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

Percentage Accurate: 68.4% → 98.4%

Time: 13.8s

Alternatives: 14

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{{\left(\sqrt[3]{y}\right)}^{2} \cdot \frac{\sqrt[3]{y}}{y + x}}{\frac{x + \left(y + 1\right)}{x} \cdot \left(y + x\right)} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(N[(N[Power[N[Power[y, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[y, 1/3], $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.2%

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

   1. associate-*l* 64.2%

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

   2. *-commutative 64.2%

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

   3. times-frac 93.9%

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

   4. +-commutative 93.9%

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

   5. +-commutative 93.9%

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

   6. distribute-lft-in 93.9%

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

   7. *-rgt-identity 93.9%

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

   8. +-commutative 93.9%

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

   9. pow2 93.9%

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

   10. +-commutative 93.9%

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

4. Applied egg-rr 93.9%

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

   1. *-commutative 93.9%

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

   2. clear-num 93.2%

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

   3. frac-times 90.7%

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

   4. *-un-lft-identity 90.7%

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

   5. +-commutative 90.7%

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

   6. +-commutative 90.7%

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

   7. +-commutative 90.7%

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

6. Applied egg-rr 90.7%

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

   1. add-cube-cbrt 89.9%

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

   2. times-frac 92.4%

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

   3. pow2 92.4%

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

   4. pow2 92.4%

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

   5. +-commutative 92.4%

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

   6. *-un-lft-identity 92.4%

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

   7. distribute-rgt-out 92.4%

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

8. Applied egg-rr 92.4%

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

   1. associate-*l/ 92.4%

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

   2. +-commutative 92.4%

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

   3. associate-/l* 97.6%

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

   4. +-commutative 97.6%

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

   5. associate-+l+ 97.6%

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

10. Simplified 97.6%

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

11. Final simplification 97.6%

    \[\leadsto \frac{{\left(\sqrt[3]{y}\right)}^{2} \cdot \frac{\sqrt[3]{y}}{y + x}}{\frac{x + \left(y + 1\right)}{x} \cdot \left(y + x\right)} \]
12. Add Preprocessing

Alternative 2: 92.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{y}{x}}{\frac{t\_0}{x} \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-156}:\\ \;\;\;\;x \cdot \frac{y}{t\_0 \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-208}:\\ \;\;\;\;\frac{y \cdot \frac{x}{\left(y + x\right) \cdot \left(1 + \left(y + x\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= x -1.65e+26)
     (/ (/ y x) (* (/ t_0 x) (+ y x)))
     (if (<= x -6.6e-156)
       (* x (/ y (* t_0 (* (+ y x) (+ y x)))))
       (if (<= x -6.5e-208)
         (/ (* y (/ x (* (+ y x) (+ 1.0 (+ y x))))) x)
         (/ (/ x y) (+ y 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (x <= -1.65e+26) {
		tmp = (y / x) / ((t_0 / x) * (y + x));
	} else if (x <= -6.6e-156) {
		tmp = x * (y / (t_0 * ((y + x) * (y + x))));
	} else if (x <= -6.5e-208) {
		tmp = (y * (x / ((y + x) * (1.0 + (y + x))))) / 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) :: t_0
    real(8) :: tmp
    t_0 = x + (y + 1.0d0)
    if (x <= (-1.65d+26)) then
        tmp = (y / x) / ((t_0 / x) * (y + x))
    else if (x <= (-6.6d-156)) then
        tmp = x * (y / (t_0 * ((y + x) * (y + x))))
    else if (x <= (-6.5d-208)) then
        tmp = (y * (x / ((y + x) * (1.0d0 + (y + x))))) / 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 t_0 = x + (y + 1.0);
	double tmp;
	if (x <= -1.65e+26) {
		tmp = (y / x) / ((t_0 / x) * (y + x));
	} else if (x <= -6.6e-156) {
		tmp = x * (y / (t_0 * ((y + x) * (y + x))));
	} else if (x <= -6.5e-208) {
		tmp = (y * (x / ((y + x) * (1.0 + (y + x))))) / x;
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x + (y + 1.0)
	tmp = 0
	if x <= -1.65e+26:
		tmp = (y / x) / ((t_0 / x) * (y + x))
	elif x <= -6.6e-156:
		tmp = x * (y / (t_0 * ((y + x) * (y + x))))
	elif x <= -6.5e-208:
		tmp = (y * (x / ((y + x) * (1.0 + (y + x))))) / x
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (x <= -1.65e+26)
		tmp = Float64(Float64(y / x) / Float64(Float64(t_0 / x) * Float64(y + x)));
	elseif (x <= -6.6e-156)
		tmp = Float64(x * Float64(y / Float64(t_0 * Float64(Float64(y + x) * Float64(y + x)))));
	elseif (x <= -6.5e-208)
		tmp = Float64(Float64(y * Float64(x / Float64(Float64(y + x) * Float64(1.0 + Float64(y + x))))) / 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)
	t_0 = x + (y + 1.0);
	tmp = 0.0;
	if (x <= -1.65e+26)
		tmp = (y / x) / ((t_0 / x) * (y + x));
	elseif (x <= -6.6e-156)
		tmp = x * (y / (t_0 * ((y + x) * (y + x))));
	elseif (x <= -6.5e-208)
		tmp = (y * (x / ((y + x) * (1.0 + (y + x))))) / 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_] := Block[{t$95$0 = N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.65e+26], N[(N[(y / x), $MachinePrecision] / N[(N[(t$95$0 / x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6.6e-156], N[(x * N[(y / N[(t$95$0 * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6.5e-208], N[(N[(y * N[(x / N[(N[(y + x), $MachinePrecision] * N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.64999999999999997e26

