Octave 3.8, jcobi/3

Average Error: 3.5 → 0.1

Time: 22.5s

Precision: binary64

Cost: 1600

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

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

Math

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

↓

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

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 (+ alpha (+ 2.0 beta))))
   (/
    (/ (- -1.0 alpha) t_0)
    (/ t_0 (/ (- -1.0 beta) (+ alpha (+ beta 3.0)))))))

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 = alpha + (2.0 + beta);
	return ((-1.0 - alpha) / t_0) / (t_0 / ((-1.0 - beta) / (alpha + (beta + 3.0))));
}

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 = alpha + (2.0d0 + beta)
    code = (((-1.0d0) - alpha) / t_0) / (t_0 / (((-1.0d0) - beta) / (alpha + (beta + 3.0d0))))
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 = alpha + (2.0 + beta);
	return ((-1.0 - alpha) / t_0) / (t_0 / ((-1.0 - beta) / (alpha + (beta + 3.0))));
}

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 = alpha + (2.0 + beta)
	return ((-1.0 - alpha) / t_0) / (t_0 / ((-1.0 - beta) / (alpha + (beta + 3.0))))

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(alpha + Float64(2.0 + beta))
	return Float64(Float64(Float64(-1.0 - alpha) / t_0) / Float64(t_0 / Float64(Float64(-1.0 - beta) / Float64(alpha + Float64(beta + 3.0)))))
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 = alpha + (2.0 + beta);
	tmp = ((-1.0 - alpha) / t_0) / (t_0 / ((-1.0 - beta) / (alpha + (beta + 3.0))));
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[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(-1.0 - alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 / N[(N[(-1.0 - beta), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 3.5

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

2. Simplified 0.1

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

   [Start] 3.5	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

3. Applied egg-rr 0.1

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

4. Simplified 0.1

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

   [Start] 0.1	\[ \frac{\frac{1}{2 + \left(\alpha + \beta\right)} \cdot \frac{\alpha + 1}{\frac{-2 - \left(\alpha + \beta\right)}{-1 - \beta}}}{\alpha + \left(\beta + 3\right)} \]
   +-commutative [=>] 0.1	\[ \frac{\frac{1}{2 + \left(\alpha + \beta\right)} \cdot \frac{\color{blue}{1 + \alpha}}{\frac{-2 - \left(\alpha + \beta\right)}{-1 - \beta}}}{\alpha + \left(\beta + 3\right)} \]
   associate-*r/ [=>] 0.1	\[ \frac{\color{blue}{\frac{\frac{1}{2 + \left(\alpha + \beta\right)} \cdot \left(1 + \alpha\right)}{\frac{-2 - \left(\alpha + \beta\right)}{-1 - \beta}}}}{\alpha + \left(\beta + 3\right)} \]
   associate-+r+ [=>] 0.1	\[ \frac{\frac{\frac{1}{\color{blue}{\left(2 + \alpha\right) + \beta}} \cdot \left(1 + \alpha\right)}{\frac{-2 - \left(\alpha + \beta\right)}{-1 - \beta}}}{\alpha + \left(\beta + 3\right)} \]
   +-commutative [=>] 0.1	\[ \frac{\frac{\frac{1}{\left(2 + \alpha\right) + \beta} \cdot \left(1 + \alpha\right)}{\frac{-2 - \color{blue}{\left(\beta + \alpha\right)}}{-1 - \beta}}}{\alpha + \left(\beta + 3\right)} \]

