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

Percentage Accurate: 69.7% → 99.8%

Time: 18.1s

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: 69.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.3%

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

   1. associate-*l* 70.3%

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

   2. times-frac 92.5%

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

   3. +-commutative 92.5%

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

   4. +-commutative 92.5%

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

   5. associate-+r+ 92.5%

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

   6. +-commutative 92.5%

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

   7. associate-+l+ 92.5%

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

4. Applied egg-rr 92.5%

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

   1. *-un-lft-identity 92.5%

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

   2. times-frac 99.8%

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

   3. +-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

   1. associate-*l/ 99.8%

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

   2. *-lft-identity 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{-261}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{x + y}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-207}:\\ \;\;\;\;t\_0 \cdot \frac{\frac{y}{y + 1}}{x + y}\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+29}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(x + y\right) \cdot \frac{y + 1}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))))
   (if (<= y -2.2e-261)
     (/ (/ y (+ y (+ x 1.0))) (+ x y))
     (if (<= y 7.8e-207)
       (* t_0 (/ (/ y (+ y 1.0)) (+ x y)))
       (if (<= y 8.6e+29)
         (* x (/ y (* (* (+ x y) (+ x y)) (+ x (+ y 1.0)))))
         (/ t_0 (* (+ x y) (/ (+ y 1.0) y))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= -2.2e-261) {
		tmp = (y / (y + (x + 1.0))) / (x + y);
	} else if (y <= 7.8e-207) {
		tmp = t_0 * ((y / (y + 1.0)) / (x + y));
	} else if (y <= 8.6e+29) {
		tmp = x * (y / (((x + y) * (x + y)) * (x + (y + 1.0))));
	} else {
		tmp = t_0 / ((x + y) * ((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 = x / (x + y)
    if (y <= (-2.2d-261)) then
        tmp = (y / (y + (x + 1.0d0))) / (x + y)
    else if (y <= 7.8d-207) then
        tmp = t_0 * ((y / (y + 1.0d0)) / (x + y))
    else if (y <= 8.6d+29) then
        tmp = x * (y / (((x + y) * (x + y)) * (x + (y + 1.0d0))))
    else
        tmp = t_0 / ((x + y) * ((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 = x / (x + y);
	double tmp;
	if (y <= -2.2e-261) {
		tmp = (y / (y + (x + 1.0))) / (x + y);
	} else if (y <= 7.8e-207) {
		tmp = t_0 * ((y / (y + 1.0)) / (x + y));
	} else if (y <= 8.6e+29) {
		tmp = x * (y / (((x + y) * (x + y)) * (x + (y + 1.0))));
	} else {
		tmp = t_0 / ((x + y) * ((y + 1.0) / y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (x + y)
	tmp = 0
	if y <= -2.2e-261:
		tmp = (y / (y + (x + 1.0))) / (x + y)
	elif y <= 7.8e-207:
		tmp = t_0 * ((y / (y + 1.0)) / (x + y))
	elif y <= 8.6e+29:
		tmp = x * (y / (((x + y) * (x + y)) * (x + (y + 1.0))))
	else:
		tmp = t_0 / ((x + y) * ((y + 1.0) / y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	tmp = 0.0
	if (y <= -2.2e-261)
		tmp = Float64(Float64(y / Float64(y + Float64(x + 1.0))) / Float64(x + y));
	elseif (y <= 7.8e-207)
		tmp = Float64(t_0 * Float64(Float64(y / Float64(y + 1.0)) / Float64(x + y)));
	elseif (y <= 8.6e+29)
		tmp = Float64(x * Float64(y / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(x + Float64(y + 1.0)))));
	else
		tmp = Float64(t_0 / Float64(Float64(x + y) * Float64(Float64(y + 1.0) / y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	tmp = 0.0;
	if (y <= -2.2e-261)
		tmp = (y / (y + (x + 1.0))) / (x + y);
	elseif (y <= 7.8e-207)
		tmp = t_0 * ((y / (y + 1.0)) / (x + y));
	elseif (y <= 8.6e+29)
		tmp = x * (y / (((x + y) * (x + y)) * (x + (y + 1.0))));
	else
		tmp = t_0 / ((x + y) * ((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[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.2e-261], N[(N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.8e-207], N[(t$95$0 * N[(N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.6e+29], N[(x * N[(y / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(x + y), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.2000000000000002e-261