   1. Initial program 45.8%

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

      1. associate-*l* 45.8%

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

      2. *-commutative 45.8%

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

      3. times-frac 78.3%

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

      4. +-commutative 78.3%

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

      5. +-commutative 78.3%

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

      6. distribute-lft-in 78.3%

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

      7. *-rgt-identity 78.3%

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

      8. +-commutative 78.3%

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

      9. pow2 78.3%

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

      10. +-commutative 78.3%

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

   4. Applied egg-rr 78.3%

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

      1. unpow2 78.3%

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

      2. +-commutative 78.3%

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

      3. +-commutative 78.3%

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

   6. Applied egg-rr 78.3%

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

   7. Taylor expanded in y around 0 70.7%

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

      1. clear-num 70.8%

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

      2. +-commutative 70.8%

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

      3. pow2 70.8%

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

      4. frac-times 70.7%

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

      5. *-rgt-identity 70.7%

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

      6. pow2 70.7%

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

      7. +-commutative 70.7%

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

      8. *-un-lft-identity 70.7%

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

      9. distribute-rgt-out 70.7%

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

   9. Applied egg-rr 70.7%

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

       1. associate-/r* 70.8%

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

       2. associate-/l* 76.3%

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

       3. +-commutative 76.3%

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

       4. associate-+l+ 76.3%

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

   11. Simplified 76.3%

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

   if -1.64999999999999997e26 < x < -6.5999999999999997e-156

   1. Initial program 81.2%

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

      1. associate-/l* 96.5%

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

      2. associate-+l+ 96.5%

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

   3. Simplified 96.5%

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

   if -6.5999999999999997e-156 < x < -6.4999999999999998e-208

   1. Initial program 67.2%

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

      1. associate-*l* 67.2%

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

      2. *-commutative 67.2%

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

      3. times-frac 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-lft-in 99.8%

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

      7. *-rgt-identity 99.8%

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

      8. +-commutative 99.8%

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

      9. pow2 99.8%

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

      10. +-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. unpow2 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around 0 34.5%

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

      1. associate-*l/ 67.8%

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

      2. +-commutative 67.8%

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

      3. +-commutative 67.8%

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

      4. *-un-lft-identity 67.8%

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

      5. distribute-rgt-out 67.8%

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

   9. Applied egg-rr 67.8%

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

   if -6.4999999999999998e-208 < x

   1. Initial program 65.7%

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

      1. associate-/l* 78.7%

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

      2. associate-+l+ 78.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in x around 0 55.9%

