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

Percentage Accurate: 68.7% → 99.8%

Time: 14.4s

Alternatives: 16

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.7% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

   1. associate-+r+ 67.9%

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

   2. *-commutative 67.9%

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

   3. frac-times 87.0%

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

   4. associate-*l/ 80.4%

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

   5. times-frac 99.8%

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

   6. associate-+r+ 99.8%

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

   7. +-commutative 99.8%

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

   8. associate-+l+ 99.8%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 87.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{y + x}\\ \mathbf{if}\;y \leq 5.5 \cdot 10^{-280}:\\ \;\;\;\;t_0 \cdot \frac{1}{x + 1}\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-17}:\\ \;\;\;\;t_0 \cdot \frac{x}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \frac{1}{y + x}\right) \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y x))))
   (if (<= y 5.5e-280)
     (* t_0 (/ 1.0 (+ x 1.0)))
     (if (<= y 5.1e-17)
       (* t_0 (/ x (+ y x)))
       (* (* t_0 (/ 1.0 (+ y x))) (/ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (y <= 5.5e-280) {
		tmp = t_0 * (1.0 / (x + 1.0));
	} else if (y <= 5.1e-17) {
		tmp = t_0 * (x / (y + x));
	} else {
		tmp = (t_0 * (1.0 / (y + x))) * (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) :: t_0
    real(8) :: tmp
    t_0 = y / (y + x)
    if (y <= 5.5d-280) then
        tmp = t_0 * (1.0d0 / (x + 1.0d0))
    else if (y <= 5.1d-17) then
        tmp = t_0 * (x / (y + x))
    else
        tmp = (t_0 * (1.0d0 / (y + x))) * (x / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (y <= 5.5e-280) {
		tmp = t_0 * (1.0 / (x + 1.0));
	} else if (y <= 5.1e-17) {
		tmp = t_0 * (x / (y + x));
	} else {
		tmp = (t_0 * (1.0 / (y + x))) * (x / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (y + x)
	tmp = 0
	if y <= 5.5e-280:
		tmp = t_0 * (1.0 / (x + 1.0))
	elif y <= 5.1e-17:
		tmp = t_0 * (x / (y + x))
	else:
		tmp = (t_0 * (1.0 / (y + x))) * (x / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(y + x))
	tmp = 0.0
	if (y <= 5.5e-280)
		tmp = Float64(t_0 * Float64(1.0 / Float64(x + 1.0)));
	elseif (y <= 5.1e-17)
		tmp = Float64(t_0 * Float64(x / Float64(y + x)));
	else
		tmp = Float64(Float64(t_0 * Float64(1.0 / Float64(y + x))) * Float64(x / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (y + x);
	tmp = 0.0;
	if (y <= 5.5e-280)
		tmp = t_0 * (1.0 / (x + 1.0));
	elseif (y <= 5.1e-17)
		tmp = t_0 * (x / (y + x));
	else
		tmp = (t_0 * (1.0 / (y + x))) * (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_] := Block[{t$95$0 = N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 5.5e-280], N[(t$95$0 * N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.1e-17], N[(t$95$0 * N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.50000000000000001e-280

   1. Initial program 70.1%

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

      1. associate-+r+ 70.1%

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

      2. *-commutative 70.1%

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

      3. frac-times 85.6%

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

      4. associate-*l/ 79.4%

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

      5. times-frac 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in y around 0 50.9%

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

      1. +-commutative 50.9%

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

   6. Simplified 50.9%

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

   if 5.50000000000000001e-280 < y < 5.1000000000000003e-17

   1. Initial program 73.9%

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

      1. associate-+r+ 73.9%

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

      2. *-commutative 73.9%

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

      3. frac-times 93.9%

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

      4. associate-*l/ 77.6%

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

      5. times-frac 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 75.0%

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

      1. +-commutative 75.0%

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

   6. Simplified 75.0%

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

   7. Taylor expanded in y around 0 75.0%

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

   if 5.1000000000000003e-17 < y

   1. Initial program 58.4%

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

      1. associate-*l* 58.4%

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

      2. +-commutative 58.4%

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

      3. +-commutative 58.4%

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

      4. +-commutative 58.4%

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

      5. associate-*l* 58.4%

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

      6. *-commutative 58.4%

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

      7. times-frac 84.8%

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

      8. +-commutative 84.8%

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

      9. +-commutative 84.8%

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

      10. +-commutative 84.8%

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

      11. associate-+l+ 84.8%

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

   3. Simplified 84.8%

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

   4. Taylor expanded in y around inf 76.3%

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

      1. associate-/r* 81.1%

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

      2. div-inv 81.1%

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

   6. Applied egg-rr 81.1%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-280}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x + 1}\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-17}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{y + x} \cdot \frac{1}{y + x}\right) \cdot \frac{x}{y}\\ \end{array} \]

