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

Percentage Accurate: 68.6% → 99.8%

Time: 13.8s

Alternatives: 21

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

   1. times-frac 87.9%

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

   2. associate-+r+ 87.9%

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

   3. associate-*l/ 81.4%

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

   4. times-frac 99.8%

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

   5. +-commutative 99.8%

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

   6. associate-+r+ 99.8%

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

   7. +-commutative 99.8%

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

   8. associate-+l+ 99.8%

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

   9. +-commutative 99.8%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 83.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{-y}{\left(-1 - x\right) - y}}{x + y}\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-97}:\\ \;\;\;\;x \cdot \frac{y}{\left(x \cdot x\right) \cdot \left(x + \left(1 + y \cdot 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -6.2e+157)
   (/ (/ (- y) (- (- -1.0 x) y)) (+ x y))
   (if (<= x -1.9e+34)
     (* (/ x (* (+ x y) (+ x y))) (/ y x))
     (if (<= x -2e-97)
       (* x (/ y (* (* x x) (+ x (+ 1.0 (* y 3.0))))))
       (/ (/ x y) (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -6.2e+157) {
		tmp = (-y / ((-1.0 - x) - y)) / (x + y);
	} else if (x <= -1.9e+34) {
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	} else if (x <= -2e-97) {
		tmp = x * (y / ((x * x) * (x + (1.0 + (y * 3.0)))));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-6.2d+157)) then
        tmp = (-y / (((-1.0d0) - x) - y)) / (x + y)
    else if (x <= (-1.9d+34)) then
        tmp = (x / ((x + y) * (x + y))) * (y / x)
    else if (x <= (-2d-97)) then
        tmp = x * (y / ((x * x) * (x + (1.0d0 + (y * 3.0d0)))))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -6.2e+157) {
		tmp = (-y / ((-1.0 - x) - y)) / (x + y);
	} else if (x <= -1.9e+34) {
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	} else if (x <= -2e-97) {
		tmp = x * (y / ((x * x) * (x + (1.0 + (y * 3.0)))));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -6.2e+157:
		tmp = (-y / ((-1.0 - x) - y)) / (x + y)
	elif x <= -1.9e+34:
		tmp = (x / ((x + y) * (x + y))) * (y / x)
	elif x <= -2e-97:
		tmp = x * (y / ((x * x) * (x + (1.0 + (y * 3.0)))))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -6.2e+157)
		tmp = Float64(Float64(Float64(-y) / Float64(Float64(-1.0 - x) - y)) / Float64(x + y));
	elseif (x <= -1.9e+34)
		tmp = Float64(Float64(x / Float64(Float64(x + y) * Float64(x + y))) * Float64(y / x));
	elseif (x <= -2e-97)
		tmp = Float64(x * Float64(y / Float64(Float64(x * x) * Float64(x + Float64(1.0 + Float64(y * 3.0))))));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.2e+157)
		tmp = (-y / ((-1.0 - x) - y)) / (x + y);
	elseif (x <= -1.9e+34)
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	elseif (x <= -2e-97)
		tmp = x * (y / ((x * x) * (x + (1.0 + (y * 3.0)))));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -6.2e+157], N[(N[((-y) / N[(N[(-1.0 - x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.9e+34], N[(N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2e-97], N[(x * N[(y / N[(N[(x * x), $MachinePrecision] * N[(x + N[(1.0 + N[(y * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.1999999999999994e157

   1. Initial program 53.1%

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

      1. associate-*r/ 75.9%

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

      2. *-commutative 75.9%

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

      3. distribute-rgt1-in 1.7%

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

      4. fma-def 75.9%

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

      5. cube-unmult 75.9%

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

   3. Simplified 75.9%

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

      1. associate-*r/ 53.1%

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

      2. fma-udef 0.0%

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

      3. cube-mult 0.0%

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

      4. distribute-rgt1-in 53.1%

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

      5. associate-+r+ 53.1%

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

      6. *-commutative 53.1%

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

      7. frac-times 75.9%

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

      8. associate-/r* 99.9%

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

      9. frac-2neg 99.9%

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

      10. frac-times 75.9%

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

      11. +-commutative 75.9%

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

      12. +-commutative 75.9%

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

      13. associate-+r+ 75.9%

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

      14. +-commutative 75.9%

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

      15. associate-+l+ 75.9%

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

   5. Applied egg-rr 75.9%

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

      1. *-commutative 75.9%

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

      2. *-commutative 75.9%

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

      3. associate-/r* 99.9%

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

      4. distribute-neg-in 99.9%

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

      5. +-commutative 99.9%

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

      6. unsub-neg 99.9%

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

      7. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in y around 0 85.8%

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

      1. mul-1-neg 85.8%

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

   10. Simplified 85.8%

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

   if -6.1999999999999994e157 < x < -1.9000000000000001e34

   1. Initial program 56.0%

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

      1. times-frac 95.7%

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

      2. /-rgt-identity 95.7%

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

      3. associate-/l/ 95.7%

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

      4. *-lft-identity 95.7%

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

      5. associate-+l+ 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in x around inf 83.6%

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

   if -1.9000000000000001e34 < x < -2.00000000000000007e-97

   1. Initial program 84.8%

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

      1. associate-*r/ 96.8%

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

      2. *-commutative 96.8%

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

      3. distribute-rgt1-in 87.4%

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

      4. fma-def 96.8%

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

      5. cube-unmult 96.8%

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

   3. Simplified 96.8%

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

   4. Taylor expanded in x around inf 66.7%

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(1 + \left(2 \cdot y + y\right)\right) \cdot {x}^{2} + {x}^{3}}} \]
   5. Step-by-step derivation

      1. +-commutative 66.7%

         \[\leadsto x \cdot \frac{y}{\color{blue}{{x}^{3} + \left(1 + \left(2 \cdot y + y\right)\right) \cdot {x}^{2}}} \]

      2. cube-mult 66.7%

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

      3. unpow2 66.7%

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

      4. distribute-rgt-out 66.7%

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

      5. unpow2 66.7%

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

      6. distribute-lft1-in 66.7%

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

      7. metadata-eval 66.7%

         \[\leadsto x \cdot \frac{y}{\left(x \cdot x\right) \cdot \left(x + \left(1 + \color{blue}{3} \cdot y\right)\right)} \]

   6. Simplified 66.7%

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(x \cdot x\right) \cdot \left(x + \left(1 + 3 \cdot y\right)\right)}} \]

   if -2.00000000000000007e-97 < x

   1. Initial program 69.2%

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

      1. times-frac 86.8%

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

      2. /-rgt-identity 86.8%

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

      3. associate-/l/ 86.8%

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

      4. *-lft-identity 86.8%

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

      5. associate-+l+ 86.8%

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

   3. Simplified 86.8%

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

   4. Taylor expanded in x around 0 55.3%

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

      1. associate-/r* 57.7%

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

      2. +-commutative 57.7%

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

   6. Simplified 57.7%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{-y}{\left(-1 - x\right) - y}}{x + y}\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-97}:\\ \;\;\;\;x \cdot \frac{y}{\left(x \cdot x\right) \cdot \left(x + \left(1 + y \cdot 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 3: 94.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{-y}{\left(-1 - x\right) - y}}{x + y}\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \frac{y}{\left(x \cdot x\right) \cdot \left(x + \left(1 + y \cdot 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{\frac{y}{y + 1}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -6.2e+157)
   (/ (/ (- y) (- (- -1.0 x) y)) (+ x y))
   (if (<= x -1.9e+34)
     (* (/ x (* (+ x y) (+ x y))) (/ y x))
     (if (<= x -1.15e-9)
       (* x (/ y (* (* x x) (+ x (+ 1.0 (* y 3.0))))))
       (* (/ x (+ x y)) (/ (/ y (+ y 1.0)) (+ x y)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -6.2e+157) {
		tmp = (-y / ((-1.0 - x) - y)) / (x + y);
	} else if (x <= -1.9e+34) {
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	} else if (x <= -1.15e-9) {
		tmp = x * (y / ((x * x) * (x + (1.0 + (y * 3.0)))));
	} else {
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (x + y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-6.2d+157)) then
        tmp = (-y / (((-1.0d0) - x) - y)) / (x + y)
    else if (x <= (-1.9d+34)) then
        tmp = (x / ((x + y) * (x + y))) * (y / x)
    else if (x <= (-1.15d-9)) then
        tmp = x * (y / ((x * x) * (x + (1.0d0 + (y * 3.0d0)))))
    else
        tmp = (x / (x + y)) * ((y / (y + 1.0d0)) / (x + y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -6.2e+157) {
		tmp = (-y / ((-1.0 - x) - y)) / (x + y);
	} else if (x <= -1.9e+34) {
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	} else if (x <= -1.15e-9) {
		tmp = x * (y / ((x * x) * (x + (1.0 + (y * 3.0)))));
	} else {
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (x + y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -6.2e+157:
		tmp = (-y / ((-1.0 - x) - y)) / (x + y)
	elif x <= -1.9e+34:
		tmp = (x / ((x + y) * (x + y))) * (y / x)
	elif x <= -1.15e-9:
		tmp = x * (y / ((x * x) * (x + (1.0 + (y * 3.0)))))
	else:
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (x + y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -6.2e+157)
		tmp = Float64(Float64(Float64(-y) / Float64(Float64(-1.0 - x) - y)) / Float64(x + y));
	elseif (x <= -1.9e+34)
		tmp = Float64(Float64(x / Float64(Float64(x + y) * Float64(x + y))) * Float64(y / x));
	elseif (x <= -1.15e-9)
		tmp = Float64(x * Float64(y / Float64(Float64(x * x) * Float64(x + Float64(1.0 + Float64(y * 3.0))))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(Float64(y / Float64(y + 1.0)) / Float64(x + y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.2e+157)
		tmp = (-y / ((-1.0 - x) - y)) / (x + y);
	elseif (x <= -1.9e+34)
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	elseif (x <= -1.15e-9)
		tmp = x * (y / ((x * x) * (x + (1.0 + (y * 3.0)))));
	else
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -6.2e+157], N[(N[((-y) / N[(N[(-1.0 - x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.9e+34], N[(N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.15e-9], N[(x * N[(y / N[(N[(x * x), $MachinePrecision] * N[(x + N[(1.0 + N[(y * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.1999999999999994e157

