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

Percentage Accurate: 68.9% → 91.2%

Time: 13.2s

Alternatives: 16

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: 68.9% 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: 91.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;y \leq 10^{-159}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+71}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(x, 2, y\right)}}{\frac{t\_0}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= y 1e-159)
     (/ (/ y (+ x 1.0)) x)
     (if (<= y 2.4e+71)
       (* x (/ y (* (* (+ y x) (+ y x)) t_0)))
       (/ (/ 1.0 (fma x 2.0 y)) (/ t_0 x))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (y <= 1e-159) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 2.4e+71) {
		tmp = x * (y / (((y + x) * (y + x)) * t_0));
	} else {
		tmp = (1.0 / fma(x, 2.0, y)) / (t_0 / x);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (y <= 1e-159)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	elseif (y <= 2.4e+71)
		tmp = Float64(x * Float64(y / Float64(Float64(Float64(y + x) * Float64(y + x)) * t_0)));
	else
		tmp = Float64(Float64(1.0 / fma(x, 2.0, y)) / Float64(t_0 / x));
	end
	return 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[y, 1e-159], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 2.4e+71], N[(x * N[(y / N[(N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x * 2.0 + y), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 9.99999999999999989e-160

   1. Initial program 72.3%

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

   3. Taylor expanded in y around 0 72.3%

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

   4. Taylor expanded in y around 0 59.9%

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

      1. +-commutative 59.9%

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

   6. Simplified 59.9%

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

      1. *-un-lft-identity 59.9%

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

      2. times-frac 59.7%

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

   8. Applied egg-rr 59.7%

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

      1. associate-*l/ 59.8%

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

      2. *-lft-identity 59.8%

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

   10. Simplified 59.8%

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

   if 9.99999999999999989e-160 < y < 2.39999999999999981e71

   1. Initial program 88.4%

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

      1. associate-/l* 91.4%

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

      2. associate-+l+ 91.4%

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

   3. Simplified 91.4%

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

   if 2.39999999999999981e71 < y

   1. Initial program 49.3%

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

      1. associate-/l* 76.9%

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

      2. associate-+l+ 76.9%

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

   3. Simplified 76.9%

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

   5. Taylor expanded in x around 0 62.9%

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

      1. associate-*r* 62.9%

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

      2. +-commutative 62.9%

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

      3. unpow2 62.9%

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

      4. distribute-rgt-in 76.9%

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

      5. *-commutative 76.9%

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

   7. Simplified 76.9%

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

      1. associate-*r/ 49.3%

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

      2. associate-+r+ 49.3%

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

      3. *-commutative 49.3%

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

      4. associate-+r+ 49.3%

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

      5. +-commutative 49.3%

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

      6. fma-define 49.3%

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

   9. Applied egg-rr 49.3%

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

       1. times-frac 85.4%

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

       2. associate-/r* 90.1%

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

       3. *-inverses 90.1%

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

   11. Simplified 90.1%

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

       1. *-commutative 90.1%

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

       2. clear-num 90.0%

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

       3. un-div-inv 90.1%

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

   13. Applied egg-rr 90.1%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-159}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+71}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(x, 2, y\right)}}{\frac{x + \left(y + 1\right)}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;y \leq 1.5 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+71}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(x, 2, y\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= y 1.5e-158)
     (/ (/ y (+ x 1.0)) x)
     (if (<= y 2.4e+71)
       (* x (/ y (* (* (+ y x) (+ y x)) t_0)))
       (/ (/ x (fma x 2.0 y)) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (y <= 1.5e-158) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 2.4e+71) {
		tmp = x * (y / (((y + x) * (y + x)) * t_0));
	} else {
		tmp = (x / fma(x, 2.0, y)) / t_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 (y <= 1.5e-158)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	elseif (y <= 2.4e+71)
		tmp = Float64(x * Float64(y / Float64(Float64(Float64(y + x) * Float64(y + x)) * t_0)));
	else
		tmp = Float64(Float64(x / fma(x, 2.0, y)) / t_0);
	end
	return 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[y, 1.5e-158], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 2.4e+71], N[(x * N[(y / N[(N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x * 2.0 + y), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.5e-158