Alternative 3: 91.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{y + x}\\ \mathbf{if}\;y \leq 5.6 \cdot 10^{-176}:\\ \;\;\;\;t_0 \cdot \frac{1}{x + 1}\\ \mathbf{elif}\;y \leq 13:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \frac{1}{y + x}\right) \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y x))))
   (if (<= y 5.6e-176)
     (* t_0 (/ 1.0 (+ x 1.0)))
     (if (<= y 13.0)
       (* (/ y (* (+ y x) (+ y x))) (/ x (+ x 1.0)))
       (* (* t_0 (/ 1.0 (+ y x))) (/ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (y <= 5.6e-176) {
		tmp = t_0 * (1.0 / (x + 1.0));
	} else if (y <= 13.0) {
		tmp = (y / ((y + x) * (y + x))) * (x / (x + 1.0));
	} else {
		tmp = (t_0 * (1.0 / (y + x))) * (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) :: t_0
    real(8) :: tmp
    t_0 = y / (y + x)
    if (y <= 5.6d-176) then
        tmp = t_0 * (1.0d0 / (x + 1.0d0))
    else if (y <= 13.0d0) then
        tmp = (y / ((y + x) * (y + x))) * (x / (x + 1.0d0))
    else
        tmp = (t_0 * (1.0d0 / (y + x))) * (x / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (y <= 5.6e-176) {
		tmp = t_0 * (1.0 / (x + 1.0));
	} else if (y <= 13.0) {
		tmp = (y / ((y + x) * (y + x))) * (x / (x + 1.0));
	} else {
		tmp = (t_0 * (1.0 / (y + x))) * (x / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (y + x)
	tmp = 0
	if y <= 5.6e-176:
		tmp = t_0 * (1.0 / (x + 1.0))
	elif y <= 13.0:
		tmp = (y / ((y + x) * (y + x))) * (x / (x + 1.0))
	else:
		tmp = (t_0 * (1.0 / (y + x))) * (x / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(y + x))
	tmp = 0.0
	if (y <= 5.6e-176)
		tmp = Float64(t_0 * Float64(1.0 / Float64(x + 1.0)));
	elseif (y <= 13.0)
		tmp = Float64(Float64(y / Float64(Float64(y + x) * Float64(y + x))) * Float64(x / Float64(x + 1.0)));
	else
		tmp = Float64(Float64(t_0 * Float64(1.0 / Float64(y + x))) * Float64(x / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (y + x);
	tmp = 0.0;
	if (y <= 5.6e-176)
		tmp = t_0 * (1.0 / (x + 1.0));
	elseif (y <= 13.0)
		tmp = (y / ((y + x) * (y + x))) * (x / (x + 1.0));
	else
		tmp = (t_0 * (1.0 / (y + x))) * (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_] := Block[{t$95$0 = N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 5.6e-176], N[(t$95$0 * N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 13.0], N[(N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.6000000000000003e-176

   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. Step-by-step derivation

      1. associate-+r+ 71.3%

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

      2. *-commutative 71.3%

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

      3. frac-times 86.6%

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

      4. associate-*l/ 79.4%

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

      5. times-frac 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in y around 0 56.2%

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

      1. +-commutative 56.2%

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

   6. Simplified 56.2%

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

   if 5.6000000000000003e-176 < y < 13

   1. Initial program 72.5%

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

      1. associate-*l* 72.5%

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

      2. +-commutative 72.5%

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

      3. +-commutative 72.5%

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

      4. +-commutative 72.5%

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

      5. associate-*l* 72.5%

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

      6. *-commutative 72.5%

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

      7. times-frac 92.9%

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

      8. +-commutative 92.9%

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

      9. +-commutative 92.9%

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

      10. +-commutative 92.9%

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

      11. associate-+l+ 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in y around 0 90.9%