   1. Initial program 53.1%

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

      1. associate-*r/ 75.9%

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

      2. *-commutative 75.9%

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

      3. distribute-rgt1-in 1.7%

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

      4. fma-def 75.9%

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

      5. cube-unmult 75.9%

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

   3. Simplified 75.9%

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

      1. associate-*r/ 53.1%

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

      2. fma-udef 0.0%

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

      3. cube-mult 0.0%

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

      4. distribute-rgt1-in 53.1%

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

      5. associate-+r+ 53.1%

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

      6. *-commutative 53.1%

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

      7. frac-times 75.9%

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

      8. associate-/r* 99.9%

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

      9. frac-2neg 99.9%

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

      10. frac-times 75.9%

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

      11. +-commutative 75.9%

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

      12. +-commutative 75.9%

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

      13. associate-+r+ 75.9%

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

      14. +-commutative 75.9%

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

      15. associate-+l+ 75.9%

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

   5. Applied egg-rr 75.9%

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

      1. *-commutative 75.9%

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

      2. *-commutative 75.9%

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

      3. associate-/r* 99.9%

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

      4. distribute-neg-in 99.9%

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

      5. +-commutative 99.9%

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

      6. unsub-neg 99.9%

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

      7. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in y around 0 85.8%

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

      1. mul-1-neg 85.8%

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

   10. Simplified 85.8%

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

   if -6.1999999999999994e157 < x < -1.9000000000000001e34

   1. Initial program 56.0%

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

      1. times-frac 95.7%

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

      2. /-rgt-identity 95.7%

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

      3. associate-/l/ 95.7%

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

      4. *-lft-identity 95.7%

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

      5. associate-+l+ 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in x around inf 83.6%

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

   if -1.9000000000000001e34 < x < -1.15e-9

   1. Initial program 84.0%

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

      1. associate-*r/ 92.2%

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

      2. *-commutative 92.2%

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

      3. distribute-rgt1-in 83.8%

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

      4. fma-def 92.2%

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

      5. cube-unmult 92.2%

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

   3. Simplified 92.2%

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

   4. Taylor expanded in x around inf 76.2%

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(1 + \left(2 \cdot y + y\right)\right) \cdot {x}^{2} + {x}^{3}}} \]
   5. Step-by-step derivation

      1. +-commutative 76.2%

         \[\leadsto x \cdot \frac{y}{\color{blue}{{x}^{3} + \left(1 + \left(2 \cdot y + y\right)\right) \cdot {x}^{2}}} \]

      2. cube-mult 76.2%

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

      3. unpow2 76.2%

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

      4. distribute-rgt-out 76.2%

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

      5. unpow2 76.2%

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

      6. distribute-lft1-in 76.2%

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

      7. metadata-eval 76.2%

         \[\leadsto x \cdot \frac{y}{\left(x \cdot x\right) \cdot \left(x + \left(1 + \color{blue}{3} \cdot y\right)\right)} \]

   6. Simplified 76.2%

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(x \cdot x\right) \cdot \left(x + \left(1 + 3 \cdot y\right)\right)}} \]

   if -1.15e-9 < x

   1. Initial program 70.9%

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

      1. times-frac 88.2%

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

      2. associate-+r+ 88.2%

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

      3. associate-*l/ 79.4%

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

      4. times-frac 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

      9. +-commutative 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 83.8%

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

      1. +-commutative 83.8%

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

   6. Simplified 83.8%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{-y}{\left(-1 - x\right) - y}}{x + y}\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \frac{y}{\left(x \cdot x\right) \cdot \left(x + \left(1 + y \cdot 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{\frac{y}{y + 1}}{x + y}\\ \end{array} \]

Alternative 4: 94.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{-y}{\left(-1 - x\right) - y}}{x + y}\\ \mathbf{elif}\;x \leq -1.02 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{\frac{y}{y + 1}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.4e+110)
   (/ (/ (- y) (- (- -1.0 x) y)) (+ x y))
   (if (<= x -1.02e-16)
     (* x (/ y (* (+ x y) (* (+ x y) (+ y (+ x 1.0))))))
     (* (/ x (+ x y)) (/ (/ y (+ y 1.0)) (+ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.4e+110) {
		tmp = (-y / ((-1.0 - x) - y)) / (x + y);
	} else if (x <= -1.02e-16) {
		tmp = x * (y / ((x + y) * ((x + y) * (y + (x + 1.0)))));
	} else {
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (x + y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.4d+110)) then
        tmp = (-y / (((-1.0d0) - x) - y)) / (x + y)
    else if (x <= (-1.02d-16)) then
        tmp = x * (y / ((x + y) * ((x + y) * (y + (x + 1.0d0)))))
    else
        tmp = (x / (x + y)) * ((y / (y + 1.0d0)) / (x + y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -3.4e+110) {
		tmp = (-y / ((-1.0 - x) - y)) / (x + y);
	} else if (x <= -1.02e-16) {
		tmp = x * (y / ((x + y) * ((x + y) * (y + (x + 1.0)))));
	} else {
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (x + y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.4e+110:
		tmp = (-y / ((-1.0 - x) - y)) / (x + y)
	elif x <= -1.02e-16:
		tmp = x * (y / ((x + y) * ((x + y) * (y + (x + 1.0)))))
	else:
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (x + y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.4e+110)
		tmp = Float64(Float64(Float64(-y) / Float64(Float64(-1.0 - x) - y)) / Float64(x + y));
	elseif (x <= -1.02e-16)
		tmp = Float64(x * Float64(y / Float64(Float64(x + y) * Float64(Float64(x + y) * Float64(y + Float64(x + 1.0))))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(Float64(y / Float64(y + 1.0)) / Float64(x + y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.4e+110)
		tmp = (-y / ((-1.0 - x) - y)) / (x + y);
	elseif (x <= -1.02e-16)
		tmp = x * (y / ((x + y) * ((x + y) * (y + (x + 1.0)))));
	else
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -3.4e+110], N[(N[((-y) / N[(N[(-1.0 - x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.02e-16], N[(x * N[(y / N[(N[(x + y), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4000000000000001e110

   1. Initial program 47.5%

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

      1. associate-*r/ 64.7%

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

      2. *-commutative 64.7%

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

      3. distribute-rgt1-in 9.3%

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

      4. fma-def 64.7%

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

      5. cube-unmult 64.7%

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

   3. Simplified 64.7%

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

      1. associate-*r/ 47.5%

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

      2. fma-udef 8.0%

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

      3. cube-mult 8.0%

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

      4. distribute-rgt1-in 47.5%

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

      5. associate-+r+ 47.5%

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

      6. *-commutative 47.5%

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

      7. frac-times 79.8%

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

      8. associate-/r* 99.8%

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

      9. frac-2neg 99.8%

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

      10. frac-times 79.9%

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

      11. +-commutative 79.9%

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

      12. +-commutative 79.9%

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

      13. associate-+r+ 79.9%

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

      14. +-commutative 79.9%

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

      15. associate-+l+ 79.9%

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

   5. Applied egg-rr 79.9%

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

      1. *-commutative 79.9%

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

      2. *-commutative 79.9%

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

      3. associate-/r* 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. +-commutative 99.8%

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

      6. unsub-neg 99.8%

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

      7. +-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in y around 0 84.9%

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

      1. mul-1-neg 84.9%

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

   10. Simplified 84.9%

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

   if -3.4000000000000001e110 < x < -1.0200000000000001e-16

   1. Initial program 82.3%

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

      1. associate-*r/ 91.7%

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

      2. *-commutative 91.7%

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

      3. distribute-rgt1-in 84.6%

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

      4. fma-def 91.8%

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

      5. cube-unmult 91.8%

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

   3. Simplified 91.8%

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

      1. fma-udef 84.6%

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

      2. cube-mult 84.6%

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

      3. distribute-rgt1-in 91.7%

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

      4. associate-+r+ 91.7%

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

      5. *-commutative 91.7%

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

      6. associate-*l* 91.7%

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

      7. +-commutative 91.7%

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

      8. +-commutative 91.7%

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

      9. associate-+r+ 91.7%

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

      10. +-commutative 91.7%

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

      11. associate-+l+ 91.7%

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

   5. Applied egg-rr 91.7%

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

   if -1.0200000000000001e-16 < x

   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 88.0%

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

      2. associate-+r+ 88.0%

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

      3. associate-*l/ 79.1%

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

      4. times-frac 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

      9. +-commutative 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 83.7%

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

      1. +-commutative 83.7%

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

   6. Simplified 83.7%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{-y}{\left(-1 - x\right) - y}}{x + y}\\ \mathbf{elif}\;x \leq -1.02 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{\frac{y}{y + 1}}{x + y}\\ \end{array} \]