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

      1. associate-/r* 56.3%

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

      2. +-commutative 56.3%

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

   7. Simplified 56.3%

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

      1. associate-*r/ 56.7%

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

      2. div-inv 56.7%

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

   9. Applied egg-rr 56.7%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{y}{x}}{\frac{x + \left(y + 1\right)}{x} \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-156}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-208}:\\ \;\;\;\;\frac{y \cdot \frac{x}{\left(y + x\right) \cdot \left(1 + \left(y + x\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{y + x}{y}}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-208}:\\ \;\;\;\;\frac{y \cdot \frac{x}{\left(y + x\right) \cdot \left(1 + \left(y + x\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.2e+117)
   (/ (/ 1.0 x) (/ (+ y x) y))
   (if (<= x -7e-208)
     (/ (* y (/ x (* (+ y x) (+ 1.0 (+ y x))))) x)
     (/ (/ x y) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.2e+117) {
		tmp = (1.0 / x) / ((y + x) / y);
	} else if (x <= -7e-208) {
		tmp = (y * (x / ((y + x) * (1.0 + (y + x))))) / 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+117)) then
        tmp = (1.0d0 / x) / ((y + x) / y)
    else if (x <= (-7d-208)) then
        tmp = (y * (x / ((y + x) * (1.0d0 + (y + x))))) / 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+117) {
		tmp = (1.0 / x) / ((y + x) / y);
	} else if (x <= -7e-208) {
		tmp = (y * (x / ((y + x) * (1.0 + (y + x))))) / 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+117:
		tmp = (1.0 / x) / ((y + x) / y)
	elif x <= -7e-208:
		tmp = (y * (x / ((y + x) * (1.0 + (y + x))))) / 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+117)
		tmp = Float64(Float64(1.0 / x) / Float64(Float64(y + x) / y));
	elseif (x <= -7e-208)
		tmp = Float64(Float64(y * Float64(x / Float64(Float64(y + x) * Float64(1.0 + Float64(y + x))))) / 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+117)
		tmp = (1.0 / x) / ((y + x) / y);
	elseif (x <= -7e-208)
		tmp = (y * (x / ((y + x) * (1.0 + (y + x))))) / 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+117], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7e-208], N[(N[(y * N[(x / N[(N[(y + x), $MachinePrecision] * N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1999999999999999e117

   1. Initial program 39.6%

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

      1. associate-*l* 39.6%

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

      2. *-commutative 39.6%

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

      3. times-frac 71.3%

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

      4. +-commutative 71.3%

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

      5. +-commutative 71.3%

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

      6. distribute-lft-in 71.3%

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

      7. *-rgt-identity 71.3%

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

      8. +-commutative 71.3%

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

      9. pow2 71.3%

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

      10. +-commutative 71.3%

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

   4. Applied egg-rr 71.3%

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

   5. Taylor expanded in x around inf 64.5%

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

      1. *-commutative 64.5%

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

      2. clear-num 64.5%

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

      3. un-div-inv 64.5%

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

      4. +-commutative 64.5%

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

   7. Applied egg-rr 64.5%

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

   if -1.1999999999999999e117 < x < -6.99999999999999982e-208

   1. Initial program 75.1%

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

      1. associate-*l* 75.0%

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

      2. *-commutative 75.0%

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

      3. times-frac 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-lft-in 99.7%

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

      7. *-rgt-identity 99.7%

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

      8. +-commutative 99.7%

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

      9. pow2 99.7%

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

      10. +-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. unpow2 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around 0 65.9%

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

      1. associate-*l/ 75.8%

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

      2. +-commutative 75.8%

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

      3. +-commutative 75.8%

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

      4. *-un-lft-identity 75.8%

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

      5. distribute-rgt-out 75.9%

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

   9. Applied egg-rr 75.9%

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

   if -6.99999999999999982e-208 < x

   1. Initial program 65.7%

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

      1. associate-/l* 78.7%

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

      2. associate-+l+ 78.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in x around 0 55.9%

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

      1. associate-/r* 56.3%

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

      2. +-commutative 56.3%

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

   7. Simplified 56.3%

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

      1. associate-*r/ 56.7%

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

      2. div-inv 56.7%

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

   9. Applied egg-rr 56.7%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{y + x}{y}}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-208}:\\ \;\;\;\;\frac{y \cdot \frac{x}{\left(y + x\right) \cdot \left(1 + \left(y + x\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{+150}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) + \left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 5.5e+150)
   (* (/ y (+ y x)) (/ x (+ (+ y x) (* (+ y x) (+ y x)))))
   (* (/ x y) (/ 1.0 (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 5.5e+150) {
		tmp = (y / (y + x)) * (x / ((y + x) + ((y + x) * (y + x))));
	} else {
		tmp = (x / y) * (1.0 / (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5.5d+150) then
        tmp = (y / (y + x)) * (x / ((y + x) + ((y + x) * (y + x))))
    else
        tmp = (x / y) * (1.0d0 / (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 5.5e+150) {
		tmp = (y / (y + x)) * (x / ((y + x) + ((y + x) * (y + x))));
	} else {
		tmp = (x / y) * (1.0 / (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 5.5e+150:
		tmp = (y / (y + x)) * (x / ((y + x) + ((y + x) * (y + x))))
	else:
		tmp = (x / y) * (1.0 / (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 5.5e+150)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(Float64(y + x) + Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.5e+150)
		tmp = (y / (y + x)) * (x / ((y + x) + ((y + x) * (y + x))));
	else
		tmp = (x / y) * (1.0 / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 5.5e+150], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(y + x), $MachinePrecision] + N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.50000000000000017e150