   1. Initial program 69.0%

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

      1. associate-*l* 69.0%

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

      2. times-frac 90.6%

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

      3. +-commutative 90.6%

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

      4. +-commutative 90.6%

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

      5. associate-+r+ 90.6%

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

      6. +-commutative 90.6%

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

      7. associate-+l+ 90.6%

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

   4. Applied egg-rr 90.6%

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

      1. *-un-lft-identity 90.6%

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

      2. times-frac 99.7%

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

      3. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around inf 48.1%

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

   if -2.2000000000000002e-261 < y < 7.80000000000000042e-207

   1. Initial program 60.2%

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

      1. associate-*l* 60.2%

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

      2. times-frac 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+l+ 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.9%

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

      2. *-lft-identity 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around 0 94.8%

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

       1. +-commutative 94.8%

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

   11. Simplified 94.8%

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

   if 7.80000000000000042e-207 < y < 8.6000000000000006e29

   1. Initial program 84.9%

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

      1. associate-/l* 90.0%

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

      2. associate-+l+ 90.0%

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

   3. Simplified 90.0%

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

   if 8.6000000000000006e29 < y

   1. Initial program 66.1%

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

      1. associate-*l* 66.1%

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

      2. times-frac 84.8%

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

      3. +-commutative 84.8%

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

      4. +-commutative 84.8%

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

      5. associate-+r+ 84.8%

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

      6. +-commutative 84.8%

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

      7. associate-+l+ 84.8%

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

   4. Applied egg-rr 84.8%

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

      1. *-un-lft-identity 84.8%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

   8. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. un-div-inv 99.8%

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

      3. +-commutative 99.8%

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

      4. div-inv 99.8%

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

      5. clear-num 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. +-commutative 99.8%

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

      9. associate-+l+ 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in x around 0 88.5%

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

       1. +-commutative 88.5%

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

   13. Simplified 88.5%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-261}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{x + y}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-207}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{\frac{y}{y + 1}}{x + y}\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+29}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(x + y\right) \cdot \frac{y + 1}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ t_1 := y + \left(x + 1\right)\\ \mathbf{if}\;y \leq -4 \cdot 10^{+124}:\\ \;\;\;\;\frac{\frac{y}{t\_1}}{x + y}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+74}:\\ \;\;\;\;t\_0 \cdot \frac{y}{\left(x + y\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))) (t_1 (+ y (+ x 1.0))))
   (if (<= y -4e+124)
     (/ (/ y t_1) (+ x y))
     (if (<= y 6.2e+74) (* t_0 (/ y (* (+ x y) t_1))) (/ t_0 (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = y + (x + 1.0);
	double tmp;
	if (y <= -4e+124) {
		tmp = (y / t_1) / (x + y);
	} else if (y <= 6.2e+74) {
		tmp = t_0 * (y / ((x + y) * t_1));
	} else {
		tmp = t_0 / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x / (x + y)
    t_1 = y + (x + 1.0d0)
    if (y <= (-4d+124)) then
        tmp = (y / t_1) / (x + y)
    else if (y <= 6.2d+74) then
        tmp = t_0 * (y / ((x + y) * t_1))
    else
        tmp = t_0 / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = y + (x + 1.0);
	double tmp;
	if (y <= -4e+124) {
		tmp = (y / t_1) / (x + y);
	} else if (y <= 6.2e+74) {
		tmp = t_0 * (y / ((x + y) * t_1));
	} else {
		tmp = t_0 / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (x + y)
	t_1 = y + (x + 1.0)
	tmp = 0
	if y <= -4e+124:
		tmp = (y / t_1) / (x + y)
	elif y <= 6.2e+74:
		tmp = t_0 * (y / ((x + y) * t_1))
	else:
		tmp = t_0 / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	t_1 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (y <= -4e+124)
		tmp = Float64(Float64(y / t_1) / Float64(x + y));
	elseif (y <= 6.2e+74)
		tmp = Float64(t_0 * Float64(y / Float64(Float64(x + y) * t_1)));
	else
		tmp = Float64(t_0 / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	t_1 = y + (x + 1.0);
	tmp = 0.0;
	if (y <= -4e+124)
		tmp = (y / t_1) / (x + y);
	elseif (y <= 6.2e+74)
		tmp = t_0 * (y / ((x + y) * t_1));
	else
		tmp = t_0 / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4e+124], N[(N[(y / t$95$1), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.2e+74], N[(t$95$0 * N[(y / N[(N[(x + y), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.99999999999999979e124

   1. Initial program 53.0%

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

      1. associate-*l* 53.0%

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

      2. times-frac 81.0%

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

      3. +-commutative 81.0%

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

      4. +-commutative 81.0%

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

      5. associate-+r+ 81.0%

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

      6. +-commutative 81.0%

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

      7. associate-+l+ 81.0%

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

   4. Applied egg-rr 81.0%

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

      1. *-un-lft-identity 81.0%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.9%