Alternative 5: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{-y}{\left(-1 - x\right) - y}}{x + y}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{\frac{y}{y + 1}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -6.2e+157)
   (/ (/ (- y) (- (- -1.0 x) y)) (+ x y))
   (if (<= x -2e-24)
     (* (/ x (* (+ x y) (+ x y))) (/ y (+ x (+ y 1.0))))
     (* (/ x (+ x y)) (/ (/ y (+ y 1.0)) (+ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -6.2e+157) {
		tmp = (-y / ((-1.0 - x) - y)) / (x + y);
	} else if (x <= -2e-24) {
		tmp = (x / ((x + y) * (x + y))) * (y / (x + (y + 1.0)));
	} else {
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (x + y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-6.2d+157)) then
        tmp = (-y / (((-1.0d0) - x) - y)) / (x + y)
    else if (x <= (-2d-24)) then
        tmp = (x / ((x + y) * (x + y))) * (y / (x + (y + 1.0d0)))
    else
        tmp = (x / (x + y)) * ((y / (y + 1.0d0)) / (x + y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -6.2e+157) {
		tmp = (-y / ((-1.0 - x) - y)) / (x + y);
	} else if (x <= -2e-24) {
		tmp = (x / ((x + y) * (x + y))) * (y / (x + (y + 1.0)));
	} else {
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (x + y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -6.2e+157:
		tmp = (-y / ((-1.0 - x) - y)) / (x + y)
	elif x <= -2e-24:
		tmp = (x / ((x + y) * (x + y))) * (y / (x + (y + 1.0)))
	else:
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (x + y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -6.2e+157)
		tmp = Float64(Float64(Float64(-y) / Float64(Float64(-1.0 - x) - y)) / Float64(x + y));
	elseif (x <= -2e-24)
		tmp = Float64(Float64(x / Float64(Float64(x + y) * Float64(x + y))) * Float64(y / Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(Float64(y / Float64(y + 1.0)) / Float64(x + y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.2e+157)
		tmp = (-y / ((-1.0 - x) - y)) / (x + y);
	elseif (x <= -2e-24)
		tmp = (x / ((x + y) * (x + y))) * (y / (x + (y + 1.0)));
	else
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -6.2e+157], N[(N[((-y) / N[(N[(-1.0 - x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2e-24], N[(N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.1999999999999994e157

   1. Initial program 53.1%

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

      1. associate-*r/ 75.9%

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

      2. *-commutative 75.9%

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

      3. distribute-rgt1-in 1.7%

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

      4. fma-def 75.9%

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

      5. cube-unmult 75.9%

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

   3. Simplified 75.9%

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

      1. associate-*r/ 53.1%

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

      2. fma-udef 0.0%

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

      3. cube-mult 0.0%

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

      4. distribute-rgt1-in 53.1%

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

      5. associate-+r+ 53.1%

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

      6. *-commutative 53.1%

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

      7. frac-times 75.9%

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

      8. associate-/r* 99.9%

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

      9. frac-2neg 99.9%

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

      10. frac-times 75.9%

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

      11. +-commutative 75.9%

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

      12. +-commutative 75.9%

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

      13. associate-+r+ 75.9%

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

      14. +-commutative 75.9%

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

      15. associate-+l+ 75.9%

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

   5. Applied egg-rr 75.9%

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

      1. *-commutative 75.9%

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

      2. *-commutative 75.9%

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

      3. associate-/r* 99.9%

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

      4. distribute-neg-in 99.9%

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

      5. +-commutative 99.9%

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

      6. unsub-neg 99.9%

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

      7. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in y around 0 85.8%

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

      1. mul-1-neg 85.8%

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

   10. Simplified 85.8%

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

   if -6.1999999999999994e157 < x < -1.99999999999999985e-24

   1. Initial program 70.1%

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

      1. times-frac 97.3%

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

      2. /-rgt-identity 97.3%

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

      3. associate-/l/ 97.3%

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

      4. *-lft-identity 97.3%

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

      5. associate-+l+ 97.3%

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

   3. Simplified 97.3%

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

   if -1.99999999999999985e-24 < x

   1. Initial program 70.0%

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

      1. times-frac 87.8%

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

      2. associate-+r+ 87.8%

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

      3. associate-*l/ 78.7%

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

      4. times-frac 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

      9. +-commutative 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 83.5%

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

      1. +-commutative 83.5%

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

   6. Simplified 83.5%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{-y}{\left(-1 - x\right) - y}}{x + y}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{\frac{y}{y + 1}}{x + y}\\ \end{array} \]

Alternative 6: 88.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-167}:\\ \;\;\;\;\frac{\frac{-y}{\left(-1 - x\right) - y}}{x + y}\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+140}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.3e-167)
   (/ (/ (- y) (- (- -1.0 x) y)) (+ x y))
   (if (<= y 1.12e+140)
     (* (/ x (* (+ x y) (+ x y))) (/ y (+ y 1.0)))
     (* (/ x (+ x y)) (/ 1.0 (+ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.3e-167) {
		tmp = (-y / ((-1.0 - x) - y)) / (x + y);
	} else if (y <= 1.12e+140) {
		tmp = (x / ((x + y) * (x + y))) * (y / (y + 1.0));
	} else {
		tmp = (x / (x + y)) * (1.0 / (x + y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.3d-167) then
        tmp = (-y / (((-1.0d0) - x) - y)) / (x + y)
    else if (y <= 1.12d+140) then
        tmp = (x / ((x + y) * (x + y))) * (y / (y + 1.0d0))
    else
        tmp = (x / (x + y)) * (1.0d0 / (x + y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.3e-167) {
		tmp = (-y / ((-1.0 - x) - y)) / (x + y);
	} else if (y <= 1.12e+140) {
		tmp = (x / ((x + y) * (x + y))) * (y / (y + 1.0));
	} else {
		tmp = (x / (x + y)) * (1.0 / (x + y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.3e-167:
		tmp = (-y / ((-1.0 - x) - y)) / (x + y)
	elif y <= 1.12e+140:
		tmp = (x / ((x + y) * (x + y))) * (y / (y + 1.0))
	else:
		tmp = (x / (x + y)) * (1.0 / (x + y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.3e-167)
		tmp = Float64(Float64(Float64(-y) / Float64(Float64(-1.0 - x) - y)) / Float64(x + y));
	elseif (y <= 1.12e+140)
		tmp = Float64(Float64(x / Float64(Float64(x + y) * Float64(x + y))) * Float64(y / Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(1.0 / Float64(x + y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.3e-167)
		tmp = (-y / ((-1.0 - x) - y)) / (x + y);
	elseif (y <= 1.12e+140)
		tmp = (x / ((x + y) * (x + y))) * (y / (y + 1.0));
	else
		tmp = (x / (x + y)) * (1.0 / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 2.3e-167], N[(N[((-y) / N[(N[(-1.0 - x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.12e+140], N[(N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.3000000000000001e-167

   1. Initial program 63.2%

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

      1. associate-*r/ 77.8%

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

      2. *-commutative 77.8%

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

      3. distribute-rgt1-in 52.9%

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

      4. fma-def 77.8%

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

      5. cube-unmult 77.8%

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

   3. Simplified 77.8%

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

      1. associate-*r/ 63.2%

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

      2. fma-udef 44.3%

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

      3. cube-mult 44.2%

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

      4. distribute-rgt1-in 63.2%

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

      5. associate-+r+ 63.2%

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

      6. *-commutative 63.2%

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

      7. frac-times 85.2%

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

      8. associate-/r* 99.8%

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

      9. frac-2neg 99.8%

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

      10. frac-times 93.3%

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

      11. +-commutative 93.3%

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

      12. +-commutative 93.3%

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

      13. associate-+r+ 93.3%

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

      14. +-commutative 93.3%

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

      15. associate-+l+ 93.3%

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

   5. Applied egg-rr 93.3%

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

      1. *-commutative 93.3%

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

      2. *-commutative 93.3%

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

      3. associate-/r* 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. +-commutative 99.8%

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

      6. unsub-neg 99.8%

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

      7. +-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in y around 0 61.1%

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

      1. mul-1-neg 61.1%

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

   10. Simplified 61.1%

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

   if 2.3000000000000001e-167 < y < 1.1199999999999999e140

   1. Initial program 79.6%

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

      1. times-frac 95.2%

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

      2. /-rgt-identity 95.2%

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

      3. associate-/l/ 95.2%

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

      4. *-lft-identity 95.2%

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

      5. associate-+l+ 95.2%

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

   3. Simplified 95.2%

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

   4. Taylor expanded in x around 0 74.9%

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

      1. +-commutative 74.9%

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

   6. Simplified 74.9%

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

   if 1.1199999999999999e140 < y

   1. Initial program 68.1%

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

      1. times-frac 86.6%

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

      2. associate-+r+ 86.6%

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

      3. associate-*l/ 86.6%

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

      4. times-frac 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-+r+ 100.0%

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

      7. +-commutative 100.0%

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

      8. associate-+l+ 100.0%

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

      9. +-commutative 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in y around inf 90.1%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-167}:\\ \;\;\;\;\frac{\frac{-y}{\left(-1 - x\right) - y}}{x + y}\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+140}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{x + y}\\ \end{array} \]