   1. Initial program 72.3%

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

   3. Taylor expanded in y around 0 72.3%

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

   4. Taylor expanded in y around 0 59.9%

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

      1. +-commutative 59.9%

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

   6. Simplified 59.9%

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

      1. *-un-lft-identity 59.9%

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

      2. times-frac 59.7%

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

   8. Applied egg-rr 59.7%

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

      1. associate-*l/ 59.8%

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

      2. *-lft-identity 59.8%

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

   10. Simplified 59.8%

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

   if 1.5e-158 < y < 2.39999999999999981e71

   1. Initial program 88.4%

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

      1. associate-/l* 91.4%

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

      2. associate-+l+ 91.4%

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

   3. Simplified 91.4%

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

   if 2.39999999999999981e71 < y

   1. Initial program 49.3%

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

      1. associate-/l* 76.9%

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

      2. associate-+l+ 76.9%

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

   3. Simplified 76.9%

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

   5. Taylor expanded in x around 0 62.9%

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

      1. associate-*r* 62.9%

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

      2. +-commutative 62.9%

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

      3. unpow2 62.9%

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

      4. distribute-rgt-in 76.9%

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

      5. *-commutative 76.9%

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

   7. Simplified 76.9%

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

      1. associate-*r/ 49.3%

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

      2. associate-+r+ 49.3%

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

      3. *-commutative 49.3%

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

      4. associate-+r+ 49.3%

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

      5. +-commutative 49.3%

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

      6. fma-define 49.3%

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

   9. Applied egg-rr 49.3%

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

       1. times-frac 85.4%

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

       2. associate-/r* 90.1%

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

       3. *-inverses 90.1%

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

   11. Simplified 90.1%

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

       1. associate-*l/ 90.1%

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

       2. un-div-inv 90.1%

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

   13. Applied egg-rr 90.1%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+71}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(x, 2, y\right)}}{x + \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{if}\;y \leq 6.8 \cdot 10^{-112}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 880000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+147}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* x (+ x 1.0)))))
   (if (<= y 6.8e-112)
     t_0
     (if (<= y 4.6e-24)
       (/ x y)
       (if (<= y 880000.0)
         t_0
         (if (<= y 6e+147) (/ x (* y (+ y 1.0))) (* (/ x y) (/ 1.0 y))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x * (x + 1.0));
	double tmp;
	if (y <= 6.8e-112) {
		tmp = t_0;
	} else if (y <= 4.6e-24) {
		tmp = x / y;
	} else if (y <= 880000.0) {
		tmp = t_0;
	} else if (y <= 6e+147) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (x * (x + 1.0d0))
    if (y <= 6.8d-112) then
        tmp = t_0
    else if (y <= 4.6d-24) then
        tmp = x / y
    else if (y <= 880000.0d0) then
        tmp = t_0
    else if (y <= 6d+147) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (x * (x + 1.0));
	double tmp;
	if (y <= 6.8e-112) {
		tmp = t_0;
	} else if (y <= 4.6e-24) {
		tmp = x / y;
	} else if (y <= 880000.0) {
		tmp = t_0;
	} else if (y <= 6e+147) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (x * (x + 1.0))
	tmp = 0
	if y <= 6.8e-112:
		tmp = t_0
	elif y <= 4.6e-24:
		tmp = x / y
	elif y <= 880000.0:
		tmp = t_0
	elif y <= 6e+147:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(x * Float64(x + 1.0)))
	tmp = 0.0
	if (y <= 6.8e-112)
		tmp = t_0;
	elseif (y <= 4.6e-24)
		tmp = Float64(x / y);
	elseif (y <= 880000.0)
		tmp = t_0;
	elseif (y <= 6e+147)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (x * (x + 1.0));
	tmp = 0.0;
	if (y <= 6.8e-112)
		tmp = t_0;
	elseif (y <= 4.6e-24)
		tmp = x / y;
	elseif (y <= 880000.0)
		tmp = t_0;
	elseif (y <= 6e+147)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 6.8e-112], t$95$0, If[LessEqual[y, 4.6e-24], N[(x / y), $MachinePrecision], If[LessEqual[y, 880000.0], t$95$0, If[LessEqual[y, 6e+147], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 6.7999999999999996e-112 or 4.6000000000000002e-24 < y < 8.8e5

   1. Initial program 73.4%

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

      1. associate-/l* 85.7%

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

      2. associate-+l+ 85.7%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in y around 0 60.7%

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

   if 6.7999999999999996e-112 < y < 4.6000000000000002e-24

   1. Initial program 99.5%

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

      1. associate-/l* 96.7%

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

      2. associate-+l+ 96.7%

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

   3. Simplified 96.7%

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

   5. Taylor expanded in x around 0 32.3%

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

      1. +-commutative 32.3%

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

   7. Simplified 32.3%

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

   8. Taylor expanded in y around 0 32.3%

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

   if 8.8e5 < y < 5.99999999999999987e147

   1. Initial program 63.3%

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

      1. associate-/l* 72.3%

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

      2. associate-+l+ 72.3%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in x around 0 64.2%

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

      1. +-commutative 64.2%

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

   7. Simplified 64.2%

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

   if 5.99999999999999987e147 < y

   1. Initial program 49.1%

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

      1. associate-/l* 81.6%

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

      2. associate-+l+ 81.6%

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

   3. Simplified 81.6%

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

   5. Taylor expanded in x around 0 81.6%

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

      1. +-commutative 81.6%

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

   7. Simplified 81.6%

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

      1. *-un-lft-identity 81.6%

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

      2. *-commutative 81.6%

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

      3. times-frac 92.3%

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

   9. Applied egg-rr 92.3%

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

   10. Taylor expanded in y around inf 92.3%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-112}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 880000:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+147}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;y \leq 2 \cdot 10^{-159}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+76}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t\_0} \cdot \frac{1 + -2 \cdot \frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= y 2e-159)
     (/ (/ y (+ x 1.0)) x)
     (if (<= y 6.8e+76)
       (* x (/ y (* (* (+ y x) (+ y x)) t_0)))
       (* (/ x t_0) (/ (+ 1.0 (* -2.0 (/ x y))) y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (y <= 2e-159) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 6.8e+76) {
		tmp = x * (y / (((y + x) * (y + x)) * t_0));
	} else {
		tmp = (x / t_0) * ((1.0 + (-2.0 * (x / y))) / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (y + 1.0d0)
    if (y <= 2d-159) then
        tmp = (y / (x + 1.0d0)) / x
    else if (y <= 6.8d+76) then
        tmp = x * (y / (((y + x) * (y + x)) * t_0))
    else
        tmp = (x / t_0) * ((1.0d0 + ((-2.0d0) * (x / y))) / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (y <= 2e-159) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 6.8e+76) {
		tmp = x * (y / (((y + x) * (y + x)) * t_0));
	} else {
		tmp = (x / t_0) * ((1.0 + (-2.0 * (x / y))) / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x + (y + 1.0)
	tmp = 0
	if y <= 2e-159:
		tmp = (y / (x + 1.0)) / x
	elif y <= 6.8e+76:
		tmp = x * (y / (((y + x) * (y + x)) * t_0))
	else:
		tmp = (x / t_0) * ((1.0 + (-2.0 * (x / y))) / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (y <= 2e-159)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	elseif (y <= 6.8e+76)
		tmp = Float64(x * Float64(y / Float64(Float64(Float64(y + x) * Float64(y + x)) * t_0)));
	else
		tmp = Float64(Float64(x / t_0) * Float64(Float64(1.0 + Float64(-2.0 * Float64(x / y))) / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x + (y + 1.0);
	tmp = 0.0;
	if (y <= 2e-159)
		tmp = (y / (x + 1.0)) / x;
	elseif (y <= 6.8e+76)
		tmp = x * (y / (((y + x) * (y + x)) * t_0));
	else
		tmp = (x / t_0) * ((1.0 + (-2.0 * (x / y))) / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 2e-159], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 6.8e+76], N[(x * N[(y / N[(N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / t$95$0), $MachinePrecision] * N[(N[(1.0 + N[(-2.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.99999999999999998e-159