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

      2. *-lft-identity 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around inf 27.3%

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

   if -3.99999999999999979e124 < y < 6.20000000000000043e74

   1. Initial program 76.4%

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

      1. associate-*l* 76.4%

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

      2. times-frac 97.7%

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

      3. +-commutative 97.7%

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

      4. +-commutative 97.7%

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

      5. associate-+r+ 97.7%

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

      6. +-commutative 97.7%

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

      7. associate-+l+ 97.7%

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

   4. Applied egg-rr 97.7%

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

   if 6.20000000000000043e74 < y

   1. Initial program 63.0%

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

      1. associate-*l* 63.0%

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

      2. times-frac 83.6%

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

      3. +-commutative 83.6%

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

      4. +-commutative 83.6%

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

      5. associate-+r+ 83.6%

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

      6. +-commutative 83.6%

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

      7. associate-+l+ 83.6%

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

   4. Applied egg-rr 83.6%

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

      1. *-un-lft-identity 83.6%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

   8. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. un-div-inv 99.9%

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

      3. +-commutative 99.9%

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

      4. div-inv 99.9%

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

      5. clear-num 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+r+ 99.9%

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

      8. +-commutative 99.9%

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

      9. associate-+l+ 99.9%

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

   10. Applied egg-rr 99.9%

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

   11. Taylor expanded in x around 0 91.6%

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

       1. +-commutative 91.6%

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

   13. Simplified 91.6%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+124}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{x + y}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+74}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;y \leq -6 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{x + y}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+74}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(x + y\right) \cdot t\_0}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

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 1.0))))
   (if (<= y -6e-133)
     (/ (/ y t_0) (+ x y))
     (if (<= y 6.2e+74)
       (* x (/ (/ y (* (+ x y) t_0)) (+ x y)))
       (/ (/ x (+ x y)) (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= -6e-133) {
		tmp = (y / t_0) / (x + y);
	} else if (y <= 6.2e+74) {
		tmp = x * ((y / ((x + y) * t_0)) / (x + y));
	} else {
		tmp = (x / (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 + 1.0d0)
    if (y <= (-6d-133)) then
        tmp = (y / t_0) / (x + y)
    else if (y <= 6.2d+74) then
        tmp = x * ((y / ((x + y) * t_0)) / (x + y))
    else
        tmp = (x / (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 + 1.0);
	double tmp;
	if (y <= -6e-133) {
		tmp = (y / t_0) / (x + y);
	} else if (y <= 6.2e+74) {
		tmp = x * ((y / ((x + y) * t_0)) / (x + y));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if y <= -6e-133:
		tmp = (y / t_0) / (x + y)
	elif y <= 6.2e+74:
		tmp = x * ((y / ((x + y) * t_0)) / (x + y))
	else:
		tmp = (x / (x + y)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (y <= -6e-133)
		tmp = Float64(Float64(y / t_0) / Float64(x + y));
	elseif (y <= 6.2e+74)
		tmp = Float64(x * Float64(Float64(y / Float64(Float64(x + y) * t_0)) / Float64(x + y)));
	else
		tmp = Float64(Float64(x / 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 + 1.0);
	tmp = 0.0;
	if (y <= -6e-133)
		tmp = (y / t_0) / (x + y);
	elseif (y <= 6.2e+74)
		tmp = x * ((y / ((x + y) * t_0)) / (x + y));
	else
		tmp = (x / (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[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6e-133], N[(N[(y / t$95$0), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.2e+74], N[(x * N[(N[(y / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.00000000000000038e-133

   1. Initial program 69.8%

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

      1. associate-*l* 69.8%

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

      2. times-frac 88.1%

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

      3. +-commutative 88.1%

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

      4. +-commutative 88.1%

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

      5. associate-+r+ 88.1%

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

      6. +-commutative 88.1%

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

      7. associate-+l+ 88.1%

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

   4. Applied egg-rr 88.1%

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

      1. *-un-lft-identity 88.1%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around inf 39.6%