Alternative 7: 83.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{\frac{-y}{\left(-1 - x\right) - y}}{x + y}\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{+157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-97}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ (- y) (- (- -1.0 x) y)) (+ x y))))
   (if (<= x -6.5e+157)
     t_0
     (if (<= x -1.9e+34)
       (* (/ x (* (+ x y) (+ x y))) (/ y x))
       (if (<= x -2.2e-97) t_0 (/ (/ x y) (+ y 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (-y / ((-1.0 - x) - y)) / (x + y);
	double tmp;
	if (x <= -6.5e+157) {
		tmp = t_0;
	} else if (x <= -1.9e+34) {
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	} else if (x <= -2.2e-97) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (-y / (((-1.0d0) - x) - y)) / (x + y)
    if (x <= (-6.5d+157)) then
        tmp = t_0
    else if (x <= (-1.9d+34)) then
        tmp = (x / ((x + y) * (x + y))) * (y / x)
    else if (x <= (-2.2d-97)) then
        tmp = t_0
    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 t_0 = (-y / ((-1.0 - x) - y)) / (x + y);
	double tmp;
	if (x <= -6.5e+157) {
		tmp = t_0;
	} else if (x <= -1.9e+34) {
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	} else if (x <= -2.2e-97) {
		tmp = t_0;
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (-y / ((-1.0 - x) - y)) / (x + y)
	tmp = 0
	if x <= -6.5e+157:
		tmp = t_0
	elif x <= -1.9e+34:
		tmp = (x / ((x + y) * (x + y))) * (y / x)
	elif x <= -2.2e-97:
		tmp = t_0
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(Float64(-y) / Float64(Float64(-1.0 - x) - y)) / Float64(x + y))
	tmp = 0.0
	if (x <= -6.5e+157)
		tmp = t_0;
	elseif (x <= -1.9e+34)
		tmp = Float64(Float64(x / Float64(Float64(x + y) * Float64(x + y))) * Float64(y / x));
	elseif (x <= -2.2e-97)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (-y / ((-1.0 - x) - y)) / (x + y);
	tmp = 0.0;
	if (x <= -6.5e+157)
		tmp = t_0;
	elseif (x <= -1.9e+34)
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	elseif (x <= -2.2e-97)
		tmp = t_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_] := Block[{t$95$0 = N[(N[((-y) / N[(N[(-1.0 - x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.5e+157], t$95$0, If[LessEqual[x, -1.9e+34], N[(N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.2e-97], t$95$0, N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.5e157 or -1.9000000000000001e34 < x < -2.1999999999999999e-97

   1. Initial program 69.0%

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

      1. associate-*r/ 86.3%

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

      2. *-commutative 86.3%

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

      3. distribute-rgt1-in 44.6%

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

      4. fma-def 86.3%

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

      5. cube-unmult 86.3%

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

   3. Simplified 86.3%

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

      1. associate-*r/ 68.9%

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

      2. fma-udef 39.2%

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

      3. cube-mult 39.2%

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

      4. distribute-rgt1-in 69.0%

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

      5. associate-+r+ 69.0%

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

      6. *-commutative 69.0%

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

      7. frac-times 87.7%

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

      8. associate-/r* 99.8%

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

      9. frac-2neg 99.8%

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

      10. frac-times 87.9%

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

      11. +-commutative 87.9%

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

      12. +-commutative 87.9%

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

      13. associate-+r+ 87.9%

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

      14. +-commutative 87.9%

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

      15. associate-+l+ 87.9%

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

   5. Applied egg-rr 87.9%

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

      1. *-commutative 87.9%

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

      2. *-commutative 87.9%

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

      3. associate-/r* 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. +-commutative 99.8%

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

      6. unsub-neg 99.8%

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

      7. +-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in y around 0 75.3%

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

      1. mul-1-neg 75.3%

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

   10. Simplified 75.3%

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

   if -6.5e157 < x < -1.9000000000000001e34

   1. Initial program 56.0%

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

      1. times-frac 95.7%

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

      2. /-rgt-identity 95.7%

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

      3. associate-/l/ 95.7%

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

      4. *-lft-identity 95.7%

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

      5. associate-+l+ 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in x around inf 83.6%

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

   if -2.1999999999999999e-97 < x

   1. Initial program 69.2%

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

      1. times-frac 86.8%

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

      2. /-rgt-identity 86.8%

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

      3. associate-/l/ 86.8%

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

      4. *-lft-identity 86.8%

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

      5. associate-+l+ 86.8%

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

   3. Simplified 86.8%

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

   4. Taylor expanded in x around 0 55.3%

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

      1. associate-/r* 57.7%

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

      2. +-commutative 57.7%

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

   6. Simplified 57.7%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{-y}{\left(-1 - x\right) - y}}{x + y}\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{\frac{-y}{\left(-1 - x\right) - y}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 8: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+157}:\\ \;\;\;\;\frac{y \cdot \frac{1}{x}}{y + \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -2.02 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -9e+157)
   (/ (* y (/ 1.0 x)) (+ y (+ x 1.0)))
   (if (<= x -1.9e+34)
     (* (/ x (* (+ x y) (+ x y))) (/ y x))
     (if (<= x -2.02e-97)
       (* (/ y (+ x (+ y 1.0))) (/ 1.0 x))
       (/ (/ x y) (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -9e+157) {
		tmp = (y * (1.0 / x)) / (y + (x + 1.0));
	} else if (x <= -1.9e+34) {
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	} else if (x <= -2.02e-97) {
		tmp = (y / (x + (y + 1.0))) * (1.0 / x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-9d+157)) then
        tmp = (y * (1.0d0 / x)) / (y + (x + 1.0d0))
    else if (x <= (-1.9d+34)) then
        tmp = (x / ((x + y) * (x + y))) * (y / x)
    else if (x <= (-2.02d-97)) then
        tmp = (y / (x + (y + 1.0d0))) * (1.0d0 / x)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -9e+157) {
		tmp = (y * (1.0 / x)) / (y + (x + 1.0));
	} else if (x <= -1.9e+34) {
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	} else if (x <= -2.02e-97) {
		tmp = (y / (x + (y + 1.0))) * (1.0 / x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -9e+157:
		tmp = (y * (1.0 / x)) / (y + (x + 1.0))
	elif x <= -1.9e+34:
		tmp = (x / ((x + y) * (x + y))) * (y / x)
	elif x <= -2.02e-97:
		tmp = (y / (x + (y + 1.0))) * (1.0 / x)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -9e+157)
		tmp = Float64(Float64(y * Float64(1.0 / x)) / Float64(y + Float64(x + 1.0)));
	elseif (x <= -1.9e+34)
		tmp = Float64(Float64(x / Float64(Float64(x + y) * Float64(x + y))) * Float64(y / x));
	elseif (x <= -2.02e-97)
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) * Float64(1.0 / x));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9e+157)
		tmp = (y * (1.0 / x)) / (y + (x + 1.0));
	elseif (x <= -1.9e+34)
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	elseif (x <= -2.02e-97)
		tmp = (y / (x + (y + 1.0))) * (1.0 / x);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -9e+157], N[(N[(y * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.9e+34], N[(N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.02e-97], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.9999999999999997e157

   1. Initial program 54.8%

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

   2. Taylor expanded in x around inf 54.8%

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

      1. unpow2 54.8%

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

   4. Simplified 54.8%

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

      1. expm1-log1p-u 54.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot y}{\left(x \cdot x\right) \cdot \left(\left(x + y\right) + 1\right)}\right)\right)} \]

      2. expm1-udef 54.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot y}{\left(x \cdot x\right) \cdot \left(\left(x + y\right) + 1\right)}\right)} - 1} \]

      3. times-frac 78.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{x \cdot x} \cdot \frac{y}{\left(x + y\right) + 1}}\right)} - 1 \]

      4. pow1 78.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{{x}^{1}}}{x \cdot x} \cdot \frac{y}{\left(x + y\right) + 1}\right)} - 1 \]

      5. pow2 78.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{{x}^{1}}{\color{blue}{{x}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1}\right)} - 1 \]

      6. pow-div 78.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(1 - 2\right)}} \cdot \frac{y}{\left(x + y\right) + 1}\right)} - 1 \]

      7. metadata-eval 78.0%

         \[\leadsto e^{\mathsf{log1p}\left({x}^{\color{blue}{-1}} \cdot \frac{y}{\left(x + y\right) + 1}\right)} - 1 \]

      8. +-commutative 78.0%

         \[\leadsto e^{\mathsf{log1p}\left({x}^{-1} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1}\right)} - 1 \]

      9. associate-+r+ 78.0%

         \[\leadsto e^{\mathsf{log1p}\left({x}^{-1} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}}\right)} - 1 \]

   6. Applied egg-rr 78.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({x}^{-1} \cdot \frac{y}{y + \left(x + 1\right)}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 88.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-1} \cdot \frac{y}{y + \left(x + 1\right)}\right)\right)} \]