   1. Initial program 72.3%

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

   3. Taylor expanded in y around 0 72.3%

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

   4. Taylor expanded in y around 0 59.9%

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

      1. +-commutative 59.9%

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

   6. Simplified 59.9%

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

      1. *-un-lft-identity 59.9%

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

      2. times-frac 59.7%

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

   8. Applied egg-rr 59.7%

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

      1. associate-*l/ 59.8%

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

      2. *-lft-identity 59.8%

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

   10. Simplified 59.8%

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

   if 1.99999999999999998e-159 < y < 6.7999999999999994e76

   1. Initial program 88.4%

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

      1. associate-/l* 91.4%

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

      2. associate-+l+ 91.4%

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

   3. Simplified 91.4%

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

   if 6.7999999999999994e76 < y

   1. Initial program 49.3%

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

      1. associate-/l* 76.9%

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

      2. associate-+l+ 76.9%

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

   3. Simplified 76.9%

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

   5. Taylor expanded in x around 0 62.9%

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

      1. associate-*r* 62.9%

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

      2. +-commutative 62.9%

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

      3. unpow2 62.9%

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

      4. distribute-rgt-in 76.9%

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

      5. *-commutative 76.9%

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

   7. Simplified 76.9%

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

      1. associate-*r/ 49.3%

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

      2. associate-+r+ 49.3%

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

      3. *-commutative 49.3%

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

      4. associate-+r+ 49.3%

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

      5. +-commutative 49.3%

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

      6. fma-define 49.3%

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

   9. Applied egg-rr 49.3%

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

       1. times-frac 85.4%

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

       2. associate-/r* 90.1%

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

       3. *-inverses 90.1%

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

   11. Simplified 90.1%

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

   12. Taylor expanded in y around inf 89.6%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-159}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+76}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + \left(y + 1\right)} \cdot \frac{1 + -2 \cdot \frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;y \leq 1.8 \cdot 10^{-159}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+99}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\frac{t\_0}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= y 1.8e-159)
     (/ (/ y (+ x 1.0)) x)
     (if (<= y 2.5e+99)
       (* x (/ y (* (* (+ y x) (+ y x)) t_0)))
       (/ (/ 1.0 y) (/ t_0 x))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (y <= 1.8e-159) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 2.5e+99) {
		tmp = x * (y / (((y + x) * (y + x)) * t_0));
	} else {
		tmp = (1.0 / y) / (t_0 / x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (y + 1.0d0)
    if (y <= 1.8d-159) then
        tmp = (y / (x + 1.0d0)) / x
    else if (y <= 2.5d+99) then
        tmp = x * (y / (((y + x) * (y + x)) * t_0))
    else
        tmp = (1.0d0 / y) / (t_0 / x)
    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 (y <= 1.8e-159) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 2.5e+99) {
		tmp = x * (y / (((y + x) * (y + x)) * t_0));
	} else {
		tmp = (1.0 / y) / (t_0 / x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x + (y + 1.0)
	tmp = 0
	if y <= 1.8e-159:
		tmp = (y / (x + 1.0)) / x
	elif y <= 2.5e+99:
		tmp = x * (y / (((y + x) * (y + x)) * t_0))
	else:
		tmp = (1.0 / y) / (t_0 / x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (y <= 1.8e-159)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	elseif (y <= 2.5e+99)
		tmp = Float64(x * Float64(y / Float64(Float64(Float64(y + x) * Float64(y + x)) * t_0)));
	else
		tmp = Float64(Float64(1.0 / y) / Float64(t_0 / x));
	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 (y <= 1.8e-159)
		tmp = (y / (x + 1.0)) / x;
	elseif (y <= 2.5e+99)
		tmp = x * (y / (((y + x) * (y + x)) * t_0));
	else
		tmp = (1.0 / y) / (t_0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.8e-159], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 2.5e+99], N[(x * N[(y / N[(N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] / N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.80000000000000011e-159

   1. Initial program 72.3%

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

   3. Taylor expanded in y around 0 72.3%

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

   4. Taylor expanded in y around 0 59.9%

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

      1. +-commutative 59.9%

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

   6. Simplified 59.9%

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

      1. *-un-lft-identity 59.9%

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

      2. times-frac 59.7%

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

   8. Applied egg-rr 59.7%

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

      1. associate-*l/ 59.8%

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

      2. *-lft-identity 59.8%

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

   10. Simplified 59.8%

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

   if 1.80000000000000011e-159 < y < 2.50000000000000004e99

   1. Initial program 86.5%

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

      1. associate-/l* 89.5%

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

      2. associate-+l+ 89.5%

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

   3. Simplified 89.5%

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

   if 2.50000000000000004e99 < y

   1. Initial program 49.9%

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

      1. associate-/l* 78.0%

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

      2. associate-+l+ 78.0%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in x around 0 63.8%