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

      1. +-commutative 90.9%

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

   6. Simplified 90.9%

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

   if 13 < y

   1. Initial program 56.4%

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

      1. associate-*l* 56.4%

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

      2. +-commutative 56.4%

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

      3. +-commutative 56.4%

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

      4. +-commutative 56.4%

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

      5. associate-*l* 56.4%

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

      6. *-commutative 56.4%

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

      7. times-frac 84.6%

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

      8. +-commutative 84.6%

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

      9. +-commutative 84.6%

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

      10. +-commutative 84.6%

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

      11. associate-+l+ 84.6%

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

   3. Simplified 84.6%

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

   4. Taylor expanded in y around inf 78.3%

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

      1. associate-/r* 84.3%

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

      2. div-inv 84.3%

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

   6. Applied egg-rr 84.3%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{-176}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1}{x + 1}\\ \mathbf{elif}\;y \leq 13:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{y + x} \cdot \frac{1}{y + x}\right) \cdot \frac{x}{y}\\ \end{array} \]

Alternative 4: 93.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)} \cdot \left(1 - \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.6e-15)
   (* (/ y (* (+ y x) (+ y (+ x 1.0)))) (- 1.0 (/ y x)))
   (* (/ y (+ y x)) (/ (/ x (+ y 1.0)) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.6e-15) {
		tmp = (y / ((y + x) * (y + (x + 1.0)))) * (1.0 - (y / x));
	} else {
		tmp = (y / (y + x)) * ((x / (y + 1.0)) / (y + x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.6d-15)) then
        tmp = (y / ((y + x) * (y + (x + 1.0d0)))) * (1.0d0 - (y / x))
    else
        tmp = (y / (y + x)) * ((x / (y + 1.0d0)) / (y + x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.6e-15) {
		tmp = (y / ((y + x) * (y + (x + 1.0)))) * (1.0 - (y / x));
	} else {
		tmp = (y / (y + x)) * ((x / (y + 1.0)) / (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.6e-15:
		tmp = (y / ((y + x) * (y + (x + 1.0)))) * (1.0 - (y / x))
	else:
		tmp = (y / (y + x)) * ((x / (y + 1.0)) / (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.6e-15)
		tmp = Float64(Float64(y / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0)))) * Float64(1.0 - Float64(y / x)));
	else
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.6e-15)
		tmp = (y / ((y + x) * (y + (x + 1.0)))) * (1.0 - (y / x));
	else
		tmp = (y / (y + x)) * ((x / (y + 1.0)) / (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.6e-15], N[(N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6e-15

   1. Initial program 66.2%

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

      1. associate-+r+ 66.2%

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

      2. associate-*l* 66.1%

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

      3. times-frac 88.8%

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

      4. associate-+r+ 88.8%

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

      5. +-commutative 88.8%

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

      6. associate-+l+ 88.8%

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

   3. Applied egg-rr 88.8%

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

   4. Taylor expanded in x around inf 80.5%

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

      1. mul-1-neg 80.5%

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

      2. unsub-neg 80.5%

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

   6. Simplified 80.5%

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

   if -1.6e-15 < x

   1. Initial program 68.6%

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

      1. associate-+r+ 68.6%

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

      2. *-commutative 68.6%

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

      3. frac-times 86.3%

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

      4. associate-*l/ 77.2%

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

      5. times-frac 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 85.5%

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

      1. +-commutative 85.5%

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

   6. Simplified 85.5%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)} \cdot \left(1 - \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]

Alternative 5: 96.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{y + x} \cdot \frac{1}{y + x}\right) \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.4e+154)
   (* (/ x (+ y x)) (/ y (* (+ y x) (+ y (+ x 1.0)))))
   (* (* (/ y (+ y x)) (/ 1.0 (+ y x))) (/ x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.4e+154) {
		tmp = (x / (y + x)) * (y / ((y + x) * (y + (x + 1.0))));
	} else {
		tmp = ((y / (y + x)) * (1.0 / (y + x))) * (x / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.4d+154) then
        tmp = (x / (y + x)) * (y / ((y + x) * (y + (x + 1.0d0))))
    else
        tmp = ((y / (y + x)) * (1.0d0 / (y + x))) * (x / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.4e+154) {
		tmp = (x / (y + x)) * (y / ((y + x) * (y + (x + 1.0))));
	} else {
		tmp = ((y / (y + x)) * (1.0 / (y + x))) * (x / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.4e+154:
		tmp = (x / (y + x)) * (y / ((y + x) * (y + (x + 1.0))))
	else:
		tmp = ((y / (y + x)) * (1.0 / (y + x))) * (x / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.4e+154)
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0)))));
	else
		tmp = Float64(Float64(Float64(y / Float64(y + x)) * Float64(1.0 / Float64(y + x))) * Float64(x / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.4e+154)
		tmp = (x / (y + x)) * (y / ((y + x) * (y + (x + 1.0))));
	else
		tmp = ((y / (y + x)) * (1.0 / (y + x))) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.4e+154], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.4e154