      2. expm1-log1p 88.1%

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

      3. unpow-1 88.1%

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

      4. associate-*r/ 88.1%

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

   8. Simplified 88.1%

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

   if -8.9999999999999997e157 < x < -1.9000000000000001e34

   1. Initial program 53.7%

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

      1. times-frac 92.2%

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

      2. /-rgt-identity 92.2%

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

      3. associate-/l/ 92.2%

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

      4. *-lft-identity 92.2%

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

      5. associate-+l+ 92.2%

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

   3. Simplified 92.2%

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

   4. Taylor expanded in x around inf 80.6%

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

   if -1.9000000000000001e34 < x < -2.0200000000000001e-97

   1. Initial program 84.8%

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

      1. times-frac 99.5%

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

      2. /-rgt-identity 99.5%

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

      3. associate-/l/ 99.5%

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

      4. *-lft-identity 99.5%

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

      5. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 63.7%

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

   if -2.0200000000000001e-97 < x

   1. Initial program 69.2%

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

      1. times-frac 86.8%

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

      2. /-rgt-identity 86.8%

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

      3. associate-/l/ 86.8%

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

      4. *-lft-identity 86.8%

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

      5. associate-+l+ 86.8%

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

   3. Simplified 86.8%

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

   4. Taylor expanded in x around 0 55.3%

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

      1. associate-/r* 57.7%

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

      2. +-commutative 57.7%

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

   6. Simplified 57.7%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+157}:\\ \;\;\;\;\frac{y \cdot \frac{1}{x}}{y + \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -2.02 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 9: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+158}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.2e+158)
   (* (/ x (+ x y)) (/ (/ y x) (+ x y)))
   (if (<= x -1.9e+34)
     (* (/ x (* (+ x y) (+ x y))) (/ y x))
     (if (<= x -2.2e-97)
       (* (/ y (+ x (+ y 1.0))) (/ 1.0 x))
       (/ (/ x y) (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.2e+158) {
		tmp = (x / (x + y)) * ((y / x) / (x + y));
	} else if (x <= -1.9e+34) {
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	} else if (x <= -2.2e-97) {
		tmp = (y / (x + (y + 1.0))) * (1.0 / x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.2d+158)) then
        tmp = (x / (x + y)) * ((y / x) / (x + y))
    else if (x <= (-1.9d+34)) then
        tmp = (x / ((x + y) * (x + y))) * (y / x)
    else if (x <= (-2.2d-97)) then
        tmp = (y / (x + (y + 1.0d0))) * (1.0d0 / x)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.2e+158) {
		tmp = (x / (x + y)) * ((y / x) / (x + y));
	} else if (x <= -1.9e+34) {
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	} else if (x <= -2.2e-97) {
		tmp = (y / (x + (y + 1.0))) * (1.0 / x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.2e+158:
		tmp = (x / (x + y)) * ((y / x) / (x + y))
	elif x <= -1.9e+34:
		tmp = (x / ((x + y) * (x + y))) * (y / x)
	elif x <= -2.2e-97:
		tmp = (y / (x + (y + 1.0))) * (1.0 / x)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.2e+158)
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(Float64(y / x) / Float64(x + y)));
	elseif (x <= -1.9e+34)
		tmp = Float64(Float64(x / Float64(Float64(x + y) * Float64(x + y))) * Float64(y / x));
	elseif (x <= -2.2e-97)
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) * Float64(1.0 / x));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.2e+158)
		tmp = (x / (x + y)) * ((y / x) / (x + y));
	elseif (x <= -1.9e+34)
		tmp = (x / ((x + y) * (x + y))) * (y / x);
	elseif (x <= -2.2e-97)
		tmp = (y / (x + (y + 1.0))) * (1.0 / x);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.2e+158], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(y / x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.9e+34], N[(N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.2e-97], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.20000000000000004e158

   1. Initial program 54.8%

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

      1. times-frac 78.0%

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

      2. associate-+r+ 78.0%

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

      3. associate-*l/ 78.0%

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

      4. times-frac 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+r+ 99.9%

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

      7. +-commutative 99.9%

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

      8. associate-+l+ 99.9%

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

      9. +-commutative 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around inf 88.3%

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

   if -1.20000000000000004e158 < x < -1.9000000000000001e34

   1. Initial program 53.7%

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

      1. times-frac 92.2%

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

      2. /-rgt-identity 92.2%

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

      3. associate-/l/ 92.2%

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

      4. *-lft-identity 92.2%

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

      5. associate-+l+ 92.2%

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

   3. Simplified 92.2%

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

   4. Taylor expanded in x around inf 80.6%

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

   if -1.9000000000000001e34 < x < -2.1999999999999999e-97

   1. Initial program 84.8%

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

      1. times-frac 99.5%

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

      2. /-rgt-identity 99.5%

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

      3. associate-/l/ 99.5%

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

      4. *-lft-identity 99.5%

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

      5. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 63.7%

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

   if -2.1999999999999999e-97 < x

   1. Initial program 69.2%

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

      1. times-frac 86.8%

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

      2. /-rgt-identity 86.8%

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

      3. associate-/l/ 86.8%

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

      4. *-lft-identity 86.8%

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

      5. associate-+l+ 86.8%

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

   3. Simplified 86.8%

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

   4. Taylor expanded in x around 0 55.3%

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

      1. associate-/r* 57.7%

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

      2. +-commutative 57.7%

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

   6. Simplified 57.7%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+158}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 10: 82.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{y \cdot \frac{1}{x}}{y + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.2e-97)
   (/ (* y (/ 1.0 x)) (+ y (+ x 1.0)))
   (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.2e-97) {
		tmp = (y * (1.0 / x)) / (y + (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.2d-97)) then
        tmp = (y * (1.0d0 / x)) / (y + (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.2e-97) {
		tmp = (y * (1.0 / x)) / (y + (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.2e-97:
		tmp = (y * (1.0 / x)) / (y + (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.2e-97)
		tmp = Float64(Float64(y * Float64(1.0 / x)) / Float64(y + Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.2e-97)
		tmp = (y * (1.0 / x)) / (y + (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.2e-97], N[(N[(y * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1999999999999999e-97

   1. Initial program 65.4%

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

   2. Taylor expanded in x around inf 52.2%

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

      1. unpow2 52.2%

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

   4. Simplified 52.2%

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

      1. expm1-log1p-u 51.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot y}{\left(x \cdot x\right) \cdot \left(\left(x + y\right) + 1\right)}\right)\right)} \]

      2. expm1-udef 29.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot y}{\left(x \cdot x\right) \cdot \left(\left(x + y\right) + 1\right)}\right)} - 1} \]

      3. times-frac 40.8%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{x \cdot x} \cdot \frac{y}{\left(x + y\right) + 1}}\right)} - 1 \]

      4. pow1 40.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{{x}^{1}}}{x \cdot x} \cdot \frac{y}{\left(x + y\right) + 1}\right)} - 1 \]

      5. pow2 40.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{{x}^{1}}{\color{blue}{{x}^{2}}} \cdot \frac{y}{\left(x + y\right) + 1}\right)} - 1 \]

      6. pow-div 40.8%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(1 - 2\right)}} \cdot \frac{y}{\left(x + y\right) + 1}\right)} - 1 \]

      7. metadata-eval 40.8%

         \[\leadsto e^{\mathsf{log1p}\left({x}^{\color{blue}{-1}} \cdot \frac{y}{\left(x + y\right) + 1}\right)} - 1 \]

      8. +-commutative 40.8%

         \[\leadsto e^{\mathsf{log1p}\left({x}^{-1} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1}\right)} - 1 \]

      9. associate-+r+ 40.8%

         \[\leadsto e^{\mathsf{log1p}\left({x}^{-1} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}}\right)} - 1 \]

   6. Applied egg-rr 40.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({x}^{-1} \cdot \frac{y}{y + \left(x + 1\right)}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 73.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-1} \cdot \frac{y}{y + \left(x + 1\right)}\right)\right)} \]

      2. expm1-log1p 73.6%

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

      3. unpow-1 73.6%

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

      4. associate-*r/ 73.7%

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

   8. Simplified 73.7%

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

   if -2.1999999999999999e-97 < x

   1. Initial program 69.2%

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

      1. times-frac 86.8%

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

      2. /-rgt-identity 86.8%

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

      3. associate-/l/ 86.8%

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

      4. *-lft-identity 86.8%

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

      5. associate-+l+ 86.8%

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

   3. Simplified 86.8%

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

   4. Taylor expanded in x around 0 55.3%

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

      1. associate-/r* 57.7%

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

      2. +-commutative 57.7%

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

   6. Simplified 57.7%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{y \cdot \frac{1}{x}}{y + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 11: 68.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -180000000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-203} \lor \neg \left(x \leq 2.2 \cdot 10^{-241}\right):\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -180000000000.0)
   (/ y (* x x))
   (if (or (<= x -8e-203) (not (<= x 2.2e-241))) (/ x (* y y)) (/ x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -180000000000.0) {
		tmp = y / (x * x);
	} else if ((x <= -8e-203) || !(x <= 2.2e-241)) {
		tmp = x / (y * y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-180000000000.0d0)) then
        tmp = y / (x * x)
    else if ((x <= (-8d-203)) .or. (.not. (x <= 2.2d-241))) then
        tmp = x / (y * y)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -180000000000.0) {
		tmp = y / (x * x);
	} else if ((x <= -8e-203) || !(x <= 2.2e-241)) {
		tmp = x / (y * y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -180000000000.0:
		tmp = y / (x * x)
	elif (x <= -8e-203) or not (x <= 2.2e-241):
		tmp = x / (y * y)
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -180000000000.0)
		tmp = Float64(y / Float64(x * x));
	elseif ((x <= -8e-203) || !(x <= 2.2e-241))
		tmp = Float64(x / Float64(y * y));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -180000000000.0)
		tmp = y / (x * x);
	elseif ((x <= -8e-203) || ~((x <= 2.2e-241)))
		tmp = x / (y * y);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -180000000000.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -8e-203], N[Not[LessEqual[x, 2.2e-241]], $MachinePrecision]], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.8e11