   1. Initial program 66.8%

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

      1. associate-*l* 66.8%

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

      2. *-commutative 66.8%

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

      3. times-frac 96.2%

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

      4. +-commutative 96.2%

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

      5. +-commutative 96.2%

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

      6. distribute-lft-in 96.1%

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

      7. *-rgt-identity 96.1%

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

      8. +-commutative 96.1%

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

      9. pow2 96.1%

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

      10. +-commutative 96.1%

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

   4. Applied egg-rr 96.1%

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

      1. unpow2 96.1%

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

      2. +-commutative 96.1%

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

      3. +-commutative 96.1%

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

   6. Applied egg-rr 96.1%

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

   if 5.50000000000000017e150 < y

   1. Initial program 47.1%

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

      1. associate-/l* 79.0%

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

      2. associate-+l+ 79.0%

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

   3. Simplified 79.0%

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

   5. Taylor expanded in x around 0 79.0%

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

      1. +-commutative 79.0%

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

   7. Simplified 79.0%

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

      1. *-un-lft-identity 79.0%

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

      2. *-commutative 79.0%

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

      3. times-frac 88.8%

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

   9. Applied egg-rr 88.8%

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

       1. *-commutative 88.8%

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

   11. Simplified 88.8%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{+150}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) + \left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{+144}:\\ \;\;\;\;y \cdot \frac{\frac{x}{\left(y + x\right) \cdot \left(1 + \left(y + x\right)\right)}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.8e+144)
   (* y (/ (/ x (* (+ y x) (+ 1.0 (+ y x)))) (+ y x)))
   (* (/ x y) (/ 1.0 (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.8e+144) {
		tmp = y * ((x / ((y + x) * (1.0 + (y + x)))) / (y + x));
	} else {
		tmp = (x / y) * (1.0 / (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.8d+144) then
        tmp = y * ((x / ((y + x) * (1.0d0 + (y + x)))) / (y + x))
    else
        tmp = (x / y) * (1.0d0 / (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.8e+144) {
		tmp = y * ((x / ((y + x) * (1.0 + (y + x)))) / (y + x));
	} else {
		tmp = (x / y) * (1.0 / (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.8e+144:
		tmp = y * ((x / ((y + x) * (1.0 + (y + x)))) / (y + x))
	else:
		tmp = (x / y) * (1.0 / (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.8e+144)
		tmp = Float64(y * Float64(Float64(x / Float64(Float64(y + x) * Float64(1.0 + Float64(y + x)))) / Float64(y + x)));
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.8e+144)
		tmp = y * ((x / ((y + x) * (1.0 + (y + x)))) / (y + x));
	else
		tmp = (x / y) * (1.0 / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.8e+144], N[(y * N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.7999999999999999e144