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

      1. associate-*r* 63.8%

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

      2. +-commutative 63.8%

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

      3. unpow2 63.8%

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

      4. distribute-rgt-in 78.0%

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

      5. *-commutative 78.0%

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

   7. Simplified 78.0%

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

      1. associate-*r/ 49.9%

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

      2. associate-+r+ 49.9%

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

      3. *-commutative 49.9%

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

      4. associate-+r+ 49.9%

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

      5. +-commutative 49.9%

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

      6. fma-define 49.9%

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

   9. Applied egg-rr 49.9%

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

       1. times-frac 86.6%

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

       2. associate-/r* 91.4%

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

       3. *-inverses 91.4%

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

   11. Simplified 91.4%

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

       1. *-commutative 91.4%

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

       2. clear-num 91.3%

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

       3. un-div-inv 91.4%

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

   13. Applied egg-rr 91.4%

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

   14. Taylor expanded in x around 0 90.1%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-159}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+99}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\frac{x + \left(y + 1\right)}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;y \leq 2.2 \cdot 10^{-157}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+99}:\\ \;\;\;\;x \cdot \frac{y}{t\_0 \cdot \left(y \cdot \left(y + x \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\frac{t\_0}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= y 2.2e-157)
     (/ (/ y (+ x 1.0)) x)
     (if (<= y 2.5e+99)
       (* x (/ y (* t_0 (* y (+ y (* x 2.0))))))
       (/ (/ 1.0 y) (/ t_0 x))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (y <= 2.2e-157) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 2.5e+99) {
		tmp = x * (y / (t_0 * (y * (y + (x * 2.0)))));
	} else {
		tmp = (1.0 / y) / (t_0 / x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (y + 1.0d0)
    if (y <= 2.2d-157) then
        tmp = (y / (x + 1.0d0)) / x
    else if (y <= 2.5d+99) then
        tmp = x * (y / (t_0 * (y * (y + (x * 2.0d0)))))
    else
        tmp = (1.0d0 / y) / (t_0 / x)
    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 (y <= 2.2e-157) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 2.5e+99) {
		tmp = x * (y / (t_0 * (y * (y + (x * 2.0)))));
	} else {
		tmp = (1.0 / y) / (t_0 / x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x + (y + 1.0)
	tmp = 0
	if y <= 2.2e-157:
		tmp = (y / (x + 1.0)) / x
	elif y <= 2.5e+99:
		tmp = x * (y / (t_0 * (y * (y + (x * 2.0)))))
	else:
		tmp = (1.0 / y) / (t_0 / x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (y <= 2.2e-157)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	elseif (y <= 2.5e+99)
		tmp = Float64(x * Float64(y / Float64(t_0 * Float64(y * Float64(y + Float64(x * 2.0))))));
	else
		tmp = Float64(Float64(1.0 / y) / Float64(t_0 / x));
	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 (y <= 2.2e-157)
		tmp = (y / (x + 1.0)) / x;
	elseif (y <= 2.5e+99)
		tmp = x * (y / (t_0 * (y * (y + (x * 2.0)))));
	else
		tmp = (1.0 / y) / (t_0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 2.2e-157], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 2.5e+99], N[(x * N[(y / N[(t$95$0 * N[(y * N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] / N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.2000000000000001e-157

   1. Initial program 72.3%

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

   3. Taylor expanded in y around 0 72.3%

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

   4. Taylor expanded in y around 0 59.9%

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

      1. +-commutative 59.9%

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

   6. Simplified 59.9%

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

      1. *-un-lft-identity 59.9%

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

      2. times-frac 59.7%

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

   8. Applied egg-rr 59.7%

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

      1. associate-*l/ 59.8%

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

      2. *-lft-identity 59.8%

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

   10. Simplified 59.8%

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

   if 2.2000000000000001e-157 < y < 2.50000000000000004e99

   1. Initial program 86.5%

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

      1. associate-/l* 89.5%

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

      2. associate-+l+ 89.5%

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

   3. Simplified 89.5%

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

   5. Taylor expanded in x around 0 67.6%

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

      1. associate-*r* 67.6%

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

      2. +-commutative 67.6%

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

      3. unpow2 67.6%

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

      4. distribute-rgt-in 67.6%

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

      5. *-commutative 67.6%

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

   7. Simplified 67.6%

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

   if 2.50000000000000004e99 < y

   1. Initial program 49.9%

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

      1. associate-/l* 78.0%

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

      2. associate-+l+ 78.0%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in x around 0 63.8%