   1. Initial program 57.0%

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

      1. associate-*r/ 72.7%

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

      2. *-commutative 72.7%

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

      3. distribute-rgt1-in 32.3%

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

      4. fma-def 72.7%

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

      5. cube-unmult 72.7%

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

   3. Simplified 72.7%

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

   4. Taylor expanded in x around inf 72.0%

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

      1. unpow2 72.0%

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

   6. Simplified 72.0%

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

   if -1.8e11 < x < -8.0000000000000003e-203 or 2.1999999999999999e-241 < x

   1. Initial program 73.8%

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

      1. associate-*r/ 86.4%

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

      2. *-commutative 86.4%

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

      3. distribute-rgt1-in 74.8%

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

      4. fma-def 86.4%

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

      5. cube-unmult 86.5%

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

   3. Simplified 86.5%

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

   4. Taylor expanded in y around inf 35.7%

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

      1. unpow2 35.7%

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

   6. Simplified 35.7%

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

   if -8.0000000000000003e-203 < x < 2.1999999999999999e-241

   1. Initial program 61.3%

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

      1. times-frac 77.7%

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

      2. /-rgt-identity 77.7%

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

      3. associate-/l/ 77.7%

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

      4. *-lft-identity 77.7%

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

      5. associate-+l+ 77.7%

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

   3. Simplified 77.7%

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

   4. Taylor expanded in x around 0 89.3%

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

      1. associate-/r* 89.3%

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

      2. +-commutative 89.3%

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

   6. Simplified 89.3%

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

   7. Taylor expanded in y around 0 86.6%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -180000000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-203} \lor \neg \left(x \leq 2.2 \cdot 10^{-241}\right):\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 12: 70.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -15200000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-201}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-240}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -15200000000.0)
   (/ y (* x x))
   (if (<= x -1.45e-201)
     (/ x (* y y))
     (if (<= x 1.65e-240) (/ x y) (/ (/ x y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -15200000000.0) {
		tmp = y / (x * x);
	} else if (x <= -1.45e-201) {
		tmp = x / (y * y);
	} else if (x <= 1.65e-240) {
		tmp = x / y;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-15200000000.0d0)) then
        tmp = y / (x * x)
    else if (x <= (-1.45d-201)) then
        tmp = x / (y * y)
    else if (x <= 1.65d-240) then
        tmp = x / y
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -15200000000.0) {
		tmp = y / (x * x);
	} else if (x <= -1.45e-201) {
		tmp = x / (y * y);
	} else if (x <= 1.65e-240) {
		tmp = x / y;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -15200000000.0:
		tmp = y / (x * x)
	elif x <= -1.45e-201:
		tmp = x / (y * y)
	elif x <= 1.65e-240:
		tmp = x / y
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -15200000000.0)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -1.45e-201)
		tmp = Float64(x / Float64(y * y));
	elseif (x <= 1.65e-240)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -15200000000.0)
		tmp = y / (x * x);
	elseif (x <= -1.45e-201)
		tmp = x / (y * y);
	elseif (x <= 1.65e-240)
		tmp = x / y;
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -15200000000.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.45e-201], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.65e-240], N[(x / y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.52e10

   1. Initial program 57.0%

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

      1. associate-*r/ 72.7%

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

      2. *-commutative 72.7%

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

      3. distribute-rgt1-in 32.3%

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

      4. fma-def 72.7%

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

      5. cube-unmult 72.7%

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

   3. Simplified 72.7%

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

   4. Taylor expanded in x around inf 72.0%

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

      1. unpow2 72.0%

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

   6. Simplified 72.0%

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

   if -1.52e10 < x < -1.4500000000000001e-201

   1. Initial program 82.8%

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

      1. associate-*r/ 95.4%

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

      2. *-commutative 95.4%

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

      3. distribute-rgt1-in 84.6%

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

      4. fma-def 95.5%

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

      5. cube-unmult 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in y around inf 40.5%

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

      1. unpow2 40.5%

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

   6. Simplified 40.5%

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

   if -1.4500000000000001e-201 < x < 1.6500000000000001e-240

   1. Initial program 62.4%

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

      1. times-frac 78.3%

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

      2. /-rgt-identity 78.3%

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

      3. associate-/l/ 78.3%

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

      4. *-lft-identity 78.3%

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

      5. associate-+l+ 78.3%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in x around 0 89.6%

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

      1. associate-/r* 89.6%

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

      2. +-commutative 89.6%

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

   6. Simplified 89.6%

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

   7. Taylor expanded in y around 0 87.0%

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

   if 1.6500000000000001e-240 < x

   1. Initial program 69.8%

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

      1. associate-*r/ 82.5%

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

      2. *-commutative 82.5%

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

      3. distribute-rgt1-in 70.5%

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

      4. fma-def 82.5%

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

      5. cube-unmult 82.5%

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

   3. Simplified 82.5%

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

      1. associate-*r/ 69.8%

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

      2. fma-udef 60.6%

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

      3. cube-mult 60.6%

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

      4. distribute-rgt1-in 69.8%

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

      5. associate-+r+ 69.8%

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

      6. *-commutative 69.8%

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

      7. frac-times 88.8%

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

      8. associate-/r* 99.8%

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

      9. frac-2neg 99.8%

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

      10. frac-times 91.5%

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

      11. +-commutative 91.5%

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

      12. +-commutative 91.5%

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

      13. associate-+r+ 91.5%

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

      14. +-commutative 91.5%

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

      15. associate-+l+ 91.5%

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

   5. Applied egg-rr 91.5%

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

      1. *-commutative 91.5%

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

      2. *-commutative 91.5%

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

      3. associate-/r* 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. +-commutative 99.8%

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

      6. unsub-neg 99.8%

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

      7. +-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in y around inf 34.0%

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

      1. unpow2 34.0%

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

      2. associate-/r* 37.6%

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

   10. Simplified 37.6%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -15200000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-201}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-240}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 13: 77.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -11600000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -11600000000.0)
   (/ y (* x x))
   (if (<= x 2.1e+16) (/ x (+ y (* y y))) (/ (/ x y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -11600000000.0) {
		tmp = y / (x * x);
	} else if (x <= 2.1e+16) {
		tmp = x / (y + (y * y));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-11600000000.0d0)) then
        tmp = y / (x * x)
    else if (x <= 2.1d+16) then
        tmp = x / (y + (y * y))
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -11600000000.0) {
		tmp = y / (x * x);
	} else if (x <= 2.1e+16) {
		tmp = x / (y + (y * y));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -11600000000.0:
		tmp = y / (x * x)
	elif x <= 2.1e+16:
		tmp = x / (y + (y * y))
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -11600000000.0)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= 2.1e+16)
		tmp = Float64(x / Float64(y + Float64(y * y)));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -11600000000.0)
		tmp = y / (x * x);
	elseif (x <= 2.1e+16)
		tmp = x / (y + (y * y));
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -11600000000.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.1e+16], N[(x / N[(y + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.16e10

   1. Initial program 57.0%

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

      1. associate-*r/ 72.7%

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

      2. *-commutative 72.7%

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

      3. distribute-rgt1-in 32.3%

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

      4. fma-def 72.7%

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

      5. cube-unmult 72.7%

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

   3. Simplified 72.7%

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

   4. Taylor expanded in x around inf 72.0%

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

      1. unpow2 72.0%

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

   6. Simplified 72.0%

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

   if -1.16e10 < x < 2.1e16

   1. Initial program 76.2%

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

      1. times-frac 89.8%

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

      2. /-rgt-identity 89.8%

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

      3. associate-/l/ 89.8%

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

      4. *-lft-identity 89.8%

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

      5. associate-+l+ 89.8%

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

   3. Simplified 89.8%

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

   4. Taylor expanded in x around 0 70.3%

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

      1. distribute-lft-in 70.3%

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

      2. *-rgt-identity 70.3%

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

   6. Simplified 70.3%

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

   if 2.1e16 < x

   1. Initial program 62.2%

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

      1. associate-*r/ 76.2%

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

      2. *-commutative 76.2%

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

      3. distribute-rgt1-in 69.6%

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

      4. fma-def 76.2%

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

      5. cube-unmult 76.2%

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

   3. Simplified 76.2%

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

      1. associate-*r/ 62.2%

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

      2. fma-udef 60.4%

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

      3. cube-mult 60.4%

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

      4. distribute-rgt1-in 62.2%

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

      5. associate-+r+ 62.2%

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

      6. *-commutative 62.2%

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

      7. frac-times 85.6%

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

      8. associate-/r* 99.8%

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

      9. frac-2neg 99.8%

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

      10. frac-times 85.6%

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

      11. +-commutative 85.6%

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

      12. +-commutative 85.6%

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

      13. associate-+r+ 85.6%

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

      14. +-commutative 85.6%

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

      15. associate-+l+ 85.6%

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

   5. Applied egg-rr 85.6%

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

      1. *-commutative 85.6%

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

      2. *-commutative 85.6%

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

      3. associate-/r* 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. +-commutative 99.7%