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

   if -6.00000000000000038e-133 < y < 6.20000000000000043e74

   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* 84.1%

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

      2. associate-+l+ 84.1%

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

   3. Simplified 84.1%

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

      1. *-un-lft-identity 84.1%

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

      2. associate-+r+ 84.1%

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

      3. associate-*l* 84.2%

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

      4. times-frac 96.5%

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

      5. +-commutative 96.5%

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

      6. +-commutative 96.5%

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

      7. associate-+r+ 96.5%

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

      8. +-commutative 96.5%

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

      9. associate-+l+ 96.5%

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

   6. Applied egg-rr 96.5%

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

      1. associate-*l/ 96.7%

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

      2. *-lft-identity 96.7%

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

      3. +-commutative 96.7%

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

   8. Simplified 96.7%

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

   if 6.20000000000000043e74 < y

   1. Initial program 63.0%

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

      1. associate-*l* 63.0%

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

      2. times-frac 83.6%

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

      3. +-commutative 83.6%

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

      4. +-commutative 83.6%

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

      5. associate-+r+ 83.6%

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

      6. +-commutative 83.6%

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

      7. associate-+l+ 83.6%

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

   4. Applied egg-rr 83.6%

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

      1. *-un-lft-identity 83.6%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

   8. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. un-div-inv 99.9%

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

      3. +-commutative 99.9%

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

      4. div-inv 99.9%

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

      5. clear-num 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+r+ 99.9%

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

      8. +-commutative 99.9%

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

      9. associate-+l+ 99.9%

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

   10. Applied egg-rr 99.9%

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

   11. Taylor expanded in x around 0 91.6%

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

       1. +-commutative 91.6%

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

   13. Simplified 91.6%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{x + y}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+74}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + 1}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-183}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-188}:\\ \;\;\;\;\frac{y}{x} \cdot \left(1 - \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -9.2e-151)
   (/ (/ y (+ x y)) (+ x 1.0))
   (if (<= x -1.55e-183)
     (/ x y)
     (if (<= x -1.05e-188)
       (* (/ y x) (- 1.0 (/ y x)))
       (/ (/ x (+ y 1.0)) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -9.2e-151) {
		tmp = (y / (x + y)) / (x + 1.0);
	} else if (x <= -1.55e-183) {
		tmp = x / y;
	} else if (x <= -1.05e-188) {
		tmp = (y / x) * (1.0 - (y / x));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-9.2d-151)) then
        tmp = (y / (x + y)) / (x + 1.0d0)
    else if (x <= (-1.55d-183)) then
        tmp = x / y
    else if (x <= (-1.05d-188)) then
        tmp = (y / x) * (1.0d0 - (y / x))
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -9.2e-151) {
		tmp = (y / (x + y)) / (x + 1.0);
	} else if (x <= -1.55e-183) {
		tmp = x / y;
	} else if (x <= -1.05e-188) {
		tmp = (y / x) * (1.0 - (y / x));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -9.2e-151:
		tmp = (y / (x + y)) / (x + 1.0)
	elif x <= -1.55e-183:
		tmp = x / y
	elif x <= -1.05e-188:
		tmp = (y / x) * (1.0 - (y / x))
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -9.2e-151)
		tmp = Float64(Float64(y / Float64(x + y)) / Float64(x + 1.0));
	elseif (x <= -1.55e-183)
		tmp = Float64(x / y);
	elseif (x <= -1.05e-188)
		tmp = Float64(Float64(y / x) * Float64(1.0 - Float64(y / x)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9.2e-151)
		tmp = (y / (x + y)) / (x + 1.0);
	elseif (x <= -1.55e-183)
		tmp = x / y;
	elseif (x <= -1.05e-188)
		tmp = (y / x) * (1.0 - (y / x));
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -9.2e-151], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.55e-183], N[(x / y), $MachinePrecision], If[LessEqual[x, -1.05e-188], N[(N[(y / x), $MachinePrecision] * N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -9.19999999999999984e-151

   1. Initial program 66.6%

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

      1. associate-*l* 66.6%

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

      2. times-frac 88.3%

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

      3. +-commutative 88.3%

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

      4. +-commutative 88.3%

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

      5. associate-+r+ 88.3%

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

      6. +-commutative 88.3%

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

      7. associate-+l+ 88.3%

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

   4. Applied egg-rr 88.3%

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

      1. clear-num 88.3%

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

      2. associate-/r* 99.7%

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

      3. frac-times 98.0%

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

      4. *-un-lft-identity 98.0%

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

      5. +-commutative 98.0%

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

   6. Applied egg-rr 98.0%

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

   7. Taylor expanded in y around 0 66.3%

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

      1. +-commutative 66.3%

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

   9. Simplified 66.3%

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

   if -9.19999999999999984e-151 < x < -1.55e-183

   1. Initial program 99.6%

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

      1. associate-/l* 98.4%

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

      2. associate-+l+ 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around 0 76.0%