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

      1. associate-*r* 63.8%

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

      2. +-commutative 63.8%

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

      3. unpow2 63.8%

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

      4. distribute-rgt-in 78.0%

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

      5. *-commutative 78.0%

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

   7. Simplified 78.0%

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

      1. associate-*r/ 49.9%

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

      2. associate-+r+ 49.9%

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

      3. *-commutative 49.9%

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

      4. associate-+r+ 49.9%

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

      5. +-commutative 49.9%

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

      6. fma-define 49.9%

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

   9. Applied egg-rr 49.9%

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

       1. times-frac 86.6%

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

       2. associate-/r* 91.4%

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

       3. *-inverses 91.4%

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

   11. Simplified 91.4%

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

       1. *-commutative 91.4%

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

       2. clear-num 91.3%

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

       3. un-div-inv 91.4%

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

   13. Applied egg-rr 91.4%

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

   14. Taylor expanded in x around 0 90.1%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-157}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+99}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(y \cdot \left(y + x \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\frac{x + \left(y + 1\right)}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;y \leq 6.8 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \mathbf{elif}\;y \leq 3800000:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\frac{t\_0}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= y 6.8e-112)
     (/ (/ y (+ x 1.0)) x)
     (if (<= y 4.5e-24)
       (/ (/ x y) t_0)
       (if (<= y 3800000.0) (/ (/ y x) (+ x 1.0)) (/ (/ 1.0 y) (/ t_0 x)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (y <= 6.8e-112) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 4.5e-24) {
		tmp = (x / y) / t_0;
	} else if (y <= 3800000.0) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (1.0 / y) / (t_0 / x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (y + 1.0d0)
    if (y <= 6.8d-112) then
        tmp = (y / (x + 1.0d0)) / x
    else if (y <= 4.5d-24) then
        tmp = (x / y) / t_0
    else if (y <= 3800000.0d0) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (1.0d0 / y) / (t_0 / x)
    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 (y <= 6.8e-112) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 4.5e-24) {
		tmp = (x / y) / t_0;
	} else if (y <= 3800000.0) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (1.0 / y) / (t_0 / x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x + (y + 1.0)
	tmp = 0
	if y <= 6.8e-112:
		tmp = (y / (x + 1.0)) / x
	elif y <= 4.5e-24:
		tmp = (x / y) / t_0
	elif y <= 3800000.0:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (1.0 / y) / (t_0 / x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (y <= 6.8e-112)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	elseif (y <= 4.5e-24)
		tmp = Float64(Float64(x / y) / t_0);
	elseif (y <= 3800000.0)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(1.0 / y) / Float64(t_0 / x));
	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 (y <= 6.8e-112)
		tmp = (y / (x + 1.0)) / x;
	elseif (y <= 4.5e-24)
		tmp = (x / y) / t_0;
	elseif (y <= 3800000.0)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (1.0 / y) / (t_0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 6.8e-112], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 4.5e-24], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 3800000.0], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] / N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 6.7999999999999996e-112

   1. Initial program 72.4%

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

   3. Taylor expanded in y around 0 72.4%

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

   4. Taylor expanded in y around 0 60.2%

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

      1. +-commutative 60.2%

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

   6. Simplified 60.2%

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

      1. *-un-lft-identity 60.2%

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

      2. times-frac 60.0%

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

   8. Applied egg-rr 60.0%

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

      1. associate-*l/ 60.1%

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

      2. *-lft-identity 60.1%

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

   10. Simplified 60.1%

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

   if 6.7999999999999996e-112 < y < 4.4999999999999997e-24

   1. Initial program 99.5%

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

      1. associate-/l* 96.7%

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

      2. associate-+l+ 96.7%

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

   3. Simplified 96.7%

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

   5. Taylor expanded in x around 0 69.9%

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

      1. associate-*r* 69.9%

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

      2. +-commutative 69.9%

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

      3. unpow2 69.9%

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

      4. distribute-rgt-in 69.9%

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

      5. *-commutative 69.9%

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

   7. Simplified 69.9%

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

      1. associate-*r/ 70.0%

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

      2. associate-+r+ 70.0%

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

      3. *-commutative 70.0%

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

      4. associate-+r+ 70.0%

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

      5. +-commutative 70.0%

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

      6. fma-define 70.0%

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

   9. Applied egg-rr 70.0%

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

       1. times-frac 33.9%

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

       2. associate-/r* 34.1%

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

       3. *-inverses 34.1%

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

   11. Simplified 34.1%

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

       1. associate-*l/ 34.1%

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

       2. un-div-inv 34.2%

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

   13. Applied egg-rr 34.2%

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

   14. Taylor expanded in x around 0 32.9%

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

   if 4.4999999999999997e-24 < y < 3.8e6

   1. Initial program 97.4%

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

      1. associate-/l* 97.4%

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

      2. associate-+l+ 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in y around 0 72.2%

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

      1. associate-/r* 72.2%

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

      2. +-commutative 72.2%

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

   7. Simplified 72.2%

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

   if 3.8e6 < y

   1. Initial program 55.1%

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

      1. associate-/l* 77.7%

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

      2. associate-+l+ 77.7%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in x around 0 63.8%

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

      1. associate-*r* 63.8%

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

      2. +-commutative 63.8%

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

      3. unpow2 63.8%

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

      4. distribute-rgt-in 75.2%

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

      5. *-commutative 75.2%

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

   7. Simplified 75.2%

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

      1. associate-*r/ 52.6%

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

      2. associate-+r+ 52.6%

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

      3. *-commutative 52.6%

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

      4. associate-+r+ 52.6%

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

      5. +-commutative 52.6%

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

      6. fma-define 52.6%

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

   9. Applied egg-rr 52.6%

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

       1. times-frac 78.6%

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

       2. associate-/r* 82.4%

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

       3. *-inverses 82.4%

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

   11. Simplified 82.4%

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

       1. *-commutative 82.4%

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

       2. clear-num 82.4%

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

       3. un-div-inv 82.5%

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

   13. Applied egg-rr 82.5%

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

   14. Taylor expanded in x around 0 80.8%

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

Alternative 8: 81.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-24} \lor \neg \left(y \leq 4200000\right):\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 6.8e-112)
   (/ (/ y (+ x 1.0)) x)
   (if (or (<= y 5.2e-24) (not (<= y 4200000.0)))
     (/ (/ x y) (+ x (+ y 1.0)))
     (/ (/ y x) (+ x 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 6.8e-112) {
		tmp = (y / (x + 1.0)) / x;
	} else if ((y <= 5.2e-24) || !(y <= 4200000.0)) {
		tmp = (x / y) / (x + (y + 1.0));
	} else {
		tmp = (y / x) / (x + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 6.8d-112) then
        tmp = (y / (x + 1.0d0)) / x
    else if ((y <= 5.2d-24) .or. (.not. (y <= 4200000.0d0))) then
        tmp = (x / y) / (x + (y + 1.0d0))
    else
        tmp = (y / x) / (x + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 6.8e-112) {
		tmp = (y / (x + 1.0)) / x;
	} else if ((y <= 5.2e-24) || !(y <= 4200000.0)) {
		tmp = (x / y) / (x + (y + 1.0));
	} else {
		tmp = (y / x) / (x + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 6.8e-112:
		tmp = (y / (x + 1.0)) / x
	elif (y <= 5.2e-24) or not (y <= 4200000.0):
		tmp = (x / y) / (x + (y + 1.0))
	else:
		tmp = (y / x) / (x + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 6.8e-112)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	elseif ((y <= 5.2e-24) || !(y <= 4200000.0))
		tmp = Float64(Float64(x / y) / Float64(x + Float64(y + 1.0)));
	else
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.8e-112)
		tmp = (y / (x + 1.0)) / x;
	elseif ((y <= 5.2e-24) || ~((y <= 4200000.0)))
		tmp = (x / y) / (x + (y + 1.0));
	else
		tmp = (y / x) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 6.8e-112], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[Or[LessEqual[y, 5.2e-24], N[Not[LessEqual[y, 4200000.0]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.7999999999999996e-112

