Octave 3.8, jcobi/3

Average Error: 3.7 → 0.1

Time: 1.6min

Precision: binary64

Cost: 1600

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

Math

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

↓

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

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 (+ 2.0 (+ beta alpha))))
   (/
    (* (/ (- -1.0 beta) t_0) (/ (- -1.0 alpha) t_0))
    (+ beta (+ alpha 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 = 2.0 + (beta + alpha);
	return (((-1.0 - beta) / t_0) * ((-1.0 - alpha) / t_0)) / (beta + (alpha + 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 = 2.0d0 + (beta + alpha)
    code = ((((-1.0d0) - beta) / t_0) * (((-1.0d0) - alpha) / t_0)) / (beta + (alpha + 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 = 2.0 + (beta + alpha);
	return (((-1.0 - beta) / t_0) * ((-1.0 - alpha) / t_0)) / (beta + (alpha + 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 = 2.0 + (beta + alpha)
	return (((-1.0 - beta) / t_0) * ((-1.0 - alpha) / t_0)) / (beta + (alpha + 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(2.0 + Float64(beta + alpha))
	return Float64(Float64(Float64(Float64(-1.0 - beta) / t_0) * Float64(Float64(-1.0 - alpha) / t_0)) / Float64(beta + Float64(alpha + 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 = 2.0 + (beta + alpha);
	tmp = (((-1.0 - beta) / t_0) * ((-1.0 - alpha) / t_0)) / (beta + (alpha + 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[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(-1.0 - beta), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(-1.0 - alpha), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 3.7

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

2. Simplified 4.5

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

   [Start] 3.7	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   rational_best-simplify-53 [=>] 4.5	\[ \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)}} \]
   rational_best-simplify-1 [=>] 4.5	\[ \frac{\frac{\left(\left(\alpha + \beta\right) + \color{blue}{\alpha \cdot \beta}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]
   metadata-eval [=>] 4.5	\[ \frac{\frac{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1}{\left(\alpha + \beta\right) + \color{blue}{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]
   metadata-eval [=>] 4.5	\[ \frac{\frac{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + \color{blue}{2}\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right)} \]
   rational_best-simplify-3 [=>] 4.5	\[ \frac{\frac{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \color{blue}{\left(1 + \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]
   metadata-eval [=>] 4.5	\[ \frac{\frac{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1}{\left(\alpha + \beta\right) + 2}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(1 + \left(\left(\alpha + \beta\right) + \color{blue}{2}\right)\right)} \]

3. Applied egg-rr 4.5

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

4. Applied egg-rr 4.6

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

5. Simplified 0.2

   \[\leadsto \color{blue}{\left(\frac{\beta + 1}{2 + \left(\beta + \alpha\right)} \cdot \frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}\right) \cdot \left(2 \cdot \frac{0.5}{\beta + \left(3 + \alpha\right)}\right)} \]
   Proof