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

   if -1.55e-183 < x < -1.05e-188

   1. Initial program 0.0%

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

      1. associate-*l* 0.0%

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

      2. times-frac 100.0%

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

      3. +-commutative 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-+r+ 100.0%

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

      6. +-commutative 100.0%

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

      7. associate-+l+ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 24.3%

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

      1. clear-num 24.3%

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

      2. inv-pow 24.3%

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

      3. distribute-lft-in 24.3%

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

      4. *-rgt-identity 24.3%

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

      5. pow2 24.3%

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

   7. Applied egg-rr 24.3%

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

      1. unpow-1 24.3%

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

      2. *-lft-identity 24.3%

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

      3. unpow2 24.3%

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

      4. distribute-rgt-in 24.3%

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

      5. associate-/l* 24.3%

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

      6. +-commutative 24.3%

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

   9. Simplified 24.3%

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

   10. Taylor expanded in x around 0 24.3%

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

   11. Taylor expanded in x around inf 24.4%

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

       1. mul-1-neg 24.4%

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

       2. unsub-neg 24.4%

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

   13. Simplified 24.4%

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

   if -1.05e-188 < x

   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* 82.6%

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

      2. associate-+l+ 82.6%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in x around 0 59.7%

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

      1. +-commutative 59.7%

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

   7. Simplified 59.7%

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

      1. *-un-lft-identity 59.7%

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

      2. times-frac 60.6%

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

   9. Applied egg-rr 60.6%

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

       1. associate-*l/ 60.6%

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

       2. *-lft-identity 60.6%

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

   11. Simplified 60.6%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + 1}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-183}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-188}:\\ \;\;\;\;\frac{y}{x} \cdot \left(1 - \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{\frac{y}{y + 1}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.8e+16)
   (/ (/ y (+ y (+ x 1.0))) (+ x y))
   (* (/ x (+ x y)) (/ (/ y (+ y 1.0)) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.8e+16) {
		tmp = (y / (y + (x + 1.0))) / (x + y);
	} else {
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (x + y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.8d+16)) then
        tmp = (y / (y + (x + 1.0d0))) / (x + y)
    else
        tmp = (x / (x + y)) * ((y / (y + 1.0d0)) / (x + y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -3.8e+16) {
		tmp = (y / (y + (x + 1.0))) / (x + y);
	} else {
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (x + y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.8e+16:
		tmp = (y / (y + (x + 1.0))) / (x + y)
	else:
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (x + y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.8e+16)
		tmp = Float64(Float64(y / Float64(y + Float64(x + 1.0))) / Float64(x + y));
	else
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(Float64(y / Float64(y + 1.0)) / Float64(x + y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.8e+16)
		tmp = (y / (y + (x + 1.0))) / (x + y);
	else
		tmp = (x / (x + y)) * ((y / (y + 1.0)) / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -3.8e+16], N[(N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.8e16

   1. Initial program 59.9%

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

      1. associate-*l* 59.9%

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

      2. times-frac 81.4%

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

      3. +-commutative 81.4%

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

      4. +-commutative 81.4%

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

      5. associate-+r+ 81.4%

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

      6. +-commutative 81.4%

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

      7. associate-+l+ 81.4%

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

   4. Applied egg-rr 81.4%

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

      1. *-un-lft-identity 81.4%

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

      2. times-frac 99.7%

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

      3. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.9%

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

      2. *-lft-identity 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around inf 77.9%

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

   if -3.8e16 < x

   1. Initial program 73.5%

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

      1. associate-*l* 73.5%

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

      2. times-frac 96.0%

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

      3. +-commutative 96.0%

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

      4. +-commutative 96.0%

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

      5. associate-+r+ 96.0%

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

      6. +-commutative 96.0%

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

      7. associate-+l+ 96.0%

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

   4. Applied egg-rr 96.0%

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

      1. *-un-lft-identity 96.0%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around 0 85.5%

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

       1. +-commutative 85.5%

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

   11. Simplified 85.5%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{\frac{y}{y + 1}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-150}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-246}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.02e-150)
   (/ (/ y x) (+ x 1.0))
   (if (<= x -8.5e-246) (/ (/ x (+ y 1.0)) y) (/ (/ x (+ x y)) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.02e-150) {
		tmp = (y / x) / (x + 1.0);
	} else if (x <= -8.5e-246) {
		tmp = (x / (y + 1.0)) / y;
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.02d-150)) then
        tmp = (y / x) / (x + 1.0d0)
    else if (x <= (-8.5d-246)) then
        tmp = (x / (y + 1.0d0)) / y
    else
        tmp = (x / (x + y)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.02e-150) {
		tmp = (y / x) / (x + 1.0);
	} else if (x <= -8.5e-246) {
		tmp = (x / (y + 1.0)) / y;
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.02e-150:
		tmp = (y / x) / (x + 1.0)
	elif x <= -8.5e-246:
		tmp = (x / (y + 1.0)) / y
	else:
		tmp = (x / (x + y)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.02e-150)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (x <= -8.5e-246)
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.02e-150)
		tmp = (y / x) / (x + 1.0);
	elseif (x <= -8.5e-246)
		tmp = (x / (y + 1.0)) / y;
	else
		tmp = (x / (x + y)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.02e-150], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8.5e-246], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.0199999999999999e-150