5. Applied egg-rr 0.2

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

6. Simplified 0.1

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

   [Start] 0.2	\[ \frac{1 + \alpha}{\left(\beta + 2\right) + \alpha} \cdot \frac{1}{\frac{\left(-2 - \beta\right) - \alpha}{-1 - \beta} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   *-commutative [=>] 0.2	\[ \color{blue}{\frac{1}{\frac{\left(-2 - \beta\right) - \alpha}{-1 - \beta} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \cdot \frac{1 + \alpha}{\left(\beta + 2\right) + \alpha}} \]
   associate-*l/ [=>] 0.1	\[ \color{blue}{\frac{1 \cdot \frac{1 + \alpha}{\left(\beta + 2\right) + \alpha}}{\frac{\left(-2 - \beta\right) - \alpha}{-1 - \beta} \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   associate-*r/ [=>] 0.1	\[ \frac{\color{blue}{\frac{1 \cdot \left(1 + \alpha\right)}{\left(\beta + 2\right) + \alpha}}}{\frac{\left(-2 - \beta\right) - \alpha}{-1 - \beta} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   associate-*l/ [<=] 0.1	\[ \frac{\color{blue}{\frac{1}{\left(\beta + 2\right) + \alpha} \cdot \left(1 + \alpha\right)}}{\frac{\left(-2 - \beta\right) - \alpha}{-1 - \beta} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   associate-/r/ [<=] 0.1	\[ \frac{\color{blue}{\frac{1}{\frac{\left(\beta + 2\right) + \alpha}{1 + \alpha}}}}{\frac{\left(-2 - \beta\right) - \alpha}{-1 - \beta} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   associate-+l+ [=>] 0.1	\[ \frac{\frac{1}{\frac{\color{blue}{\beta + \left(2 + \alpha\right)}}{1 + \alpha}}}{\frac{\left(-2 - \beta\right) - \alpha}{-1 - \beta} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   associate--l- [=>] 0.1	\[ \frac{\frac{1}{\frac{\beta + \left(2 + \alpha\right)}{1 + \alpha}}}{\frac{\color{blue}{-2 - \left(\beta + \alpha\right)}}{-1 - \beta} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   +-commutative [=>] 0.1	\[ \frac{\frac{1}{\frac{\beta + \left(2 + \alpha\right)}{1 + \alpha}}}{\frac{-2 - \left(\beta + \alpha\right)}{-1 - \beta} \cdot \color{blue}{\left(\left(\beta + 3\right) + \alpha\right)}} \]
   associate-+l+ [=>] 0.1	\[ \frac{\frac{1}{\frac{\beta + \left(2 + \alpha\right)}{1 + \alpha}}}{\frac{-2 - \left(\beta + \alpha\right)}{-1 - \beta} \cdot \color{blue}{\left(\beta + \left(3 + \alpha\right)\right)}} \]

7. Applied egg-rr 0.2

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

8. Simplified 0.1

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

   [Start] 0.2	\[ \left(-\frac{1 + \alpha}{\beta + \left(2 + \alpha\right)}\right) \cdot \frac{1}{\frac{-2 - \left(\beta + \alpha\right)}{-1 - \beta} \cdot \left(-\left(\beta + \left(\alpha + 3\right)\right)\right)} \]
   distribute-lft-neg-out [=>] 0.2	\[ \color{blue}{-\frac{1 + \alpha}{\beta + \left(2 + \alpha\right)} \cdot \frac{1}{\frac{-2 - \left(\beta + \alpha\right)}{-1 - \beta} \cdot \left(-\left(\beta + \left(\alpha + 3\right)\right)\right)}} \]
   associate-*r/ [=>] 0.1	\[ -\color{blue}{\frac{\frac{1 + \alpha}{\beta + \left(2 + \alpha\right)} \cdot 1}{\frac{-2 - \left(\beta + \alpha\right)}{-1 - \beta} \cdot \left(-\left(\beta + \left(\alpha + 3\right)\right)\right)}} \]
   *-rgt-identity [=>] 0.1	\[ -\frac{\color{blue}{\frac{1 + \alpha}{\beta + \left(2 + \alpha\right)}}}{\frac{-2 - \left(\beta + \alpha\right)}{-1 - \beta} \cdot \left(-\left(\beta + \left(\alpha + 3\right)\right)\right)} \]
   distribute-neg-frac [=>] 0.1	\[ \color{blue}{\frac{-\frac{1 + \alpha}{\beta + \left(2 + \alpha\right)}}{\frac{-2 - \left(\beta + \alpha\right)}{-1 - \beta} \cdot \left(-\left(\beta + \left(\alpha + 3\right)\right)\right)}} \]
   distribute-neg-frac [=>] 0.1	\[ \frac{\color{blue}{\frac{-\left(1 + \alpha\right)}{\beta + \left(2 + \alpha\right)}}}{\frac{-2 - \left(\beta + \alpha\right)}{-1 - \beta} \cdot \left(-\left(\beta + \left(\alpha + 3\right)\right)\right)} \]
   mul-1-neg [<=] 0.1	\[ \frac{\frac{\color{blue}{-1 \cdot \left(1 + \alpha\right)}}{\beta + \left(2 + \alpha\right)}}{\frac{-2 - \left(\beta + \alpha\right)}{-1 - \beta} \cdot \left(-\left(\beta + \left(\alpha + 3\right)\right)\right)} \]
   distribute-lft-in [=>] 0.1	\[ \frac{\frac{\color{blue}{-1 \cdot 1 + -1 \cdot \alpha}}{\beta + \left(2 + \alpha\right)}}{\frac{-2 - \left(\beta + \alpha\right)}{-1 - \beta} \cdot \left(-\left(\beta + \left(\alpha + 3\right)\right)\right)} \]
   metadata-eval [=>] 0.1	\[ \frac{\frac{\color{blue}{-1} + -1 \cdot \alpha}{\beta + \left(2 + \alpha\right)}}{\frac{-2 - \left(\beta + \alpha\right)}{-1 - \beta} \cdot \left(-\left(\beta + \left(\alpha + 3\right)\right)\right)} \]
   neg-mul-1 [<=] 0.1	\[ \frac{\frac{-1 + \color{blue}{\left(-\alpha\right)}}{\beta + \left(2 + \alpha\right)}}{\frac{-2 - \left(\beta + \alpha\right)}{-1 - \beta} \cdot \left(-\left(\beta + \left(\alpha + 3\right)\right)\right)} \]
   unsub-neg [=>] 0.1	\[ \frac{\frac{\color{blue}{-1 - \alpha}}{\beta + \left(2 + \alpha\right)}}{\frac{-2 - \left(\beta + \alpha\right)}{-1 - \beta} \cdot \left(-\left(\beta + \left(\alpha + 3\right)\right)\right)} \]
   associate-+r+ [=>] 0.1	\[ \frac{\frac{-1 - \alpha}{\color{blue}{\left(\beta + 2\right) + \alpha}}}{\frac{-2 - \left(\beta + \alpha\right)}{-1 - \beta} \cdot \left(-\left(\beta + \left(\alpha + 3\right)\right)\right)} \]
   +-commutative [=>] 0.1	\[ \frac{\frac{-1 - \alpha}{\color{blue}{\alpha + \left(\beta + 2\right)}}}{\frac{-2 - \left(\beta + \alpha\right)}{-1 - \beta} \cdot \left(-\left(\beta + \left(\alpha + 3\right)\right)\right)} \]
   +-commutative [=>] 0.1	\[ \frac{\frac{-1 - \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\frac{-2 - \left(\beta + \alpha\right)}{-1 - \beta} \cdot \left(-\left(\beta + \left(\alpha + 3\right)\right)\right)} \]
   associate-*l/ [=>] 2.2	\[ \frac{\frac{-1 - \alpha}{\alpha + \left(2 + \beta\right)}}{\color{blue}{\frac{\left(-2 - \left(\beta + \alpha\right)\right) \cdot \left(-\left(\beta + \left(\alpha + 3\right)\right)\right)}{-1 - \beta}}} \]

9. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.2
Cost	1732

\[\begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 10^{+45}:\\ \;\;\;\;\frac{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{3 + \left(\alpha + \beta\right)}}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-1 - \alpha\right) \cdot \frac{-1}{-3 - \left(\alpha + \beta\right)}}{-3 + \left(\alpha \cdot -2 - \beta\right)}\\ \end{array} \]

Alternative 2
Error	0.1
Cost	1600

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

Alternative 3
Error	0.1
Cost	1600

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

Alternative 4
Error	0.9
Cost	1476

\[\begin{array}{l} \mathbf{if}\;\beta \leq 10000000000:\\ \;\;\;\;\frac{\frac{1}{2 + \beta}}{\frac{-2 - \left(\alpha + \beta\right)}{-1 - \beta} \cdot \left(\beta + \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-1 - \alpha\right) \cdot \frac{-1}{-3 - \left(\alpha + \beta\right)}}{-3 + \left(\alpha \cdot -2 - \beta\right)}\\ \end{array} \]

Alternative 5
Error	1.1
Cost	1348

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

Alternative 6
Error	0.9
Cost	1348

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.6 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{\beta + 1}{2 + \beta}}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(3 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-1 - \alpha\right) \cdot \frac{-1}{-3 - \left(\alpha + \beta\right)}}{-3 + \left(\alpha \cdot -2 - \beta\right)}\\ \end{array} \]

Alternative 7
Error	1.9
Cost	1220

\[\begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 17:\\ \;\;\;\;\frac{\alpha \cdot 0.2222222222222222 + 0.3333333333333333}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 8
Error	1.4
Cost	1220

\[\begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 6.4:\\ \;\;\;\;\frac{\alpha \cdot 0.2222222222222222 + 0.3333333333333333}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 9
Error	1.1
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.96:\\ \;\;\;\;\frac{\frac{-1 - \alpha}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + 3\right) \cdot \left(-2 - \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\beta + 3\right) + \alpha \cdot 2}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 10
Error	2.3
Cost	964

\[\begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 6.6:\\ \;\;\;\;\frac{0.3333333333333333}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 11
Error	23.7
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.8 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{\left(\beta + 3\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]

Alternative 12
Error	23.7
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{\left(\beta + 3\right) \cdot \left(\beta + \left(\alpha + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 13
Error	26.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.2:\\ \;\;\;\;0.1111111111111111\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 14
Error	24.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.1:\\ \;\;\;\;0.1111111111111111\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]

Alternative 15
Error	27.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3:\\ \;\;\;\;0.1111111111111111\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 16
Error	27.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3:\\ \;\;\;\;0.1111111111111111\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \end{array} \]

Alternative 17
Error	56.5
Cost	320

\[\frac{0.3333333333333333}{\alpha + 3} \]

Alternative 18
Error	56.6
Cost	64

\[0.1111111111111111 \]

Reproduce

herbie shell --seed 2023027 
(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)))