   1. Initial program 67.0%

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

      1. associate-*l* 67.0%

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

      2. *-commutative 67.0%

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

      3. times-frac 96.1%

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

      4. +-commutative 96.1%

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

      5. +-commutative 96.1%

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

      6. distribute-lft-in 96.1%

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

      7. *-rgt-identity 96.1%

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

      8. +-commutative 96.1%

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

      9. pow2 96.1%

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

      10. +-commutative 96.1%

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

   4. Applied egg-rr 96.1%

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

      1. *-commutative 96.1%

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

      2. clear-num 95.4%

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

      3. frac-times 92.8%

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

      4. *-un-lft-identity 92.8%

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

      5. +-commutative 92.8%

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

      6. +-commutative 92.8%

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

      7. +-commutative 92.8%

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

   6. Applied egg-rr 92.8%

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

      1. clear-num 92.7%

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

      2. associate-/r/ 92.7%

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

      3. associate-/r* 93.6%

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

      4. +-commutative 93.6%

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

      5. pow2 93.6%

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

      6. clear-num 93.7%

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

      7. +-commutative 93.7%

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

      8. +-commutative 93.7%

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

      9. *-un-lft-identity 93.7%

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

      10. distribute-rgt-out 93.7%

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

   8. Applied egg-rr 93.7%

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

   if 1.7999999999999999e144 < y

   1. Initial program 47.4%

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

      1. associate-/l* 77.5%

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

      2. associate-+l+ 77.5%

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

   3. Simplified 77.5%

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

   5. Taylor expanded in x around 0 80.2%

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

      1. +-commutative 80.2%

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

   7. Simplified 80.2%

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

      1. *-un-lft-identity 80.2%

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

      2. *-commutative 80.2%

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

      3. times-frac 89.4%

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

   9. Applied egg-rr 89.4%

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

       1. *-commutative 89.4%

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

   11. Simplified 89.4%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{+144}:\\ \;\;\;\;y \cdot \frac{\frac{x}{\left(y + x\right) \cdot \left(1 + \left(y + x\right)\right)}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{y + x}{y}}\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-208}:\\ \;\;\;\;\frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.2e+117)
   (/ (/ 1.0 x) (/ (+ y x) y))
   (if (<= x -6.8e-208)
     (/ y (* (+ x (+ y 1.0)) (+ y x)))
     (/ (/ x y) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.2e+117) {
		tmp = (1.0 / x) / ((y + x) / y);
	} else if (x <= -6.8e-208) {
		tmp = y / ((x + (y + 1.0)) * (y + x));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.2d+117)) then
        tmp = (1.0d0 / x) / ((y + x) / y)
    else if (x <= (-6.8d-208)) then
        tmp = y / ((x + (y + 1.0d0)) * (y + x))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.2e+117) {
		tmp = (1.0 / x) / ((y + x) / y);
	} else if (x <= -6.8e-208) {
		tmp = y / ((x + (y + 1.0)) * (y + x));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.2e+117:
		tmp = (1.0 / x) / ((y + x) / y)
	elif x <= -6.8e-208:
		tmp = y / ((x + (y + 1.0)) * (y + x))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.2e+117)
		tmp = Float64(Float64(1.0 / x) / Float64(Float64(y + x) / y));
	elseif (x <= -6.8e-208)
		tmp = Float64(y / Float64(Float64(x + Float64(y + 1.0)) * Float64(y + 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+117)
		tmp = (1.0 / x) / ((y + x) / y);
	elseif (x <= -6.8e-208)
		tmp = y / ((x + (y + 1.0)) * (y + x));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.2e+117], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6.8e-208], N[(y / N[(N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1999999999999999e117

   1. Initial program 39.6%

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

      1. associate-*l* 39.6%

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

      2. *-commutative 39.6%

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

      3. times-frac 71.3%

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

      4. +-commutative 71.3%

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

      5. +-commutative 71.3%

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

      6. distribute-lft-in 71.3%

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

      7. *-rgt-identity 71.3%

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

      8. +-commutative 71.3%

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

      9. pow2 71.3%

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

      10. +-commutative 71.3%

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

   4. Applied egg-rr 71.3%

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

   5. Taylor expanded in x around inf 64.5%

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

      1. *-commutative 64.5%

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

      2. clear-num 64.5%

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

      3. un-div-inv 64.5%

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

      4. +-commutative 64.5%

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

   7. Applied egg-rr 64.5%

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

   if -1.1999999999999999e117 < x < -6.8e-208

   1. Initial program 75.1%

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

      1. associate-*l* 75.0%

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

      2. *-commutative 75.0%

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

      3. times-frac 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-lft-in 99.7%

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

      7. *-rgt-identity 99.7%

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

      8. +-commutative 99.7%

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

      9. pow2 99.7%

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

      10. +-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. unpow2 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around 0 65.9%

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

      1. clear-num 65.7%

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

      2. frac-times 60.3%

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

      3. *-un-lft-identity 60.3%

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

      4. +-commutative 60.3%

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

      5. +-commutative 60.3%

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

      6. *-un-lft-identity 60.3%

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

      7. distribute-rgt-out 60.4%

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

   9. Applied egg-rr 60.4%

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

       1. associate-/r* 64.2%

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

       2. associate-/r/ 72.1%

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

       3. *-inverses 72.1%

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

       4. *-lft-identity 72.1%

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

       5. +-commutative 72.1%

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

       6. associate-+l+ 72.1%

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

   11. Simplified 72.1%

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

   if -6.8e-208 < x

   1. Initial program 65.7%

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

      1. associate-/l* 78.7%

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

      2. associate-+l+ 78.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in x around 0 55.9%