   1. Initial program 71.5%

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

      1. associate-+r+ 71.5%

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

      2. associate-*l* 71.4%

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

      3. times-frac 96.3%

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

      4. associate-+r+ 96.3%

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

      5. +-commutative 96.3%

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

      6. associate-+l+ 96.3%

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

   3. Applied egg-rr 96.3%

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

   if 1.4e154 < y

   1. Initial program 44.6%

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

      1. associate-*l* 44.6%

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

      2. +-commutative 44.6%

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

      3. +-commutative 44.6%

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

      4. +-commutative 44.6%

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

      5. associate-*l* 44.6%

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

      6. *-commutative 44.6%

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

      7. times-frac 75.2%

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

      8. +-commutative 75.2%

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

      9. +-commutative 75.2%

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

      10. +-commutative 75.2%

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

      11. associate-+l+ 75.2%

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

   3. Simplified 75.2%

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

   4. Taylor expanded in y around inf 75.2%

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

      1. associate-/r* 85.9%

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

      2. div-inv 85.9%

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

   6. Applied egg-rr 85.9%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{y + x} \cdot \frac{1}{y + x}\right) \cdot \frac{x}{y}\\ \end{array} \]

Alternative 6: 93.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.6e-15)
   (* (/ y (* (+ y x) (+ y x))) (/ x (+ x 1.0)))
   (* (/ y (+ y x)) (/ (/ x (+ y 1.0)) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.6e-15) {
		tmp = (y / ((y + x) * (y + x))) * (x / (x + 1.0));
	} else {
		tmp = (y / (y + x)) * ((x / (y + 1.0)) / (y + x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.6d-15)) then
        tmp = (y / ((y + x) * (y + x))) * (x / (x + 1.0d0))
    else
        tmp = (y / (y + x)) * ((x / (y + 1.0d0)) / (y + x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.6e-15) {
		tmp = (y / ((y + x) * (y + x))) * (x / (x + 1.0));
	} else {
		tmp = (y / (y + x)) * ((x / (y + 1.0)) / (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.6e-15:
		tmp = (y / ((y + x) * (y + x))) * (x / (x + 1.0))
	else:
		tmp = (y / (y + x)) * ((x / (y + 1.0)) / (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.6e-15)
		tmp = Float64(Float64(y / Float64(Float64(y + x) * Float64(y + x))) * Float64(x / Float64(x + 1.0)));
	else
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.6e-15)
		tmp = (y / ((y + x) * (y + x))) * (x / (x + 1.0));
	else
		tmp = (y / (y + x)) * ((x / (y + 1.0)) / (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.6e-15], N[(N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6e-15

   1. Initial program 66.2%

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

      1. associate-*l* 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. +-commutative 66.1%