   1. Initial program 72.4%

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

   3. Taylor expanded in y around 0 72.4%

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

   4. Taylor expanded in y around 0 60.2%

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

      1. +-commutative 60.2%

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

   6. Simplified 60.2%

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

      1. *-un-lft-identity 60.2%

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

      2. times-frac 60.0%

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

   8. Applied egg-rr 60.0%

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

      1. associate-*l/ 60.1%

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

      2. *-lft-identity 60.1%

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

   10. Simplified 60.1%

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

   if 6.7999999999999996e-112 < y < 5.2e-24 or 4.2e6 < y

   1. Initial program 61.2%

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

      1. associate-/l* 80.3%

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

      2. associate-+l+ 80.3%

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

   3. Simplified 80.3%

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

   5. Taylor expanded in x around 0 64.6%

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

      1. associate-*r* 64.6%

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

      2. +-commutative 64.6%

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

      3. unpow2 64.6%

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

      4. distribute-rgt-in 74.5%

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

      5. *-commutative 74.5%

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

   7. Simplified 74.5%

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

      1. associate-*r/ 55.0%

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

      2. associate-+r+ 55.0%

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

      3. *-commutative 55.0%

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

      4. associate-+r+ 55.0%

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

      5. +-commutative 55.0%

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

      6. fma-define 55.0%

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

   9. Applied egg-rr 55.0%

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

       1. times-frac 72.4%

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

       2. associate-/r* 75.7%

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

       3. *-inverses 75.7%

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

   11. Simplified 75.7%

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

       1. associate-*l/ 75.7%

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

       2. un-div-inv 75.7%

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

   13. Applied egg-rr 75.7%

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

   14. Taylor expanded in x around 0 74.2%

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

   if 5.2e-24 < y < 4.2e6

   1. Initial program 97.4%

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

      1. associate-/l* 97.4%

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

      2. associate-+l+ 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in y around 0 72.2%

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

      1. associate-/r* 72.2%

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

      2. +-commutative 72.2%

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

   7. Simplified 72.2%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-24} \lor \neg \left(y \leq 4200000\right):\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 3200000:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 6.8e-112)
   (/ (/ y (+ x 1.0)) x)
   (if (<= y 4.4e-24)
     (/ x y)
     (if (<= y 3200000.0) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 6.8e-112) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 4.4e-24) {
		tmp = x / y;
	} else if (y <= 3200000.0) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 6.8d-112) then
        tmp = (y / (x + 1.0d0)) / x
    else if (y <= 4.4d-24) then
        tmp = x / y
    else if (y <= 3200000.0d0) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 6.8e-112) {
		tmp = (y / (x + 1.0)) / x;
	} else if (y <= 4.4e-24) {
		tmp = x / y;
	} else if (y <= 3200000.0) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 6.8e-112:
		tmp = (y / (x + 1.0)) / x
	elif y <= 4.4e-24:
		tmp = x / y
	elif y <= 3200000.0:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 6.8e-112)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	elseif (y <= 4.4e-24)
		tmp = Float64(x / y);
	elseif (y <= 3200000.0)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.8e-112)
		tmp = (y / (x + 1.0)) / x;
	elseif (y <= 4.4e-24)
		tmp = x / y;
	elseif (y <= 3200000.0)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 6.8e-112], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 4.4e-24], N[(x / y), $MachinePrecision], If[LessEqual[y, 3200000.0], 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 4 regimes

2. if y < 6.7999999999999996e-112

   1. Initial program 72.4%

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

   3. Taylor expanded in y around 0 72.4%

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

   4. Taylor expanded in y around 0 60.2%

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

      1. +-commutative 60.2%

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

   6. Simplified 60.2%

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

      1. *-un-lft-identity 60.2%

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

      2. times-frac 60.0%

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

   8. Applied egg-rr 60.0%

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

      1. associate-*l/ 60.1%

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

      2. *-lft-identity 60.1%

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

   10. Simplified 60.1%

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

   if 6.7999999999999996e-112 < y < 4.40000000000000003e-24

   1. Initial program 99.5%

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

      1. associate-/l* 96.7%

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

      2. associate-+l+ 96.7%

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

   3. Simplified 96.7%

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

   5. Taylor expanded in x around 0 32.3%

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

      1. +-commutative 32.3%

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

   7. Simplified 32.3%

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

   8. Taylor expanded in y around 0 32.3%

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

   if 4.40000000000000003e-24 < y < 3.2e6

   1. Initial program 97.4%

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

      1. associate-/l* 97.4%

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

      2. associate-+l+ 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in y around 0 72.2%

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

      1. associate-/r* 72.2%

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

      2. +-commutative 72.2%

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

   7. Simplified 72.2%

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

   if 3.2e6 < y

   1. Initial program 55.1%

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

      1. associate-/l* 77.7%

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

      2. associate-+l+ 77.7%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in x around 0 74.3%

