Octave 3.8, jcobi/3

Average Error: 3.4 → 0.1

Time: 33.7s

Precision: binary64

Cost: 1600

\[\alpha > -1 \land \beta > -1\]

\[ \begin{array}{c}[alpha, beta] = \mathsf{sort}([alpha, beta])\\ \end{array} \]

Math

\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

↓

\[\begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \frac{\beta + 1}{t_0} \cdot \frac{\frac{1 + \alpha}{\beta + \left(3 + \alpha\right)}}{t_0} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (/
  (/
   (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0)))
   (+ (+ alpha beta) (* 2.0 1.0)))
  (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha))))
   (* (/ (+ beta 1.0) t_0) (/ (/ (+ 1.0 alpha) (+ beta (+ 3.0 alpha))) t_0))))

C

double code(double alpha, double beta) {
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
}

↓

double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	return ((beta + 1.0) / t_0) * (((1.0 + alpha) / (beta + (3.0 + alpha))) / t_0);
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = (((((alpha + beta) + (beta * alpha)) + 1.0d0) / ((alpha + beta) + (2.0d0 * 1.0d0))) / ((alpha + beta) + (2.0d0 * 1.0d0))) / (((alpha + beta) + (2.0d0 * 1.0d0)) + 1.0d0)
end function

↓

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = 2.0d0 + (beta + alpha)
    code = ((beta + 1.0d0) / t_0) * (((1.0d0 + alpha) / (beta + (3.0d0 + alpha))) / t_0)
end function

Java

public static double code(double alpha, double beta) {
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
}

↓

public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	return ((beta + 1.0) / t_0) * (((1.0 + alpha) / (beta + (3.0 + alpha))) / t_0);
}

Python

def code(alpha, beta):
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0)

↓

def code(alpha, beta):
	t_0 = 2.0 + (beta + alpha)
	return ((beta + 1.0) / t_0) * (((1.0 + alpha) / (beta + (3.0 + alpha))) / t_0)

Julia

function code(alpha, beta)
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) + 1.0) / Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))) / Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))) / Float64(Float64(Float64(alpha + beta) + Float64(2.0 * 1.0)) + 1.0))
end

↓

function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	return Float64(Float64(Float64(beta + 1.0) / t_0) * Float64(Float64(Float64(1.0 + alpha) / Float64(beta + Float64(3.0 + alpha))) / t_0))
end

MATLAB

function tmp = code(alpha, beta)
	tmp = (((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
end

↓

function tmp = code(alpha, beta)
	t_0 = 2.0 + (beta + alpha);
	tmp = ((beta + 1.0) / t_0) * (((1.0 + alpha) / (beta + (3.0 + alpha))) / t_0);
end

Wolfram

code[alpha_, beta_] := N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(beta + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(N[(1.0 + alpha), $MachinePrecision] / N[(beta + N[(3.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 3.4

   \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

2. Simplified 4.2

   \[\leadsto \color{blue}{\frac{\frac{\beta + \left(\left(\alpha + \alpha \cdot \beta\right) + 1\right)}{\beta + \left(\alpha + 2\right)}}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)}} \]
   Proof

   [Start] 3.4	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   rational.json-simplify-44 [=>] 3.3	\[ \color{blue}{\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}} \]
   rational.json-simplify-1 [=>] 3.3	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1} \]
   rational.json-simplify-17 [=>] 3.3	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) - -1}}}{\left(\alpha + \beta\right) + 2 \cdot 1} \]
   rational.json-simplify-50 [=>] 3.3	\[ \frac{\color{blue}{\frac{-\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{-1 - \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1} \]
   rational.json-simplify-50 [=>] 3.3	\[ \frac{\color{blue}{\frac{-\left(-\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) - -1}}}{\left(\alpha + \beta\right) + 2 \cdot 1} \]
   rational.json-simplify-17 [<=] 3.3	\[ \frac{\frac{-\left(-\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)}{\color{blue}{1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1} \]
   rational.json-simplify-1 [<=] 3.3	\[ \frac{\frac{-\left(-\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}}}{\left(\alpha + \beta\right) + 2 \cdot 1} \]
   rational.json-simplify-47 [=>] 4.2	\[ \color{blue}{\frac{-\left(-\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]
   rational.json-simplify-2 [<=] 4.2	\[ \frac{-\left(-\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}\right)}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}} \]