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

      3. +-commutative 66.1%

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

      4. +-commutative 66.1%

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

      5. associate-*l* 66.2%

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

      6. *-commutative 66.2%

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

      7. times-frac 88.8%

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

      8. +-commutative 88.8%

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

      9. +-commutative 88.8%

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

      10. +-commutative 88.8%

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

      11. associate-+l+ 88.8%

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

   3. Simplified 88.8%

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

   4. Taylor expanded in y around 0 80.2%

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

      1. +-commutative 80.2%

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

   6. Simplified 80.2%

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

   if -1.6e-15 < x

   1. Initial program 68.6%

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

      1. associate-+r+ 68.6%

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

      2. *-commutative 68.6%

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

      3. frac-times 86.3%

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

      4. associate-*l/ 77.2%

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

      5. times-frac 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 85.5%

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

      1. +-commutative 85.5%

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

   6. Simplified 85.5%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]

Alternative 7: 86.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ t_1 := \frac{y}{y + x}\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{-15}:\\ \;\;\;\;\frac{t_1}{x + 1}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-279}:\\ \;\;\;\;t_1 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))) (t_1 (/ y (+ y x))))
   (if (<= x -1.45e-15)
     (/ t_1 (+ x 1.0))
     (if (<= x -1.6e-279) (* t_1 t_0) (* t_0 (/ 1.0 (+ y 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y + x);
	double t_1 = y / (y + x);
	double tmp;
	if (x <= -1.45e-15) {
		tmp = t_1 / (x + 1.0);
	} else if (x <= -1.6e-279) {
		tmp = t_1 * t_0;
	} else {
		tmp = t_0 * (1.0 / (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x / (y + x)
    t_1 = y / (y + x)
    if (x <= (-1.45d-15)) then
        tmp = t_1 / (x + 1.0d0)
    else if (x <= (-1.6d-279)) then
        tmp = t_1 * t_0
    else
        tmp = t_0 * (1.0d0 / (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double t_1 = y / (y + x);
	double tmp;
	if (x <= -1.45e-15) {
		tmp = t_1 / (x + 1.0);
	} else if (x <= -1.6e-279) {
		tmp = t_1 * t_0;
	} else {
		tmp = t_0 * (1.0 / (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y + x)
	t_1 = y / (y + x)
	tmp = 0
	if x <= -1.45e-15:
		tmp = t_1 / (x + 1.0)
	elif x <= -1.6e-279:
		tmp = t_1 * t_0
	else:
		tmp = t_0 * (1.0 / (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	t_1 = Float64(y / Float64(y + x))
	tmp = 0.0
	if (x <= -1.45e-15)
		tmp = Float64(t_1 / Float64(x + 1.0));
	elseif (x <= -1.6e-279)
		tmp = Float64(t_1 * t_0);
	else
		tmp = Float64(t_0 * Float64(1.0 / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	t_1 = y / (y + x);
	tmp = 0.0;
	if (x <= -1.45e-15)
		tmp = t_1 / (x + 1.0);
	elseif (x <= -1.6e-279)
		tmp = t_1 * t_0;
	else
		tmp = t_0 * (1.0 / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.45e-15], N[(t$95$1 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.6e-279], N[(t$95$1 * t$95$0), $MachinePrecision], N[(t$95$0 * N[(1.0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.45000000000000009e-15

   1. Initial program 66.2%

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

      1. associate-+r+ 66.2%

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

      2. *-commutative 66.2%

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

      3. frac-times 88.8%

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

      4. associate-*l/ 88.8%

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

      5. times-frac 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in y around 0 76.3%

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

      1. +-commutative 76.3%

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

   6. Simplified 76.3%

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

      1. un-div-inv 76.3%

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

      2. +-commutative 76.3%

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

   8. Applied egg-rr 76.3%

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

   if -1.45000000000000009e-15 < x < -1.5999999999999999e-279

   1. Initial program 78.2%

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

      1. associate-+r+ 78.2%

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

      2. *-commutative 78.2%

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

      3. frac-times 90.1%

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

      4. associate-*l/ 79.6%

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

      5. times-frac 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 99.8%

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

      1. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in y around 0 74.9%

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

   if -1.5999999999999999e-279 < x

   1. Initial program 63.8%

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

      1. associate-+r+ 63.8%

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

      2. associate-*l* 63.8%

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

      3. times-frac 93.2%

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

      4. associate-+r+ 93.2%

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

      5. +-commutative 93.2%

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

      6. associate-+l+ 93.2%

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

   3. Applied egg-rr 93.2%

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

   4. Taylor expanded in x around 0 56.8%

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

      1. +-commutative 56.8%

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

   6. Simplified 56.8%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-279}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{1}{y + 1}\\ \end{array} \]

Alternative 8: 80.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-102}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+74}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.65e-102)
   (/ y (* x (+ x 1.0)))
   (if (<= y 1.25e+74) (/ x (* y (+ y 1.0))) (* (/ x y) (/ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.65e-102) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 1.25e+74) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.65d-102) then
        tmp = y / (x * (x + 1.0d0))
    else if (y <= 1.25d+74) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.65e-102) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 1.25e+74) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.65e-102:
		tmp = y / (x * (x + 1.0))
	elif y <= 1.25e+74:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.65e-102)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (y <= 1.25e+74)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.65e-102)
		tmp = y / (x * (x + 1.0));
	elseif (y <= 1.25e+74)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.65e-102], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e+74], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.65e-102

   1. Initial program 71.4%

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

      1. associate-*l* 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. +-commutative 71.3%

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

      3. +-commutative 71.3%

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

      4. +-commutative 71.3%

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

      5. associate-*l* 71.4%

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

      6. *-commutative 71.4%

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

      7. times-frac 86.3%

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

      8. +-commutative 86.3%

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

      9. +-commutative 86.3%

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

      10. +-commutative 86.3%

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

      11. associate-+l+ 86.3%

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

   3. Simplified 86.3%

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

   4. Taylor expanded in y around 0 56.2%

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

      1. +-commutative 56.2%

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

   6. Simplified 56.2%

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

   if 1.65e-102 < y < 1.24999999999999991e74

   1. Initial program 77.4%

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

      1. associate-*l* 77.3%

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

      2. +-commutative 77.3%

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

      3. +-commutative 77.3%

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

      4. +-commutative 77.3%

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

      5. associate-*l* 77.4%

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

      6. *-commutative 77.4%

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

      7. times-frac 98.5%

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

      8. +-commutative 98.5%

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

      9. +-commutative 98.5%

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

      10. +-commutative 98.5%

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

      11. associate-+l+ 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in x around 0 58.1%