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

      6. unsub-neg 99.7%

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

      7. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around inf 19.5%

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

      1. unpow2 19.5%

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

      2. associate-/r* 25.6%

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

   10. Simplified 25.6%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -11600000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 14: 80.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.2e-97)
   (/ y (* x (+ x 1.0)))
   (if (<= x 1.55e+17) (/ x (+ y (* y y))) (/ (/ x y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.2e-97) {
		tmp = y / (x * (x + 1.0));
	} else if (x <= 1.55e+17) {
		tmp = x / (y + (y * y));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.2d-97)) then
        tmp = y / (x * (x + 1.0d0))
    else if (x <= 1.55d+17) then
        tmp = x / (y + (y * y))
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.2e-97) {
		tmp = y / (x * (x + 1.0));
	} else if (x <= 1.55e+17) {
		tmp = x / (y + (y * y));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.2e-97:
		tmp = y / (x * (x + 1.0))
	elif x <= 1.55e+17:
		tmp = x / (y + (y * y))
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.2e-97)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (x <= 1.55e+17)
		tmp = Float64(x / Float64(y + Float64(y * y)));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.2e-97)
		tmp = y / (x * (x + 1.0));
	elseif (x <= 1.55e+17)
		tmp = x / (y + (y * y));
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.2e-97], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.55e+17], N[(x / N[(y + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1999999999999999e-97

   1. Initial program 65.4%

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

      1. times-frac 89.9%

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

      2. /-rgt-identity 89.9%

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

      3. associate-/l/ 89.9%

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

      4. *-lft-identity 89.9%

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

      5. associate-+l+ 89.9%

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

   3. Simplified 89.9%

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

   4. Taylor expanded in y around 0 69.9%

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

   if -2.1999999999999999e-97 < x < 1.55e17

   1. Initial program 73.5%

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

      1. times-frac 87.6%

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

      2. /-rgt-identity 87.6%

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

      3. associate-/l/ 87.6%

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

      4. *-lft-identity 87.6%

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

      5. associate-+l+ 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in x around 0 77.3%

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

      1. distribute-lft-in 77.3%

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

      2. *-rgt-identity 77.3%

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

   6. Simplified 77.3%

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

   if 1.55e17 < x

   1. Initial program 62.2%

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

      1. associate-*r/ 76.2%

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

      2. *-commutative 76.2%

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

      3. distribute-rgt1-in 69.6%

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

      4. fma-def 76.2%

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

      5. cube-unmult 76.2%

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

   3. Simplified 76.2%

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

      1. associate-*r/ 62.2%

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

      2. fma-udef 60.4%

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

      3. cube-mult 60.4%

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

      4. distribute-rgt1-in 62.2%

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

      5. associate-+r+ 62.2%

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

      6. *-commutative 62.2%

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

      7. frac-times 85.6%

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

      8. associate-/r* 99.8%

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

      9. frac-2neg 99.8%

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

      10. frac-times 85.6%

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

      11. +-commutative 85.6%

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

      12. +-commutative 85.6%

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

      13. associate-+r+ 85.6%

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

      14. +-commutative 85.6%

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

      15. associate-+l+ 85.6%

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

   5. Applied egg-rr 85.6%

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

      1. *-commutative 85.6%

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

      2. *-commutative 85.6%

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

      3. associate-/r* 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. +-commutative 99.7%

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

      6. unsub-neg 99.7%

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

      7. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around inf 19.5%

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

      1. unpow2 19.5%

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

      2. associate-/r* 25.6%

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

   10. Simplified 25.6%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 15: 82.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -9.4 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -9.4e+14)
   (/ (/ y x) (+ x y))
   (if (<= x -2.2e-97) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -9.4e+14) {
		tmp = (y / x) / (x + y);
	} else if (x <= -2.2e-97) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-9.4d+14)) then
        tmp = (y / x) / (x + y)
    else if (x <= (-2.2d-97)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -9.4e+14) {
		tmp = (y / x) / (x + y);
	} else if (x <= -2.2e-97) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -9.4e+14:
		tmp = (y / x) / (x + y)
	elif x <= -2.2e-97:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -9.4e+14)
		tmp = Float64(Float64(y / x) / Float64(x + y));
	elseif (x <= -2.2e-97)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9.4e+14)
		tmp = (y / x) / (x + y);
	elseif (x <= -2.2e-97)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -9.4e+14], N[(N[(y / x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.2e-97], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.4e14

   1. Initial program 56.4%

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

      1. associate-*r/ 72.3%

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

      2. *-commutative 72.3%

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

      3. distribute-rgt1-in 31.3%

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

      4. fma-def 72.3%

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

      5. cube-unmult 72.3%

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

   3. Simplified 72.3%

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

      1. associate-*r/ 56.4%

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

      2. fma-udef 29.4%

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

      3. cube-mult 29.4%

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

      4. distribute-rgt1-in 56.4%

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

      5. associate-+r+ 56.4%

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

      6. *-commutative 56.4%

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

      7. frac-times 86.0%

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

      8. associate-/r* 99.6%

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

      9. frac-2neg 99.6%

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

      10. frac-times 86.1%

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

      11. +-commutative 86.1%

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

      12. +-commutative 86.1%

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

      13. associate-+r+ 86.1%

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

      14. +-commutative 86.1%

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

      15. associate-+l+ 86.1%

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

   5. Applied egg-rr 86.1%

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

      1. *-commutative 86.1%

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

      2. *-commutative 86.1%

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

      3. associate-/r* 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. +-commutative 99.7%

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

      6. unsub-neg 99.7%

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

      7. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around inf 77.1%

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

   if -9.4e14 < x < -2.1999999999999999e-97

   1. Initial program 88.3%

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

      1. times-frac 99.5%

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

      2. /-rgt-identity 99.5%

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

      3. associate-/l/ 99.5%

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

      4. *-lft-identity 99.5%

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

      5. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 64.8%

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

   if -2.1999999999999999e-97 < x

   1. Initial program 69.2%

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

      1. times-frac 86.8%

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

      2. /-rgt-identity 86.8%

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

      3. associate-/l/ 86.8%

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

      4. *-lft-identity 86.8%

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

      5. associate-+l+ 86.8%

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

   3. Simplified 86.8%

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

   4. Taylor expanded in x around 0 55.3%

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

      1. associate-/r* 57.7%

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

      2. +-commutative 57.7%

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

   6. Simplified 57.7%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.4 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 16: 82.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.2e-97) (/ (/ y (+ x 1.0)) (+ x y)) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.2e-97) {
		tmp = (y / (x + 1.0)) / (x + y);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.2d-97)) then
        tmp = (y / (x + 1.0d0)) / (x + y)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.2e-97) {
		tmp = (y / (x + 1.0)) / (x + y);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.2e-97:
		tmp = (y / (x + 1.0)) / (x + y)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.2e-97)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(x + y));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.2e-97)
		tmp = (y / (x + 1.0)) / (x + y);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.2e-97], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1999999999999999e-97

   1. Initial program 65.4%

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

      1. associate-*r/ 80.0%

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

      2. *-commutative 80.0%

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

      3. distribute-rgt1-in 48.4%

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

      4. fma-def 80.0%

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

      5. cube-unmult 80.1%

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

   3. Simplified 80.1%

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

      1. associate-*r/ 65.4%

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

      2. fma-udef 43.8%

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

      3. cube-mult 43.8%

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

      4. distribute-rgt1-in 65.4%

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

      5. associate-+r+ 65.4%

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

      6. *-commutative 65.4%

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

      7. frac-times 89.9%

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

      8. associate-/r* 99.6%

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

      9. frac-2neg 99.6%

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

      10. frac-times 90.0%

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

      11. +-commutative 90.0%

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

      12. +-commutative 90.0%

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

      13. associate-+r+ 90.0%

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

      14. +-commutative 90.0%

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

      15. associate-+l+ 90.0%

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

   5. Applied egg-rr 90.0%

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

      1. *-commutative 90.0%

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

      2. *-commutative 90.0%

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

      3. associate-/r* 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. +-commutative 99.7%

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

      6. unsub-neg 99.7%

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

      7. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around 0 73.7%

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

      1. +-commutative 73.7%

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

   10. Simplified 73.7%

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

   if -2.1999999999999999e-97 < x

   1. Initial program 69.2%

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

      1. times-frac 86.8%

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

      2. /-rgt-identity 86.8%

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

      3. associate-/l/ 86.8%

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

      4. *-lft-identity 86.8%

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

      5. associate-+l+ 86.8%

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

   3. Simplified 86.8%

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

   4. Taylor expanded in x around 0 55.3%

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

      1. associate-/r* 57.7%

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

      2. +-commutative 57.7%

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

   6. Simplified 57.7%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 17: 80.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.2e-97) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.2e-97) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.2d-97)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.2e-97) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.2e-97:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.2e-97)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.2e-97)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.2e-97], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1999999999999999e-97

   1. Initial program 65.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 89.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 89.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. associate-/l/ 89.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lft-identity 89.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      5. associate-+l+ 89.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 89.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 69.9%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]

   if -2.1999999999999999e-97 < x

   1. Initial program 69.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 86.8%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 86.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. associate-/l/ 86.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lft-identity 86.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      5. associate-+l+ 86.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 86.8%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around 0 55.3%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 57.7%

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

      2. +-commutative 57.7%

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y + 1}} \]

   6. Simplified 57.7%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 18: 47.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 0.76:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 0.76) (- (/ x y) x) (/ x (* y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 0.76) {
		tmp = (x / y) - x;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.76d0) then
        tmp = (x / y) - x
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 0.76) {
		tmp = (x / y) - x;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 0.76:
		tmp = (x / y) - x
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 0.76)
		tmp = Float64(Float64(x / y) - x);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.76)
		tmp = (x / y) - x;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 0.76], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.76000000000000001

   1. Initial program 66.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 87.7%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 87.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. associate-/l/ 87.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lft-identity 87.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      5. associate-+l+ 87.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 87.7%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around 0 37.6%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 40.2%