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

      1. associate-/r* 56.3%

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

      2. +-commutative 56.3%

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

   7. Simplified 56.3%

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

      1. associate-*r/ 56.7%

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

      2. div-inv 56.7%

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

   9. Applied egg-rr 56.7%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{y + x}{y}}\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-208}:\\ \;\;\;\;\frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-208}:\\ \;\;\;\;\frac{\frac{y}{x}}{\frac{x + \left(y + 1\right)}{x} \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -7e-208)
   (/ (/ y x) (* (/ (+ x (+ y 1.0)) x) (+ y x)))
   (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -7e-208) {
		tmp = (y / x) / (((x + (y + 1.0)) / x) * (y + x));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7d-208)) then
        tmp = (y / x) / (((x + (y + 1.0d0)) / x) * (y + x))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -7e-208) {
		tmp = (y / x) / (((x + (y + 1.0)) / x) * (y + x));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -7e-208:
		tmp = (y / x) / (((x + (y + 1.0)) / x) * (y + x))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -7e-208)
		tmp = Float64(Float64(y / x) / Float64(Float64(Float64(x + Float64(y + 1.0)) / x) * Float64(y + 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 <= -7e-208)
		tmp = (y / x) / (((x + (y + 1.0)) / x) * (y + x));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -7e-208], N[(N[(y / x), $MachinePrecision] / N[(N[(N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.99999999999999982e-208

   1. Initial program 61.1%

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

      1. associate-*l* 61.1%

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

      2. *-commutative 61.1%

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

      3. times-frac 88.6%

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

      4. +-commutative 88.6%

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

      5. +-commutative 88.6%

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

      6. distribute-lft-in 88.5%

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

      7. *-rgt-identity 88.5%

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

      8. +-commutative 88.5%

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

      9. pow2 88.5%

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

      10. +-commutative 88.5%

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

   4. Applied egg-rr 88.5%

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

      1. unpow2 88.5%

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

      2. +-commutative 88.5%

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

      3. +-commutative 88.5%

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

   6. Applied egg-rr 88.5%

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

   7. Taylor expanded in y around 0 66.9%

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

      1. clear-num 66.9%

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

      2. +-commutative 66.9%

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

      3. pow2 66.9%

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

      4. frac-times 72.8%

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

      5. *-rgt-identity 72.8%

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

      6. pow2 72.8%

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

      7. +-commutative 72.8%

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

      8. *-un-lft-identity 72.8%

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

      9. distribute-rgt-out 72.9%

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

   9. Applied egg-rr 72.9%

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

       1. associate-/r* 67.0%

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

       2. associate-/l* 69.9%

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

       3. +-commutative 69.9%

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

       4. associate-+l+ 69.9%

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

   11. Simplified 69.9%

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

   if -6.99999999999999982e-208 < x

   1. Initial program 65.7%

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

      1. associate-/l* 78.7%

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

      2. associate-+l+ 78.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in x around 0 55.9%

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

      1. associate-/r* 56.3%

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

      2. +-commutative 56.3%

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

   7. Simplified 56.3%

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

      1. associate-*r/ 56.7%

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

      2. div-inv 56.7%

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

   9. Applied egg-rr 56.7%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-208}:\\ \;\;\;\;\frac{\frac{y}{x}}{\frac{x + \left(y + 1\right)}{x} \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-172}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (* (/ y x) (/ 1.0 x))
   (if (<= x -2.2e-172) (/ y x) (/ x (* y (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -2.2e-172) {
		tmp = y / x;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (y / x) * (1.0d0 / x)
    else if (x <= (-2.2d-172)) then
        tmp = y / x
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -2.2e-172) {
		tmp = y / x;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (y / x) * (1.0 / x)
	elif x <= -2.2e-172:
		tmp = y / x
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	elseif (x <= -2.2e-172)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (y / x) * (1.0 / x);
	elseif (x <= -2.2e-172)
		tmp = y / x;
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.0], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.2e-172], N[(y / x), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 51.2%