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

      1. +-commutative 58.1%

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

   6. Simplified 58.1%

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

   if 1.24999999999999991e74 < y

   1. Initial program 47.6%

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

      1. associate-*l* 47.6%

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

      2. +-commutative 47.6%

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

      3. +-commutative 47.6%

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

      4. +-commutative 47.6%

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

      5. associate-*l* 47.6%

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

      6. *-commutative 47.6%

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

      7. times-frac 80.1%

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

      8. +-commutative 80.1%

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

      9. +-commutative 80.1%

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

      10. +-commutative 80.1%

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

      11. associate-+l+ 80.1%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in y around inf 76.1%

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

   5. Taylor expanded in y around inf 75.6%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-102}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+74}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]

Alternative 9: 80.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{-117}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.15e-117) (/ y (* x (+ x 1.0))) (/ (/ x (+ y (+ x 1.0))) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.15e-117) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + (x + 1.0))) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.15d-117) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / (y + (x + 1.0d0))) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.15e-117) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + (x + 1.0))) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.15e-117:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / (y + (x + 1.0))) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.15e-117)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.15e-117)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / (y + (x + 1.0))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 2.15e-117], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.15e-117

   1. Initial program 71.4%

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

      1. associate-*l* 71.4%

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

      2. +-commutative 71.4%

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

      3. +-commutative 71.4%

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

      4. +-commutative 71.4%

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

      5. associate-*l* 71.4%

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

      6. *-commutative 71.4%

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

      7. times-frac 86.1%

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

      8. +-commutative 86.1%

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

      9. +-commutative 86.1%

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

      10. +-commutative 86.1%

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

      11. associate-+l+ 86.1%

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

   3. Simplified 86.1%

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

   4. Taylor expanded in y around 0 56.0%

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

      1. +-commutative 56.0%

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

   6. Simplified 56.0%

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

   if 2.15e-117 < y

   1. Initial program 61.2%

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

      1. associate-*l* 61.1%

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

      2. +-commutative 61.1%

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

      3. +-commutative 61.1%

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

      4. +-commutative 61.1%

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

      5. associate-*l* 61.2%

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

      6. *-commutative 61.2%

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

      7. times-frac 88.7%

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

      8. +-commutative 88.7%

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

      9. +-commutative 88.7%

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

      10. +-commutative 88.7%

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

      11. associate-+l+ 88.7%

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

   3. Simplified 88.7%

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

   4. Taylor expanded in y around inf 67.1%

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

      1. associate-*l/ 67.1%

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

      2. *-un-lft-identity 67.1%

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

      3. associate-+r+ 67.1%

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

      4. +-commutative 67.1%

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

      5. associate-+r+ 67.1%

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

   6. Applied egg-rr 67.1%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{-117}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \]

Alternative 10: 82.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.3e-102) (/ (/ y (+ y x)) (+ x 1.0)) (/ (/ x (+ y (+ x 1.0))) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.3e-102) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = (x / (y + (x + 1.0))) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.3d-102) then
        tmp = (y / (y + x)) / (x + 1.0d0)
    else
        tmp = (x / (y + (x + 1.0d0))) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.3e-102) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = (x / (y + (x + 1.0))) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.3e-102:
		tmp = (y / (y + x)) / (x + 1.0)
	else:
		tmp = (x / (y + (x + 1.0))) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.3e-102)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.3e-102)
		tmp = (y / (y + x)) / (x + 1.0);
	else
		tmp = (x / (y + (x + 1.0))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.3e-102], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.3e-102

   1. Initial program 71.4%

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

      1. associate-+r+ 71.4%

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

      2. *-commutative 71.4%

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

      3. frac-times 86.3%

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

      4. associate-*l/ 79.0%

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

      5. times-frac 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in y around 0 57.3%