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

      2. +-commutative 40.2%

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y + 1}} \]

   6. Simplified 40.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]

   7. Taylor expanded in y around 0 16.8%

      \[\leadsto \color{blue}{\frac{x}{y} + -1 \cdot x} \]
   8. Step-by-step derivation

      1. mul-1-neg 16.8%

         \[\leadsto \frac{x}{y} + \color{blue}{\left(-x\right)} \]

      2. unsub-neg 16.8%

         \[\leadsto \color{blue}{\frac{x}{y} - x} \]

   9. Simplified 16.8%

      \[\leadsto \color{blue}{\frac{x}{y} - x} \]

   if 0.76000000000000001 < y

   1. Initial program 72.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*r/ 85.4%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. *-commutative 85.4%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      3. distribute-rgt1-in 79.8%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. fma-def 85.4%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(x + y, x + y, \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)\right)}} \]

      5. cube-unmult 85.4%

         \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(x + y\right)}^{3}}\right)} \]

   3. Simplified 85.4%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   4. Taylor expanded in y around inf 67.5%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 67.5%

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   6. Simplified 67.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.76:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 19: 28.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;\frac{0.3333333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -0.96) (/ 0.3333333333333333 x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -0.96) {
		tmp = 0.3333333333333333 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.96d0)) then
        tmp = 0.3333333333333333d0 / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -0.96) {
		tmp = 0.3333333333333333 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -0.96:
		tmp = 0.3333333333333333 / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -0.96)
		tmp = Float64(0.3333333333333333 / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.96)
		tmp = 0.3333333333333333 / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -0.96], N[(0.3333333333333333 / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.95999999999999996

   1. Initial program 59.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*r/ 73.9%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. *-commutative 73.9%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      3. distribute-rgt1-in 35.4%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. fma-def 73.9%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(x + y, x + y, \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)\right)}} \]

      5. cube-unmult 73.9%

         \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(x + y\right)}^{3}}\right)} \]

   3. Simplified 73.9%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   4. Taylor expanded in x around inf 44.0%

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(1 + \left(2 \cdot y + y\right)\right) \cdot {x}^{2} + {x}^{3}}} \]
   5. Step-by-step derivation

      1. +-commutative 44.0%

         \[\leadsto x \cdot \frac{y}{\color{blue}{{x}^{3} + \left(1 + \left(2 \cdot y + y\right)\right) \cdot {x}^{2}}} \]

      2. cube-mult 44.0%

         \[\leadsto x \cdot \frac{y}{\color{blue}{x \cdot \left(x \cdot x\right)} + \left(1 + \left(2 \cdot y + y\right)\right) \cdot {x}^{2}} \]

      3. unpow2 44.0%

         \[\leadsto x \cdot \frac{y}{x \cdot \color{blue}{{x}^{2}} + \left(1 + \left(2 \cdot y + y\right)\right) \cdot {x}^{2}} \]

      4. distribute-rgt-out 69.6%

         \[\leadsto x \cdot \frac{y}{\color{blue}{{x}^{2} \cdot \left(x + \left(1 + \left(2 \cdot y + y\right)\right)\right)}} \]

      5. unpow2 69.6%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(x \cdot x\right)} \cdot \left(x + \left(1 + \left(2 \cdot y + y\right)\right)\right)} \]

      6. distribute-lft1-in 69.6%

         \[\leadsto x \cdot \frac{y}{\left(x \cdot x\right) \cdot \left(x + \left(1 + \color{blue}{\left(2 + 1\right) \cdot y}\right)\right)} \]

      7. metadata-eval 69.6%

         \[\leadsto x \cdot \frac{y}{\left(x \cdot x\right) \cdot \left(x + \left(1 + \color{blue}{3} \cdot y\right)\right)} \]

   6. Simplified 69.6%

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(x \cdot x\right) \cdot \left(x + \left(1 + 3 \cdot y\right)\right)}} \]

   7. Taylor expanded in y around inf 6.5%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{x}} \]

   if -0.95999999999999996 < x

   1. Initial program 71.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 88.2%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 88.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. associate-/l/ 88.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lft-identity 88.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      5. associate-+l+ 88.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 88.2%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around 0 54.2%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 56.3%

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

      2. +-commutative 56.3%

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y + 1}} \]

   6. Simplified 56.3%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]

   7. Taylor expanded in y around 0 36.5%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;\frac{0.3333333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 20: 4.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{0.3333333333333333}{x} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ 0.3333333333333333 x))

C

assert(x < y);
double code(double x, double y) {
	return 0.3333333333333333 / 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 = 0.3333333333333333d0 / x
end function

Java

assert x < y;
public static double code(double x, double y) {
	return 0.3333333333333333 / x;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return 0.3333333333333333 / x

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(0.3333333333333333 / x)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 0.3333333333333333 / x;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(0.3333333333333333 / x), $MachinePrecision]

Derivation

1. Initial program 67.9%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-*r/ 81.8%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   2. *-commutative 81.8%

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   3. distribute-rgt1-in 62.6%

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   4. fma-def 81.8%

      \[\leadsto x \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(x + y, x + y, \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)\right)}} \]

   5. cube-unmult 81.8%

      \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(x + y\right)}^{3}}\right)} \]

3. Simplified 81.8%

   \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

4. Taylor expanded in x around inf 39.4%

   \[\leadsto x \cdot \frac{y}{\color{blue}{\left(1 + \left(2 \cdot y + y\right)\right) \cdot {x}^{2} + {x}^{3}}} \]
5. Step-by-step derivation

   1. +-commutative 39.4%

      \[\leadsto x \cdot \frac{y}{\color{blue}{{x}^{3} + \left(1 + \left(2 \cdot y + y\right)\right) \cdot {x}^{2}}} \]

   2. cube-mult 39.4%

      \[\leadsto x \cdot \frac{y}{\color{blue}{x \cdot \left(x \cdot x\right)} + \left(1 + \left(2 \cdot y + y\right)\right) \cdot {x}^{2}} \]

   3. unpow2 39.4%

      \[\leadsto x \cdot \frac{y}{x \cdot \color{blue}{{x}^{2}} + \left(1 + \left(2 \cdot y + y\right)\right) \cdot {x}^{2}} \]

   4. distribute-rgt-out 48.2%

      \[\leadsto x \cdot \frac{y}{\color{blue}{{x}^{2} \cdot \left(x + \left(1 + \left(2 \cdot y + y\right)\right)\right)}} \]

   5. unpow2 48.2%

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(x \cdot x\right)} \cdot \left(x + \left(1 + \left(2 \cdot y + y\right)\right)\right)} \]

   6. distribute-lft1-in 48.2%

      \[\leadsto x \cdot \frac{y}{\left(x \cdot x\right) \cdot \left(x + \left(1 + \color{blue}{\left(2 + 1\right) \cdot y}\right)\right)} \]

   7. metadata-eval 48.2%

      \[\leadsto x \cdot \frac{y}{\left(x \cdot x\right) \cdot \left(x + \left(1 + \color{blue}{3} \cdot y\right)\right)} \]

6. Simplified 48.2%

   \[\leadsto x \cdot \frac{y}{\color{blue}{\left(x \cdot x\right) \cdot \left(x + \left(1 + 3 \cdot y\right)\right)}} \]

7. Taylor expanded in y around inf 4.4%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{x}} \]

8. Final simplification 4.4%

   \[\leadsto \frac{0.3333333333333333}{x} \]

Alternative 21: 3.5% accurate, 17.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ 1 \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 1.0)

C

assert(x < y);
double code(double x, double y) {
	return 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 = 1.0d0
end function

Java

assert x < y;
public static double code(double x, double y) {
	return 1.0;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return 1.0

Julia

x, y = sort([x, y])
function code(x, y)
	return 1.0
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := 1.0

Derivation

1. Initial program 67.9%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. times-frac 87.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. associate-+r+ 87.9%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. associate-*l/ 81.4%

      \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{x + \left(y + 1\right)}}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   4. times-frac 99.8%

      \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y}} \]

   5. +-commutative 99.8%

      \[\leadsto \frac{x}{\color{blue}{y + x}} \cdot \frac{\frac{y}{x + \left(y + 1\right)}}{x + y} \]

   6. associate-+r+ 99.8%

      \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}}}{x + y} \]

   7. +-commutative 99.8%

      \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{\left(y + x\right)} + 1}}{x + y} \]

   8. associate-+l+ 99.8%

      \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{\color{blue}{y + \left(x + 1\right)}}}{x + y} \]

   9. +-commutative 99.8%

      \[\leadsto \frac{x}{y + x} \cdot \frac{\frac{y}{y + \left(x + 1\right)}}{\color{blue}{y + x}} \]

3. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{x}{y + x} \cdot \frac{\frac{y}{y + \left(x + 1\right)}}{y + x}} \]

4. Taylor expanded in x around 0 48.2%

   \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{1}{1 + y}} \]
5. Step-by-step derivation

   1. +-commutative 48.2%

      \[\leadsto \frac{x}{y + x} \cdot \frac{1}{\color{blue}{y + 1}} \]

6. Simplified 48.2%

   \[\leadsto \frac{x}{y + x} \cdot \color{blue}{\frac{1}{y + 1}} \]

7. Taylor expanded in y around 0 3.4%

   \[\leadsto \color{blue}{1} \]

8. Final simplification 3.4%

   \[\leadsto 1 \]

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 2023238 
(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))))