3. Applied egg-rr 9.7

   \[\leadsto \color{blue}{-1 \cdot \frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(-\left(\beta + \left(\alpha + 2\right)\right)\right)\right)}} \]

4. Simplified 0.5

   \[\leadsto \color{blue}{\frac{1}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\left(\alpha + \left(2 + \beta\right)\right) \cdot \frac{\frac{3 + \left(\beta + \alpha\right)}{1 + \beta}}{1 + \alpha}\right)}} \]
   Proof

   [Start] 9.7	\[ -1 \cdot \frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(-\left(\beta + \left(\alpha + 2\right)\right)\right)\right)} \]
   rational.json-simplify-46 [=>] 4.2	\[ -1 \cdot \color{blue}{\frac{\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\beta + \left(\alpha + 2\right)}}{\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(-\left(\beta + \left(\alpha + 2\right)\right)\right)}} \]
   rational.json-simplify-46 [=>] 3.3	\[ -1 \cdot \color{blue}{\frac{\frac{\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\beta + \left(\alpha + 2\right)}}{\beta + \left(\alpha + 3\right)}}{-\left(\beta + \left(\alpha + 2\right)\right)}} \]
   rational.json-simplify-46 [<=] 4.2	\[ -1 \cdot \frac{\color{blue}{\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}}{-\left(\beta + \left(\alpha + 2\right)\right)} \]
   rational.json-simplify-2 [=>] 4.2	\[ \color{blue}{\frac{\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}{-\left(\beta + \left(\alpha + 2\right)\right)} \cdot -1} \]
   rational.json-simplify-8 [<=] 4.2	\[ \color{blue}{-\frac{\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}{-\left(\beta + \left(\alpha + 2\right)\right)}} \]
   rational.json-simplify-10 [=>] 4.2	\[ \color{blue}{\frac{\frac{\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}}{-\left(\beta + \left(\alpha + 2\right)\right)}}{-1}} \]

5. Applied egg-rr 0.1

   \[\leadsto \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \beta}{\alpha + \left(2 + \beta\right)}}{\alpha + \left(2 + \beta\right)} + 0} \]

6. Simplified 0.1

   \[\leadsto \color{blue}{\frac{\beta + 1}{2 + \left(\beta + \alpha\right)} \cdot \frac{\frac{1 + \alpha}{\beta + \left(3 + \alpha\right)}}{2 + \left(\beta + \alpha\right)}} \]
   Proof

   [Start] 0.1	\[ \frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \beta}{\alpha + \left(2 + \beta\right)}}{\alpha + \left(2 + \beta\right)} + 0 \]
   rational.json-simplify-4 [=>] 0.1	\[ \color{blue}{\frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \beta}{\alpha + \left(2 + \beta\right)}}{\alpha + \left(2 + \beta\right)}} \]
   rational.json-simplify-49 [=>] 0.1	\[ \color{blue}{\frac{1 + \beta}{\alpha + \left(2 + \beta\right)} \cdot \frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(2 + \beta\right)}} \]
   rational.json-simplify-1 [<=] 0.1	\[ \frac{\color{blue}{\beta + 1}}{\alpha + \left(2 + \beta\right)} \cdot \frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(2 + \beta\right)} \]
   rational.json-simplify-41 [=>] 0.1	\[ \frac{\beta + 1}{\color{blue}{2 + \left(\beta + \alpha\right)}} \cdot \frac{\frac{1 + \alpha}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(2 + \beta\right)} \]
   rational.json-simplify-41 [=>] 0.1	\[ \frac{\beta + 1}{2 + \left(\beta + \alpha\right)} \cdot \frac{\frac{1 + \alpha}{\color{blue}{\beta + \left(3 + \alpha\right)}}}{\alpha + \left(2 + \beta\right)} \]
   rational.json-simplify-41 [=>] 0.1	\[ \frac{\beta + 1}{2 + \left(\beta + \alpha\right)} \cdot \frac{\frac{1 + \alpha}{\beta + \left(3 + \alpha\right)}}{\color{blue}{2 + \left(\beta + \alpha\right)}} \]