   1. Initial program 66.6%

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

      1. associate-/l* 80.8%

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

      2. associate-+l+ 80.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in y around 0 63.6%

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

      1. associate-/r* 66.2%

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

      2. +-commutative 66.2%

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

   7. Simplified 66.2%

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

   if -1.0199999999999999e-150 < x < -8.4999999999999998e-246

   1. Initial program 83.2%

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

      1. associate-/l* 83.4%

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

      2. associate-+l+ 83.4%

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

   3. Simplified 83.4%

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

   5. Taylor expanded in x around 0 84.7%

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

      1. +-commutative 84.7%

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

   7. Simplified 84.7%

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

      1. *-un-lft-identity 84.7%

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

      2. times-frac 84.5%

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

   9. Applied egg-rr 84.5%

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

       1. associate-*l/ 84.8%

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

       2. *-lft-identity 84.8%

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

   11. Simplified 84.8%

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

   if -8.4999999999999998e-246 < x

   1. Initial program 71.3%

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

      1. associate-*l* 71.3%

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

      2. times-frac 94.6%

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

      3. +-commutative 94.6%

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

      4. +-commutative 94.6%

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

      5. associate-+r+ 94.6%

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

      6. +-commutative 94.6%

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

      7. associate-+l+ 94.6%

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

   4. Applied egg-rr 94.6%

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

      1. *-un-lft-identity 94.6%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

   8. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. un-div-inv 99.8%

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

      3. +-commutative 99.8%

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

      4. div-inv 99.7%

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

      5. clear-num 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. +-commutative 99.8%

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

      9. associate-+l+ 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in x around 0 58.7%

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

       1. +-commutative 58.7%

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

   13. Simplified 58.7%

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

Alternative 8: 43.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-151} \lor \neg \left(x \leq -1.75 \cdot 10^{-245}\right) \land x \leq -1.7 \cdot 10^{-245}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (or (<= x -8.2e-151) (and (not (<= x -1.75e-245)) (<= x -1.7e-245)))
   (/ y x)
   (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if ((x <= -8.2e-151) || (!(x <= -1.75e-245) && (x <= -1.7e-245))) {
		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 <= (-8.2d-151)) .or. (.not. (x <= (-1.75d-245))) .and. (x <= (-1.7d-245))) 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 <= -8.2e-151) || (!(x <= -1.75e-245) && (x <= -1.7e-245))) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if (x <= -8.2e-151) or (not (x <= -1.75e-245) and (x <= -1.7e-245)):
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if ((x <= -8.2e-151) || (!(x <= -1.75e-245) && (x <= -1.7e-245)))
		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 <= -8.2e-151) || (~((x <= -1.75e-245)) && (x <= -1.7e-245)))
		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[Or[LessEqual[x, -8.2e-151], And[N[Not[LessEqual[x, -1.75e-245]], $MachinePrecision], LessEqual[x, -1.7e-245]]], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.2000000000000002e-151 or -1.75000000000000008e-245 < x < -1.7e-245