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

      1. +-commutative 57.3%

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

   6. Simplified 57.3%

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

      1. un-div-inv 57.3%

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

      2. +-commutative 57.3%

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

   8. Applied egg-rr 57.3%

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

   if 3.3e-102 < y

   1. Initial program 60.9%

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

      1. associate-*l* 60.9%

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

      2. +-commutative 60.9%

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

      3. +-commutative 60.9%

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

      4. +-commutative 60.9%

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

      5. associate-*l* 60.9%

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

      6. *-commutative 60.9%

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

      7. times-frac 88.3%

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

      8. +-commutative 88.3%

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

      9. +-commutative 88.3%

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

      10. +-commutative 88.3%

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

      11. associate-+l+ 88.3%

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

   3. Simplified 88.3%

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

   4. Taylor expanded in y around inf 68.1%

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

      1. associate-*l/ 68.1%

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

      2. *-un-lft-identity 68.1%

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

      3. associate-+r+ 68.1%

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

      4. +-commutative 68.1%

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

      5. associate-+r+ 68.1%

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

   6. Applied egg-rr 68.1%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \]

Alternative 11: 48.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 0.76:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 0.76) (- (/ x y) x) (* (/ x y) (/ 1.0 y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 0.76) {
		tmp = (x / y) - x;
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.76d0) then
        tmp = (x / 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 (y <= 0.76) {
		tmp = (x / y) - x;
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 0.76:
		tmp = (x / y) - x
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 0.76)
		tmp = Float64(Float64(x / y) - x);
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.76)
		tmp = (x / 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[y, 0.76], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.76000000000000001

   1. Initial program 71.6%

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

      1. associate-*l* 71.5%

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

      2. +-commutative 71.5%

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

      3. +-commutative 71.5%

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

      4. +-commutative 71.5%

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

      5. associate-*l* 71.6%

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

      6. *-commutative 71.6%

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

      7. times-frac 87.9%

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

      8. +-commutative 87.9%

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

      9. +-commutative 87.9%

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

      10. +-commutative 87.9%

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

      11. associate-+l+ 87.9%

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

   3. Simplified 87.9%

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

   4. Taylor expanded in x around 0 44.7%

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

      1. +-commutative 44.7%

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

   6. Simplified 44.7%

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

   7. Taylor expanded in y around 0 18.9%

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

      1. neg-mul-1 18.9%

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

      2. +-commutative 18.9%

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

      3. unsub-neg 18.9%

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

   9. Simplified 18.9%

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

   if 0.76000000000000001 < y

   1. Initial program 56.4%

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

      1. associate-*l* 56.4%

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

      2. +-commutative 56.4%

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

      3. +-commutative 56.4%

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

      4. +-commutative 56.4%

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

      5. associate-*l* 56.4%

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

      6. *-commutative 56.4%

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

      7. times-frac 84.1%

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

      8. +-commutative 84.1%

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

      9. +-commutative 84.1%

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

      10. +-commutative 84.1%

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

      11. associate-+l+ 84.1%

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

   3. Simplified 84.1%

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

   4. Taylor expanded in y around inf 74.6%

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

   5. Taylor expanded in y around inf 73.2%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.76:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]

Alternative 12: 50.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+55}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 5e+55) (/ x (* y (+ y 1.0))) (* (/ x y) (/ 1.0 y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 5e+55) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5d+55) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 5e+55) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 5e+55:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 5e+55)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5e+55)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 5e+55], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.00000000000000046e55

   1. Initial program 72.2%

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

      1. associate-*l* 72.2%

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

      2. +-commutative 72.2%

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

      3. +-commutative 72.2%

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

      4. +-commutative 72.2%

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

      5. associate-*l* 72.2%

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

      6. *-commutative 72.2%

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

      7. times-frac 88.4%

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

      8. +-commutative 88.4%

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

      9. +-commutative 88.4%

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

      10. +-commutative 88.4%

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

      11. associate-+l+ 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in x around 0 46.0%

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

      1. +-commutative 46.0%

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

   6. Simplified 46.0%

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

   if 5.00000000000000046e55 < y

   1. Initial program 49.8%

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

      1. associate-*l* 49.8%

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

      2. +-commutative 49.8%

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

      3. +-commutative 49.8%

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

      4. +-commutative 49.8%

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

      5. associate-*l* 49.8%

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

      6. *-commutative 49.8%

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

      7. times-frac 80.9%

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

      8. +-commutative 80.9%

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

      9. +-commutative 80.9%

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

      10. +-commutative 80.9%

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

      11. associate-+l+ 80.9%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in y around inf 77.0%

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

   5. Taylor expanded in y around inf 76.6%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+55}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]

Alternative 13: 80.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.35 \cdot 10^{-102}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.35e-102) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.35e-102) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.35d-102) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.35e-102) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.35e-102:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.35e-102)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.35e-102)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.35e-102], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.35e-102