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

      1. +-commutative 74.3%

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

   7. Simplified 74.3%

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

      1. *-un-lft-identity 74.3%

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

      2. *-commutative 74.3%

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

      3. times-frac 80.4%

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

   9. Applied egg-rr 80.4%

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

       1. associate-*l/ 80.5%

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

       2. *-un-lft-identity 80.5%

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

   11. Applied egg-rr 80.5%

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

Alternative 10: 81.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{\frac{y}{x}}{x + 1}\\ \mathbf{if}\;y \leq 6.5 \cdot 10^{-112}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 45000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ y x) (+ x 1.0))))
   (if (<= y 6.5e-112)
     t_0
     (if (<= y 5e-24) (/ x y) (if (<= y 45000.0) t_0 (/ (/ x y) (+ y 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y / x) / (x + 1.0);
	double tmp;
	if (y <= 6.5e-112) {
		tmp = t_0;
	} else if (y <= 5e-24) {
		tmp = x / y;
	} else if (y <= 45000.0) {
		tmp = t_0;
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y / x) / (x + 1.0d0)
    if (y <= 6.5d-112) then
        tmp = t_0
    else if (y <= 5d-24) then
        tmp = x / y
    else if (y <= 45000.0d0) then
        tmp = t_0
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y / x) / (x + 1.0);
	double tmp;
	if (y <= 6.5e-112) {
		tmp = t_0;
	} else if (y <= 5e-24) {
		tmp = x / y;
	} else if (y <= 45000.0) {
		tmp = t_0;
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y / x) / (x + 1.0)
	tmp = 0
	if y <= 6.5e-112:
		tmp = t_0
	elif y <= 5e-24:
		tmp = x / y
	elif y <= 45000.0:
		tmp = t_0
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y / x) / Float64(x + 1.0))
	tmp = 0.0
	if (y <= 6.5e-112)
		tmp = t_0;
	elseif (y <= 5e-24)
		tmp = Float64(x / y);
	elseif (y <= 45000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (y / x) / (x + 1.0);
	tmp = 0.0;
	if (y <= 6.5e-112)
		tmp = t_0;
	elseif (y <= 5e-24)
		tmp = x / y;
	elseif (y <= 45000.0)
		tmp = t_0;
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 6.5e-112], t$95$0, If[LessEqual[y, 5e-24], N[(x / y), $MachinePrecision], If[LessEqual[y, 45000.0], t$95$0, N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.49999999999999956e-112 or 4.9999999999999998e-24 < y < 45000

   1. Initial program 73.4%

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

      1. associate-/l* 85.7%

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

      2. associate-+l+ 85.7%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in y around 0 60.7%

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

      1. associate-/r* 60.6%

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

      2. +-commutative 60.6%

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

   7. Simplified 60.6%

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

   if 6.49999999999999956e-112 < y < 4.9999999999999998e-24

   1. Initial program 99.5%

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

      1. associate-/l* 96.7%

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

      2. associate-+l+ 96.7%

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

   3. Simplified 96.7%

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

   5. Taylor expanded in x around 0 32.3%

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

      1. +-commutative 32.3%

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

   7. Simplified 32.3%

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

   8. Taylor expanded in y around 0 32.3%

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

   if 45000 < y

   1. Initial program 55.1%

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

      1. associate-/l* 77.7%

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

      2. associate-+l+ 77.7%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in x around 0 74.3%

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

      1. +-commutative 74.3%

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

   7. Simplified 74.3%

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

      1. *-un-lft-identity 74.3%

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

      2. *-commutative 74.3%

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

      3. times-frac 80.4%

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

   9. Applied egg-rr 80.4%

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

       1. associate-*l/ 80.5%

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

       2. *-un-lft-identity 80.5%

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

   11. Applied egg-rr 80.5%

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

Alternative 11: 66.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-201}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+147}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.8e-201)
   (/ y x)
   (if (<= y 6e+147) (/ x (* y (+ y 1.0))) (* (/ x y) (/ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.8e-201) {
		tmp = y / x;
	} else if (y <= 6e+147) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.8d-201) then
        tmp = y / x
    else if (y <= 6d+147) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.8e-201) {
		tmp = y / x;
	} else if (y <= 6e+147) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.8e-201:
		tmp = y / x
	elif y <= 6e+147:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.8e-201)
		tmp = Float64(y / x);
	elseif (y <= 6e+147)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.8e-201)
		tmp = y / x;
	elseif (y <= 6e+147)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.8e-201], N[(y / x), $MachinePrecision], If[LessEqual[y, 6e+147], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.8e-201

   1. Initial program 72.3%

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

      1. associate-/l* 84.8%

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

      2. associate-+l+ 84.8%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in y around 0 59.4%

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

      1. associate-/r* 59.3%

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

      2. +-commutative 59.3%

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

   7. Simplified 59.3%

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

   8. Taylor expanded in x around 0 34.5%

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

   if 3.8e-201 < y < 5.99999999999999987e147

   1. Initial program 76.1%

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

      1. associate-/l* 82.8%

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

      2. associate-+l+ 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in x around 0 48.9%

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

      1. +-commutative 48.9%

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

   7. Simplified 48.9%

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

   if 5.99999999999999987e147 < y

   1. Initial program 49.1%

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

      1. associate-/l* 81.6%

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

      2. associate-+l+ 81.6%

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

   3. Simplified 81.6%

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

   5. Taylor expanded in x around 0 81.6%

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

      1. +-commutative 81.6%

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

   7. Simplified 81.6%

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

      1. *-un-lft-identity 81.6%

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

      2. *-commutative 81.6%

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

      3. times-frac 92.3%

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

   9. Applied egg-rr 92.3%

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

   10. Taylor expanded in y around inf 92.3%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-201}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+147}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-201}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.8e-201) (/ y x) (if (<= y 1.0) (/ x y) (* (/ x y) (/ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.8e-201) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.8d-201) then
        tmp = y / x
    else if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.8e-201) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.8e-201:
		tmp = y / x
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.8e-201)
		tmp = Float64(y / x);
	elseif (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.8e-201)
		tmp = y / x;
	elseif (y <= 1.0)
		tmp = x / y;
	else
		tmp = (x / y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.8e-201], N[(y / x), $MachinePrecision], If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.8e-201