   1. Initial program 65.9%

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

      1. associate-/l* 80.0%

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

      2. associate-+l+ 80.0%

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

   3. Simplified 80.0%

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

   5. Taylor expanded in y around 0 64.0%

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

      1. associate-/r* 66.5%

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

      2. +-commutative 66.5%

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

   7. Simplified 66.5%

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

   8. Taylor expanded in x around 0 33.1%

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

   if -8.2000000000000002e-151 < x < -1.75000000000000008e-245 or -1.7e-245 < x

   1. Initial program 73.1%

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

      1. associate-/l* 83.0%

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

      2. associate-+l+ 83.0%

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

   3. Simplified 83.0%

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

   5. Taylor expanded in x around 0 60.8%

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

      1. +-commutative 60.8%

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

   7. Simplified 60.8%

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

   8. Taylor expanded in y around 0 39.2%

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

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-151} \lor \neg \left(x \leq -1.75 \cdot 10^{-245}\right) \land x \leq -1.7 \cdot 10^{-245}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5.1e-141) (/ (/ y (+ y (+ x 1.0))) (+ x y)) (/ (/ x (+ y 1.0)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5.1e-141) {
		tmp = (y / (y + (x + 1.0))) / (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 (x <= (-5.1d-141)) then
        tmp = (y / (y + (x + 1.0d0))) / (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 (x <= -5.1e-141) {
		tmp = (y / (y + (x + 1.0))) / (x + y);
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -5.1e-141:
		tmp = (y / (y + (x + 1.0))) / (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 (x <= -5.1e-141)
		tmp = Float64(Float64(y / Float64(y + Float64(x + 1.0))) / Float64(x + y));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.1e-141)
		tmp = (y / (y + (x + 1.0))) / (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[x, -5.1e-141], N[(N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.09999999999999977e-141

   1. Initial program 66.9%

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

      1. associate-*l* 66.9%

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

      2. times-frac 88.1%

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

      3. +-commutative 88.1%

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

      4. +-commutative 88.1%

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

      5. associate-+r+ 88.1%

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

      6. +-commutative 88.1%

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

      7. associate-+l+ 88.1%

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

   4. Applied egg-rr 88.1%

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

      1. *-un-lft-identity 88.1%

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

      2. times-frac 99.7%

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

      3. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around inf 67.0%

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

   if -5.09999999999999977e-141 < x

   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* 82.7%

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

      2. associate-+l+ 82.7%

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

   3. Simplified 82.7%

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

   5. Taylor expanded in x around 0 60.3%

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

      1. +-commutative 60.3%

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

   7. Simplified 60.3%

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

      1. *-un-lft-identity 60.3%

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

      2. times-frac 61.1%

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

   9. Applied egg-rr 61.1%

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

       1. associate-*l/ 61.2%

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

       2. *-lft-identity 61.2%

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

   11. Simplified 61.2%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-124}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.15e-124) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ y 1.0)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.15e-124) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.15d-124)) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.15e-124) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.15e-124:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.15e-124)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.15e-124)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.15e-124], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.15000000000000006e-124

   1. Initial program 67.6%

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

      1. associate-/l* 79.0%

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

      2. associate-+l+ 79.0%

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

   3. Simplified 79.0%

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

   5. Taylor expanded in y around 0 64.1%

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

      1. associate-/r* 67.0%

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

      2. +-commutative 67.0%

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

   7. Simplified 67.0%

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

   if -1.15000000000000006e-124 < x

   1. Initial program 71.8%

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

      1. associate-/l* 83.4%

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

      2. associate-+l+ 83.4%

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

   3. Simplified 83.4%

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

   5. Taylor expanded in x around 0 59.3%

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

      1. +-commutative 59.3%

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

   7. Simplified 59.3%

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

      1. *-un-lft-identity 59.3%

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

      2. times-frac 60.0%

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

   9. Applied egg-rr 60.0%

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

       1. associate-*l/ 60.1%

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

       2. *-lft-identity 60.1%

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

   11. Simplified 60.1%

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

Alternative 11: 80.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{-141}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5.4e-141) (/ y (* x (+ x 1.0))) (/ (/ x (+ y 1.0)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5.4e-141) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.4d-141)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -5.4e-141) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -5.4e-141:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5.4e-141)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.4e-141)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -5.4e-141], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.4000000000000005e-141

   1. Initial program 66.9%

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

      1. associate-/l* 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 63.8%

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

   if -5.4000000000000005e-141 < x

   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* 82.7%

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

      2. associate-+l+ 82.7%

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

   3. Simplified 82.7%

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

   5. Taylor expanded in x around 0 60.3%

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

      1. +-commutative 60.3%

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

   7. Simplified 60.3%

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

      1. *-un-lft-identity 60.3%

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

      2. times-frac 61.1%

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

   9. Applied egg-rr 61.1%

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

       1. associate-*l/ 61.2%

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

       2. *-lft-identity 61.2%

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

   11. Simplified 61.2%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{-141}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 78.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-124}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.15e-124) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.15e-124) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.15d-124)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.15e-124) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.15e-124:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.15e-124)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.15e-124)
		tmp = y / (x * (x + 1.0));
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.15e-124], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.15000000000000006e-124