   [Start] 4.6	\[ \left(2 \cdot \frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\right) \cdot \frac{1}{2 \cdot \left(\left(\alpha + \beta\right) + 3\right)} \]
   rational_best-simplify-1 [=>] 4.6	\[ \color{blue}{\frac{1}{2 \cdot \left(\left(\alpha + \beta\right) + 3\right)} \cdot \left(2 \cdot \frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}\right)} \]
   rational_best-simplify-50 [=>] 4.6	\[ \color{blue}{\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \cdot \left(2 \cdot \frac{1}{2 \cdot \left(\left(\alpha + \beta\right) + 3\right)}\right)} \]
   rational_best-simplify-3 [<=] 4.6	\[ \frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \cdot \left(2 \cdot \frac{1}{2 \cdot \left(\left(\alpha + \beta\right) + 3\right)}\right) \]
   rational_best-simplify-78 [=>] 0.2	\[ \color{blue}{\left(\frac{\beta + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right)} \cdot \left(2 \cdot \frac{1}{2 \cdot \left(\left(\alpha + \beta\right) + 3\right)}\right) \]
   rational_best-simplify-47 [=>] 0.2	\[ \left(\frac{\beta + 1}{\color{blue}{2 + \left(\beta + \alpha\right)}} \cdot \frac{1 + \alpha}{\alpha + \left(\beta + 2\right)}\right) \cdot \left(2 \cdot \frac{1}{2 \cdot \left(\left(\alpha + \beta\right) + 3\right)}\right) \]
   rational_best-simplify-47 [=>] 0.2	\[ \left(\frac{\beta + 1}{2 + \left(\beta + \alpha\right)} \cdot \frac{1 + \alpha}{\color{blue}{2 + \left(\beta + \alpha\right)}}\right) \cdot \left(2 \cdot \frac{1}{2 \cdot \left(\left(\alpha + \beta\right) + 3\right)}\right) \]
   rational_best-simplify-54 [=>] 0.2	\[ \left(\frac{\beta + 1}{2 + \left(\beta + \alpha\right)} \cdot \frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}\right) \cdot \left(2 \cdot \color{blue}{\frac{\frac{1}{2}}{\left(\alpha + \beta\right) + 3}}\right) \]
   metadata-eval [=>] 0.2	\[ \left(\frac{\beta + 1}{2 + \left(\beta + \alpha\right)} \cdot \frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}\right) \cdot \left(2 \cdot \frac{\color{blue}{0.5}}{\left(\alpha + \beta\right) + 3}\right) \]
   rational_best-simplify-3 [=>] 0.2	\[ \left(\frac{\beta + 1}{2 + \left(\beta + \alpha\right)} \cdot \frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}\right) \cdot \left(2 \cdot \frac{0.5}{\color{blue}{3 + \left(\alpha + \beta\right)}}\right) \]
   rational_best-simplify-47 [=>] 0.2	\[ \left(\frac{\beta + 1}{2 + \left(\beta + \alpha\right)} \cdot \frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}\right) \cdot \left(2 \cdot \frac{0.5}{\color{blue}{\beta + \left(\alpha + 3\right)}}\right) \]
   rational_best-simplify-3 [<=] 0.2	\[ \left(\frac{\beta + 1}{2 + \left(\beta + \alpha\right)} \cdot \frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}\right) \cdot \left(2 \cdot \frac{0.5}{\beta + \color{blue}{\left(3 + \alpha\right)}}\right) \]

6. Applied egg-rr 0.1

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

7. Final simplification 0.1

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

Alternatives

Alternative 1
Error	19.0
Cost	1736

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 3\right)\\ \mathbf{if}\;\alpha \leq -2.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{\frac{\alpha + 1}{\beta + \left(2 + \alpha\right)}}{2 + \alpha}}{\beta + \left(3 + \alpha\right)}\\ \mathbf{elif}\;\alpha \leq 3.1 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{1}{t_0}}{\left(-2 - \beta\right) - \alpha} \cdot \left(-\frac{\beta + 1}{\beta + 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1 + \alpha}{\beta} + -1\right) \cdot \frac{-1 - \alpha}{2 + \left(\beta + \alpha\right)}}{t_0}\\ \end{array} \]

Alternative 2
Error	5.8
Cost	1732

\[\begin{array}{l} t_0 := \beta + \left(2 + \alpha\right)\\ \mathbf{if}\;\beta \leq 40000000000:\\ \;\;\;\;\frac{\left(\beta + 1\right) \cdot \left(1 + \alpha\right)}{\left(\beta + \left(3 + \alpha\right)\right) \cdot \left(t_0 \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1 + \alpha}{\beta} + -1\right) \cdot \frac{-1 - \alpha}{2 + \left(\beta + \alpha\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 3
Error	17.8
Cost	1672

\[\begin{array}{l} t_0 := \beta + \left(3 + \alpha\right)\\ \mathbf{if}\;\alpha \leq -3 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{\frac{\alpha + 1}{\beta + \left(2 + \alpha\right)}}{2 + \alpha}}{t_0}\\ \mathbf{elif}\;\alpha \leq 3.1 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{1}{\beta + \left(\alpha + 3\right)}}{\left(-2 - \beta\right) - \alpha} \cdot \left(-\frac{\beta + 1}{\beta + 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 - \alpha}{\left(-2 - \alpha\right) - \beta}}{t_0}\\ \end{array} \]