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

      1. associate-*l* 51.2%

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

      2. *-commutative 51.2%

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

      3. times-frac 80.5%

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

      4. +-commutative 80.5%

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

      5. +-commutative 80.5%

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

      6. distribute-lft-in 80.4%

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

      7. *-rgt-identity 80.4%

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

      8. +-commutative 80.4%

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

      9. pow2 80.4%

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

      10. +-commutative 80.4%

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

   4. Applied egg-rr 80.4%

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

   5. Taylor expanded in x around inf 55.7%

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

   6. Taylor expanded in y around 0 54.9%

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

   if -1 < x < -2.20000000000000009e-172

   1. Initial program 72.6%

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

      1. associate-/l* 89.3%

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

      2. associate-+l+ 89.3%

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

   3. Simplified 89.3%

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

   5. Taylor expanded in y around 0 49.8%

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

      1. associate-/r* 49.8%

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

      2. +-commutative 49.8%

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

   7. Simplified 49.8%

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

   8. Taylor expanded in x around 0 49.8%

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

   if -2.20000000000000009e-172 < x

   1. Initial program 66.3%

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

      1. associate-/l* 79.0%

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

      2. associate-+l+ 79.0%

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

   3. Simplified 79.0%

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

   5. Taylor expanded in x around 0 56.7%

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

      1. +-commutative 56.7%

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

   7. Simplified 56.7%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-172}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-172}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (* (/ y x) (/ 1.0 x))
   (if (<= x -2.4e-172) (/ y x) (/ (/ x y) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -2.4e-172) {
		tmp = y / x;
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (y / x) * (1.0d0 / x)
    else if (x <= (-2.4d-172)) then
        tmp = y / x
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -2.4e-172) {
		tmp = y / x;
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (y / x) * (1.0 / x)
	elif x <= -2.4e-172:
		tmp = y / x
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	elseif (x <= -2.4e-172)
		tmp = Float64(y / 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 <= -1.0)
		tmp = (y / x) * (1.0 / x);
	elseif (x <= -2.4e-172)
		tmp = y / x;
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.0], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.4e-172], N[(y / x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 51.2%

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

      1. associate-*l* 51.2%

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

      2. *-commutative 51.2%

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

      3. times-frac 80.5%

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

      4. +-commutative 80.5%

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

      5. +-commutative 80.5%

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

      6. distribute-lft-in 80.4%

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

      7. *-rgt-identity 80.4%

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

      8. +-commutative 80.4%

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

      9. pow2 80.4%

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

      10. +-commutative 80.4%

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

   4. Applied egg-rr 80.4%

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

   5. Taylor expanded in x around inf 55.7%

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

   6. Taylor expanded in y around 0 54.9%

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

   if -1 < x < -2.4000000000000001e-172

   1. Initial program 72.6%

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

      1. associate-/l* 89.3%

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

      2. associate-+l+ 89.3%

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

   3. Simplified 89.3%

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

   5. Taylor expanded in y around 0 49.8%

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

      1. associate-/r* 49.8%

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

      2. +-commutative 49.8%

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

   7. Simplified 49.8%

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

   8. Taylor expanded in x around 0 49.8%

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

   if -2.4000000000000001e-172 < x

   1. Initial program 66.3%

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

      1. associate-/l* 79.0%

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

      2. associate-+l+ 79.0%

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

   3. Simplified 79.0%

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

   5. Taylor expanded in x around 0 56.6%

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

      1. associate-/r* 57.0%

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

      2. +-commutative 57.0%

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

   7. Simplified 57.0%

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

      1. associate-*r/ 57.4%

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

      2. div-inv 57.5%

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

   9. Applied egg-rr 57.5%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-172}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-208}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) (* (/ y x) (/ 1.0 x)) (if (<= x -7e-208) (/ y x) (/ x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -7e-208) {
		tmp = y / 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 <= (-1.0d0)) then
        tmp = (y / x) * (1.0d0 / x)
    else if (x <= (-7d-208)) then
        tmp = y / 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 <= -1.0) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -7e-208) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (y / x) * (1.0 / x)
	elif x <= -7e-208:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	elseif (x <= -7e-208)
		tmp = Float64(y / 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 <= -1.0)
		tmp = (y / x) * (1.0 / x);
	elseif (x <= -7e-208)
		tmp = y / 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, -1.0], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7e-208], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 51.2%

