-+l+ 86.7%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

3. Simplified 86.7%

   \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 49.2%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
6. Step-by-step derivation

   1. +-commutative 49.2%

      \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

7. Simplified 49.2%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

8. Taylor expanded in y around 0 27.3%

   \[\leadsto \color{blue}{\frac{x}{y}} \]

9. Final simplification 27.3%

   \[\leadsto \frac{x}{y} \]
10. Add Preprocessing

Developer target: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x)))))

C

double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x / ((y + 1.0d0) + x)) / (y + x)) / (1.0d0 / (y / (y + x)))
end function

Java

public static double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Python

def code(x, y):
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)))

Julia

function code(x, y)
	return Float64(Float64(Float64(x / Float64(Float64(y + 1.0) + x)) / Float64(y + x)) / Float64(1.0 / Float64(y / Float64(y + x))))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
end

Wolfram

code[x_, y_] := N[(N[(N[(x / N[(N[(y + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024019 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))