Alternative 4
Error	17.9
Cost	1608

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ t_1 := \beta + \left(3 + \alpha\right)\\ \mathbf{if}\;\alpha \leq -2.9 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{\frac{\alpha + 1}{\beta + \left(2 + \alpha\right)}}{2 + \alpha}}{t_1}\\ \mathbf{elif}\;\alpha \leq 3.1 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{t_0 \cdot \left(1 + t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 - \alpha}{\left(-2 - \alpha\right) - \beta}}{t_1}\\ \end{array} \]

Alternative 5
Error	17.9
Cost	1608

\[\begin{array}{l} t_0 := \beta + \left(3 + \alpha\right)\\ \mathbf{if}\;\alpha \leq -2.9 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{\frac{\alpha + 1}{\beta + \left(2 + \alpha\right)}}{2 + \alpha}}{t_0}\\ \mathbf{elif}\;\alpha \leq 3.1 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{\beta + 1}{\beta + 2}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\left(\alpha + \left(\beta + 1\right)\right) - -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 - \alpha}{\left(-2 - \alpha\right) - \beta}}{t_0}\\ \end{array} \]

Alternative 6
Error	6.0
Cost	1604

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 1200000000:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) + 1}{t_0 \cdot \left(t_0 \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1 + \alpha}{\beta} + -1\right) \cdot \frac{-1 - \alpha}{2 + \left(\beta + \alpha\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 7
Error	4.5
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.3:\\ \;\;\;\;\frac{\frac{\alpha - -1}{2 + \alpha}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 - \alpha}{\left(-2 - \alpha\right) - \beta}}{\beta + \left(3 + \alpha\right)}\\ \end{array} \]

Alternative 8
Error	4.2
Cost	1220

\[\begin{array}{l} t_0 := \beta + \left(3 + \alpha\right)\\ \mathbf{if}\;\beta \leq 2.55:\\ \;\;\;\;\frac{\frac{\frac{\alpha + 1}{2 + \alpha}}{2 + \alpha}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 - \alpha}{\left(-2 - \alpha\right) - \beta}}{t_0}\\ \end{array} \]

Alternative 9
Error	17.2
Cost	1092

\[\begin{array}{l} t_0 := \beta + \left(3 + \alpha\right)\\ \mathbf{if}\;\beta \leq 1.68:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 - \alpha}{\left(-2 - \alpha\right) - \beta}}{t_0}\\ \end{array} \]

Alternative 10
Error	4.5
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.55:\\ \;\;\;\;\frac{\frac{\alpha - -1}{2 + \alpha}}{\left(2 + \alpha\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 - \alpha}{\left(-2 - \alpha\right) - \beta}}{\beta + \left(3 + \alpha\right)}\\ \end{array} \]

Alternative 11
Error	18.5
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 3}}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 2} \cdot \left(\alpha + 1\right)\\ \end{array} \]

Alternative 12
Error	17.8
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{\beta + \left(3 + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 2} \cdot \left(\alpha + 1\right)\\ \end{array} \]

Alternative 13
Error	17.4
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.8:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{\beta + \left(3 + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 14
Error	41.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 0.22:\\ \;\;\;\;\frac{1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta + 2}\\ \end{array} \]

Alternative 15
Error	18.6
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.5:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha - -1}{\beta \cdot \left(\beta + 2\right)}\\ \end{array} \]

Alternative 16
Error	18.6
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 3}}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha - -1}{\beta \cdot \left(\beta + 2\right)}\\ \end{array} \]

Alternative 17
Error	41.8
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.42:\\ \;\;\;\;\frac{1}{\beta + \left(3 + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 2\right)}\\ \end{array} \]

Alternative 18
Error	41.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.45:\\ \;\;\;\;\frac{1}{\beta + \left(3 + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 2}\\ \end{array} \]

Alternative 19
Error	46.4
Cost	448

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

Alternative 20
Error	61.3
Cost	192

\[\frac{1}{\alpha} \]

Alternative 21
Error	61.2
Cost	192

\[\frac{1}{\beta} \]

Reproduce

herbie shell --seed 2023099 
(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)))