   1. Initial program 72.3%

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

      1. associate-/l* 84.8%

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

      2. associate-+l+ 84.8%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in y around 0 59.4%

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

      1. associate-/r* 59.3%

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

      2. +-commutative 59.3%

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

   7. Simplified 59.3%

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

   8. Taylor expanded in x around 0 34.5%

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

   if 3.8e-201 < y < 1

   1. Initial program 90.2%

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

      1. associate-/l* 94.5%

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

      2. associate-+l+ 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in x around 0 32.6%

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

      1. +-commutative 32.6%

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

   7. Simplified 32.6%

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

   8. Taylor expanded in y around 0 29.5%

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

   if 1 < y

   1. Initial program 55.6%

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

      1. associate-/l* 77.9%

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

      2. associate-+l+ 77.9%

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

   3. Simplified 77.9%

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

   5. Taylor expanded in x around 0 73.4%

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

      1. +-commutative 73.4%

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

   7. Simplified 73.4%

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

      1. *-un-lft-identity 73.4%

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

      2. *-commutative 73.4%

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

      3. times-frac 79.5%

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

   9. Applied egg-rr 79.5%

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

   10. Taylor expanded in y around inf 78.8%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-201}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 80.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-112}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 6.8e-112) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 6.8e-112) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 6.8d-112) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 6.8e-112) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 6.8e-112:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 6.8e-112)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.8e-112)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 6.8e-112], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.7999999999999996e-112

   1. Initial program 72.4%

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

      1. associate-/l* 85.1%

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

      2. associate-+l+ 85.1%

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

   3. Simplified 85.1%

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

   5. Taylor expanded in y around 0 60.2%

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

   if 6.7999999999999996e-112 < y

   1. Initial program 63.7%

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

      1. associate-/l* 81.5%

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

      2. associate-+l+ 81.5%

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

   3. Simplified 81.5%

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

   5. Taylor expanded in x around 0 65.8%

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

      1. +-commutative 65.8%

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

   7. Simplified 65.8%

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

      1. *-un-lft-identity 65.8%

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

      2. *-commutative 65.8%

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

      3. times-frac 70.7%

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

   9. Applied egg-rr 70.7%

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

       1. associate-*l/ 70.8%

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

       2. *-un-lft-identity 70.8%

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

   11. Applied egg-rr 70.8%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-112}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-189}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -3.2e-189) (/ y x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.2e-189) {
		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 <= (-3.2d-189)) 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 <= -3.2e-189) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.2e-189:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.2e-189)
		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 <= -3.2e-189)
		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, -3.2e-189], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.2000000000000001e-189

   1. Initial program 65.1%

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

      1. associate-/l* 80.5%

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

      2. associate-+l+ 80.5%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in y around 0 59.2%

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

      1. associate-/r* 56.0%

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

      2. +-commutative 56.0%

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

   7. Simplified 56.0%

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

   8. Taylor expanded in x around 0 27.1%

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

   if -3.2000000000000001e-189 < x

   1. Initial program 71.5%

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

      1. associate-/l* 85.8%

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

      2. associate-+l+ 85.8%

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

   3. Simplified 85.8%

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

   5. Taylor expanded in x around 0 55.2%

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

      1. +-commutative 55.2%

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

   7. Simplified 55.2%

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

   8. Taylor expanded in y around 0 38.3%

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

Alternative 15: 27.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -2.5e+16) (/ 0.5 x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.5e+16) {
		tmp = 0.5 / 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 <= (-2.5d+16)) then
        tmp = 0.5d0 / 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 <= -2.5e+16) {
		tmp = 0.5 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.5e+16:
		tmp = 0.5 / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.5e+16)
		tmp = Float64(0.5 / 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 <= -2.5e+16)
		tmp = 0.5 / 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, -2.5e+16], N[(0.5 / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5e16

   1. Initial program 55.2%

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

      1. associate-/l* 75.9%

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

      2. associate-+l+ 75.9%

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

   3. Simplified 75.9%

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

   5. Taylor expanded in x around 0 44.3%

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

      1. associate-*r* 44.3%

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

      2. +-commutative 44.3%

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

      3. unpow2 44.3%

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

      4. distribute-rgt-in 57.0%

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

      5. *-commutative 57.0%

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

   7. Simplified 57.0%

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

   8. Taylor expanded in x around inf 6.0%

      \[\leadsto \color{blue}{\frac{0.5}{x}} \]

   if -2.5e16 < x

   1. Initial program 74.4%

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

      1. associate-/l* 86.8%

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

      2. associate-+l+ 86.8%

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

   3. Simplified 86.8%

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

   5. Taylor expanded in x around 0 57.9%

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

      1. +-commutative 57.9%

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

   7. Simplified 57.9%

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

   8. Taylor expanded in y around 0 35.2%

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

Alternative 16: 4.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{0.5}{x} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ 0.5 x))

C

assert(x < y);
double code(double x, double y) {
	return 0.5 / x;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.5d0 / x
end function

Java

assert x < y;
public static double code(double x, double y) {
	return 0.5 / x;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return 0.5 / x

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(0.5 / x)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 0.5 / x;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(0.5 / x), $MachinePrecision]

Derivation

1. Initial program 69.0%

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

   1. associate-/l* 83.7%

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

   2. associate-+l+ 83.7%

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

3. Simplified 83.7%

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

5. Taylor expanded in x around 0 59.6%

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

   1. associate-*r* 59.6%

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

   2. +-commutative 59.6%

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

   3. unpow2 59.6%

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

   4. distribute-rgt-in 63.2%

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

   5. *-commutative 63.2%

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

7. Simplified 63.2%

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

8. Taylor expanded in x around inf 4.2%

   \[\leadsto \color{blue}{\frac{0.5}{x}} \]
9. 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 2024087 
(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))))