   1. Initial program 71.4%

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

      1. associate-*l* 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. +-commutative 71.3%

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

      3. +-commutative 71.3%

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

      4. +-commutative 71.3%

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

      5. associate-*l* 71.4%

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

      6. *-commutative 71.4%

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

      7. times-frac 86.3%

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

      8. +-commutative 86.3%

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

      9. +-commutative 86.3%

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

      10. +-commutative 86.3%

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

      11. associate-+l+ 86.3%

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

   3. Simplified 86.3%

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

   4. Taylor expanded in y around 0 56.2%

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

      1. +-commutative 56.2%

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

   6. Simplified 56.2%

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

   if 3.35e-102 < y

   1. Initial program 60.9%

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

      1. associate-+r+ 60.9%

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

      2. *-commutative 60.9%

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

      3. frac-times 88.3%

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

      4. associate-*l/ 83.3%

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

      5. times-frac 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. div-inv 99.6%

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

      2. +-commutative 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in x around 0 65.8%

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

      1. associate-/r* 67.7%

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

      2. +-commutative 67.7%

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

   8. Simplified 67.7%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.35 \cdot 10^{-102}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 14: 27.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.5e-6) (/ 1.0 (+ x 1.0)) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.5e-6) {
		tmp = 1.0 / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.5d-6)) then
        tmp = 1.0d0 / (x + 1.0d0)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -3.5e-6) {
		tmp = 1.0 / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.5e-6:
		tmp = 1.0 / (x + 1.0)
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.5e-6)
		tmp = Float64(1.0 / Float64(x + 1.0));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.5e-6)
		tmp = 1.0 / (x + 1.0);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -3.5e-6], N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.49999999999999995e-6

   1. Initial program 65.2%

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

      1. associate-+r+ 65.2%

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

      2. *-commutative 65.2%

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

      3. frac-times 88.5%

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

      4. associate-*l/ 88.5%

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

      5. times-frac 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in y around 0 77.1%

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

      1. +-commutative 77.1%

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

   6. Simplified 77.1%

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

   7. Taylor expanded in y around inf 6.2%

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

      1. +-commutative 6.2%

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

   9. Simplified 6.2%

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

   if -3.49999999999999995e-6 < x

   1. Initial program 68.9%

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

      1. associate-*l* 68.9%

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

      2. +-commutative 68.9%

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

      3. +-commutative 68.9%

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

      4. +-commutative 68.9%

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

      5. associate-*l* 68.9%

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

      6. *-commutative 68.9%

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

      7. times-frac 86.4%

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

      8. +-commutative 86.4%

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

      9. +-commutative 86.4%

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

      10. +-commutative 86.4%

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

      11. associate-+l+ 86.4%

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

   3. Simplified 86.4%

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

   4. Taylor expanded in x around 0 61.8%

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

      1. +-commutative 61.8%

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

   6. Simplified 61.8%

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

   7. Taylor expanded in y around 0 36.8%

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

4. Final simplification 28.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 15: 4.1% accurate, 5.7× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return 1.0 / 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 = 1.0d0 / y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

   1. associate-*l* 67.9%

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

   2. +-commutative 67.9%

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

   3. +-commutative 67.9%

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

   4. +-commutative 67.9%

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

   5. associate-*l* 67.9%

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

   6. *-commutative 67.9%

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

   7. times-frac 87.0%

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

   8. +-commutative 87.0%

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

   9. +-commutative 87.0%

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

   10. +-commutative 87.0%

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

   11. associate-+l+ 87.0%

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

3. Simplified 87.0%

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

4. Taylor expanded in y around inf 52.0%

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

5. Taylor expanded in x around inf 4.1%

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

6. Final simplification 4.1%

   \[\leadsto \frac{1}{y} \]

Alternative 16: 25.8% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

   1. associate-*l* 67.9%

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

   2. +-commutative 67.9%

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

   3. +-commutative 67.9%

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

   4. +-commutative 67.9%

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

   5. associate-*l* 67.9%

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

   6. *-commutative 67.9%

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

   7. times-frac 87.0%

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

   8. +-commutative 87.0%

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

   9. +-commutative 87.0%

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

   10. +-commutative 87.0%

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

   11. associate-+l+ 87.0%

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

3. Simplified 87.0%

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

4. Taylor expanded in x around 0 51.2%

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

   1. +-commutative 51.2%

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

6. Simplified 51.2%

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

7. Taylor expanded in y around 0 27.5%

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

8. Final simplification 27.5%

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

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 2023332 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

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