7. Final simplification 0.1

   \[\leadsto \frac{\beta + 1}{2 + \left(\beta + \alpha\right)} \cdot \frac{\frac{1 + \alpha}{\beta + \left(3 + \alpha\right)}}{2 + \left(\beta + \alpha\right)} \]

Alternatives

Alternative 1
Error	0.3
Cost	1668

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\beta \leq 5400000000000:\\ \;\;\;\;\frac{\frac{\beta + \left(\alpha + 1\right)}{t_0}}{t_0 \cdot \left(\left(\alpha + \beta\right) + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(-\frac{1 + \alpha}{\beta}\right)\right) \cdot \frac{\frac{1 + \alpha}{\beta + \left(3 + \alpha\right)}}{2 + \left(\beta + \alpha\right)}\\ \end{array} \]

Alternative 2
Error	0.4
Cost	1604

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ t_1 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\beta \leq 4.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\beta + \left(\alpha + 1\right)}{t_1}}{t_1 \cdot \left(\left(\alpha + \beta\right) + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{t_0}}{t_0 + 1}\\ \end{array} \]

Alternative 3
Error	0.9
Cost	1476

\[\begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 1.05 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \frac{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}{\beta + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{t_0} \cdot \frac{\frac{1 + \alpha}{\beta}}{t_0}\\ \end{array} \]

Alternative 4
Error	0.9
Cost	1348

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \frac{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}{\beta + 1}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{-1 - \alpha}{\alpha + \left(\beta + 3\right)}}{\beta}\\ \end{array} \]

Alternative 5
Error	0.9
Cost	1348

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 7.8 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \frac{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}{\beta + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{t_0}}{t_0 + 1}\\ \end{array} \]

Alternative 6
Error	1.0
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.4 \cdot 10^{+24}:\\ \;\;\;\;\frac{\beta + 1}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\beta + 3\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{-1 - \alpha}{\alpha + \left(\beta + 3\right)}}{\beta}\\ \end{array} \]

Alternative 7
Error	1.0
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{-1 - \alpha}{\alpha + \left(\beta + 3\right)}}{\beta}\\ \end{array} \]

Alternative 8
Error	2.1
Cost	900

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.4:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \alpha}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{-1 - \alpha}{\alpha + \left(\beta + 3\right)}}{\beta}\\ \end{array} \]

Alternative 9
Error	1.8
Cost	900

\[\begin{array}{l} \mathbf{if}\;\beta \leq 10:\\ \;\;\;\;\frac{1}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(6 + \left(-\alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{-1 - \alpha}{\alpha + \left(\beta + 3\right)}}{\beta}\\ \end{array} \]

Alternative 10
Error	2.1
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 11
Error	4.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \alpha}\\ \mathbf{elif}\;\beta \leq 1.05 \cdot 10^{+160}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 12
Error	4.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.2:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \alpha}\\ \mathbf{elif}\;\beta \leq 4.2 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 13
Error	2.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 14
Error	34.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.16666666666666666}{\beta}\\ \end{array} \]

Alternative 15
Error	15.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.6 \cdot 10^{+47}:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 16
Error	34.3
Cost	324

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.16666666666666666}{\beta}\\ \end{array} \]

Alternative 17
Error	34.3
Cost	320

\[\frac{0.16666666666666666}{\beta + 2} \]

Alternative 18
Error	35.4
Cost	64

\[0.08333333333333333 \]

Reproduce

herbie shell --seed 2023073 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))