   1. Initial program 67.6%

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

      1. associate-/l* 79.0%

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

      2. associate-+l+ 79.0%

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

   3. Simplified 79.0%

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

   5. Taylor expanded in y around 0 64.1%

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

   if -1.15000000000000006e-124 < x

   1. Initial program 71.8%

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

      1. associate-/l* 83.4%

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

      2. associate-+l+ 83.4%

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

   3. Simplified 83.4%

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

   5. Taylor expanded in x around 0 59.3%

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

      1. +-commutative 59.3%

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

   7. Simplified 59.3%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-124}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{-156}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.9e-156) (/ y x) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.9e-156) {
		tmp = y / x;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.9d-156) then
        tmp = y / x
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.9e-156) {
		tmp = y / x;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.9e-156:
		tmp = y / x
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.9e-156)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.9e-156)
		tmp = y / x;
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 2.9e-156], N[(y / x), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.90000000000000021e-156

   1. Initial program 67.2%

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

      1. associate-/l* 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 54.9%

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

      1. associate-/r* 58.8%

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

      2. +-commutative 58.8%

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

   7. Simplified 58.8%

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

   8. Taylor expanded in x around 0 35.9%

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

   if 2.90000000000000021e-156 < y

   1. Initial program 75.2%

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

      1. associate-/l* 83.9%

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

      2. associate-+l+ 83.9%

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

   3. Simplified 83.9%

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

   5. Taylor expanded in x around 0 60.2%

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

      1. +-commutative 60.2%

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

   7. Simplified 60.2%

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

Alternative 14: 27.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -9e+34) {
		tmp = 1.0 / 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 <= (-9d+34)) then
        tmp = 1.0d0 / 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 <= -9e+34) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

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

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -9e+34)
		tmp = Float64(1.0 / 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 <= -9e+34)
		tmp = 1.0 / 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, -9e+34], N[(1.0 / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.0000000000000001e34

   1. Initial program 58.9%

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

      1. associate-*l* 58.9%

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

      2. times-frac 80.0%

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

      3. +-commutative 80.0%

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

      4. +-commutative 80.0%

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

      5. associate-+r+ 80.0%

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

      6. +-commutative 80.0%

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

      7. associate-+l+ 80.0%

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

   4. Applied egg-rr 80.0%

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

   5. Taylor expanded in y around 0 73.1%

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

   6. Taylor expanded in y around inf 6.2%

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

      1. +-commutative 6.2%

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

   8. Simplified 6.2%

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

   9. Taylor expanded in x around inf 6.2%

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

   if -9.0000000000000001e34 < x

   1. Initial program 73.6%

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

      1. associate-/l* 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 x around 0 58.5%

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

      1. +-commutative 58.5%

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

   7. Simplified 58.5%

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

   8. Taylor expanded in y around 0 33.6%

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

Alternative 15: 4.3% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.3%

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

   1. associate-*l* 70.3%

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

   2. times-frac 92.5%

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

   3. +-commutative 92.5%

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

   4. +-commutative 92.5%

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

   5. associate-+r+ 92.5%

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

   6. +-commutative 92.5%

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

   7. associate-+l+ 92.5%

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

4. Applied egg-rr 92.5%

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

5. Taylor expanded in y around 0 47.4%

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

6. Taylor expanded in y around inf 4.7%

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

   1. +-commutative 4.7%

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

8. Simplified 4.7%

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

9. Taylor expanded in x around inf 4.1%

   \[\leadsto \color{blue}{\frac{1}{x}} \]
10. Add Preprocessing

Alternative 16: 3.4% accurate, 17.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.3%

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

   1. associate-*l* 70.3%

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

   2. times-frac 92.5%

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

   3. +-commutative 92.5%

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

   4. +-commutative 92.5%

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

   5. associate-+r+ 92.5%

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

   6. +-commutative 92.5%

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

   7. associate-+l+ 92.5%

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

4. Applied egg-rr 92.5%

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

   1. *-un-lft-identity 92.5%

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

   2. times-frac 99.8%

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

   3. +-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

   1. associate-*l/ 99.8%

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

   2. *-lft-identity 99.8%

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

8. Simplified 99.8%

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

   1. clear-num 99.7%

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

   2. un-div-inv 99.8%

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

   3. +-commutative 99.8%

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

   4. div-inv 99.7%

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

   5. clear-num 99.7%

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

   6. +-commutative 99.7%

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

   7. associate-+r+ 99.7%

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

   8. +-commutative 99.7%

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

   9. associate-+l+ 99.7%

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

10. Applied egg-rr 99.7%

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

11. Taylor expanded in x around 0 51.9%

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

    1. +-commutative 51.9%

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

13. Simplified 51.9%

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

14. Taylor expanded in y around 0 3.4%

    \[\leadsto \color{blue}{1} \]
15. 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 2024107 
(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))))