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

      1. associate-*l* 51.2%

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

      2. *-commutative 51.2%

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

      3. times-frac 80.5%

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

      4. +-commutative 80.5%

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

      5. +-commutative 80.5%

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

      6. distribute-lft-in 80.4%

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

      7. *-rgt-identity 80.4%

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

      8. +-commutative 80.4%

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

      9. pow2 80.4%

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

      10. +-commutative 80.4%

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

   4. Applied egg-rr 80.4%

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

   5. Taylor expanded in x around inf 55.7%

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

   6. Taylor expanded in y around 0 54.9%

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

   if -1 < x < -6.99999999999999982e-208

   1. Initial program 75.0%

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

      1. associate-/l* 90.2%

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

      2. associate-+l+ 90.2%

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

   3. Simplified 90.2%

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

   5. Taylor expanded in y around 0 45.8%

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

      1. associate-/r* 45.8%

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

      2. +-commutative 45.8%

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

   7. Simplified 45.8%

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

   8. Taylor expanded in x around 0 45.8%

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

   if -6.99999999999999982e-208 < x

   1. Initial program 65.7%

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

      1. associate-/l* 78.7%

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

      2. associate-+l+ 78.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in x around 0 56.0%

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

      1. +-commutative 56.0%

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

   7. Simplified 56.0%

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

   8. Taylor expanded in y around 0 30.7%

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

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-208}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-172}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.4e-172) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.4e-172) {
		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 <= (-2.4d-172)) 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 <= -2.4e-172) {
		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 <= -2.4e-172:
		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 <= -2.4e-172)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.4e-172)
		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, -2.4e-172], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4000000000000001e-172

   1. Initial program 59.7%

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

      1. associate-/l* 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. associate-+l+ 76.2%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in y around 0 55.8%

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

      1. associate-/r* 55.2%

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

      2. +-commutative 55.2%

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

   7. Simplified 55.2%

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

   if -2.4000000000000001e-172 < x

   1. Initial program 66.3%

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

      1. associate-/l* 79.0%

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

      2. associate-+l+ 79.0%

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

   3. Simplified 79.0%

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

   5. Taylor expanded in x around 0 56.6%

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

      1. associate-/r* 57.0%

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

      2. +-commutative 57.0%

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

   7. Simplified 57.0%

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

      1. associate-*r/ 57.4%

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

      2. div-inv 57.5%

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

   9. Applied egg-rr 57.5%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-172}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-208}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -7e-208) (/ y x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -7e-208) {
		tmp = y / 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 <= (-7d-208)) then
        tmp = y / 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 <= -7e-208) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -7e-208:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -7e-208)
		tmp = Float64(y / 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 <= -7e-208)
		tmp = y / 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, -7e-208], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.99999999999999982e-208

   1. Initial program 61.1%

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

      1. associate-/l* 77.1%

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

      2. associate-+l+ 77.1%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in y around 0 53.9%

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

      1. associate-/r* 53.3%

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

      2. +-commutative 53.3%

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

   7. Simplified 53.3%

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

   8. Taylor expanded in x around 0 30.5%

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

   if -6.99999999999999982e-208 < x

   1. Initial program 65.7%

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

      1. associate-/l* 78.7%

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

      2. associate-+l+ 78.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in x around 0 56.0%

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

      1. +-commutative 56.0%

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

   7. Simplified 56.0%

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

   8. Taylor expanded in y around 0 30.7%

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

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-208}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 4.3% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.2%

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

   1. associate-*l* 64.2%

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

   2. *-commutative 64.2%

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

   3. times-frac 93.9%

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

   4. +-commutative 93.9%

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

   5. +-commutative 93.9%

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

   6. distribute-lft-in 93.9%

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

   7. *-rgt-identity 93.9%

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

   8. +-commutative 93.9%

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

   9. pow2 93.9%

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

   10. +-commutative 93.9%

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

4. Applied egg-rr 93.9%

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

5. Taylor expanded in x around inf 32.8%

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

6. Taylor expanded in y around inf 4.1%

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

7. Final simplification 4.1%

   \[\leadsto \frac{1}{x} \]
8. Add Preprocessing

Alternative 14: 27.3% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.2%

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

   1. associate-/l* 78.1%

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

   2. associate-+l+ 78.1%

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

3. Simplified 78.1%

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

5. Taylor expanded in x around 0 51.6%

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

   1. +-commutative 51.6%

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

7. Simplified 51.6%

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

8. Taylor expanded in y around 0 25.9%

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

9. Final simplification 25.9%

   \[\leadsto \frac{x}{y} \]
10. Add Preprocessing

Developer target: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024115 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))