Octave 3.8, jcobi/3

Average Error: 4.0 → 0.1

Time: 31.0s

Precision: binary64

Cost: 2368

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

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

Math

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

↓

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

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

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

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

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

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

Derivation

1. Initial program 4.0

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

2. Applied egg-rr 0.1

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

3. Simplified 0.1

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

   [Start] 0.1	\[ \frac{\frac{\frac{\alpha + \beta}{\alpha + \left(\beta + 2\right)} + \left(\alpha \cdot \frac{\beta}{\alpha + \left(\beta + 2\right)} - \frac{-1}{\alpha + \left(\beta + 2\right)}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   rational.json-simplify-21 [<=] 0.1	\[ \frac{\frac{\color{blue}{\alpha \cdot \frac{\beta}{\alpha + \left(\beta + 2\right)} + \left(\frac{\alpha + \beta}{\alpha + \left(\beta + 2\right)} - \frac{-1}{\alpha + \left(\beta + 2\right)}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   rational.json-simplify-11 [=>] 0.1	\[ \frac{\frac{\alpha \cdot \frac{\beta}{\color{blue}{\beta + \left(\alpha + 2\right)}} + \left(\frac{\alpha + \beta}{\alpha + \left(\beta + 2\right)} - \frac{-1}{\alpha + \left(\beta + 2\right)}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   rational.json-simplify-41 [<=] 0.1	\[ \frac{\frac{\alpha \cdot \frac{\beta}{\beta + \color{blue}{\left(2 + \alpha\right)}} + \left(\frac{\alpha + \beta}{\alpha + \left(\beta + 2\right)} - \frac{-1}{\alpha + \left(\beta + 2\right)}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   rational.json-simplify-32 [=>] 0.1	\[ \frac{\frac{\alpha \cdot \frac{\beta}{\beta + \left(2 + \alpha\right)} + \color{blue}{\frac{\left(\alpha + \beta\right) - -1}{\alpha + \left(\beta + 2\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   rational.json-simplify-41 [=>] 0.1	\[ \frac{\frac{\alpha \cdot \frac{\beta}{\beta + \left(2 + \alpha\right)} + \frac{\color{blue}{\left(\beta + \alpha\right)} - -1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   rational.json-simplify-11 [=>] 0.1	\[ \frac{\frac{\alpha \cdot \frac{\beta}{\beta + \left(2 + \alpha\right)} + \frac{\left(\beta + \alpha\right) - -1}{\color{blue}{\beta + \left(\alpha + 2\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   rational.json-simplify-41 [<=] 0.1	\[ \frac{\frac{\alpha \cdot \frac{\beta}{\beta + \left(2 + \alpha\right)} + \frac{\left(\beta + \alpha\right) - -1}{\beta + \color{blue}{\left(2 + \alpha\right)}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

4. Applied egg-rr 0.1

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

5. Applied egg-rr 0.1

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

6. Simplified 0.1

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

   [Start] 0.1	\[ \frac{\frac{\frac{1}{0 - \frac{\frac{-2 - \left(\beta + \alpha\right)}{\beta}}{\alpha}} + \frac{\left(\beta + \alpha\right) - -1}{\beta + \left(2 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   rational.json-simplify-67 [<=] 0.1	\[ \frac{\frac{\frac{1}{\color{blue}{-\frac{\frac{-2 - \left(\beta + \alpha\right)}{\beta}}{\alpha}}} + \frac{\left(\beta + \alpha\right) - -1}{\beta + \left(2 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   rational.json-simplify-69 [=>] 0.1	\[ \frac{\frac{\frac{1}{\color{blue}{\frac{\frac{\frac{-2 - \left(\beta + \alpha\right)}{\beta}}{\alpha}}{-1}}} + \frac{\left(\beta + \alpha\right) - -1}{\beta + \left(2 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   rational.json-simplify-15 [<=] 0.1	\[ \frac{\frac{\frac{1}{\color{blue}{\frac{\frac{-2 - \left(\beta + \alpha\right)}{\beta}}{\alpha \cdot -1}}} + \frac{\left(\beta + \alpha\right) - -1}{\beta + \left(2 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   rational.json-simplify-71 [<=] 0.1	\[ \frac{\frac{\frac{1}{\frac{\frac{-2 - \left(\beta + \alpha\right)}{\beta}}{\color{blue}{-\alpha}}} + \frac{\left(\beta + \alpha\right) - -1}{\beta + \left(2 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

7. Applied egg-rr 4.0

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

8. Simplified 0.1

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

   [Start] 4.0	\[ \frac{\frac{\frac{1}{\frac{\left(\beta + \alpha\right) + 2}{\beta \cdot \alpha} + 0} + \frac{\left(\beta + \alpha\right) - -1}{\beta + \left(2 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   rational.json-simplify-65 [=>] 4.0	\[ \frac{\frac{\frac{1}{\color{blue}{\frac{\left(\beta + \alpha\right) + 2}{\beta \cdot \alpha}}} + \frac{\left(\beta + \alpha\right) - -1}{\beta + \left(2 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   rational.json-simplify-15 [=>] 0.1	\[ \frac{\frac{\frac{1}{\color{blue}{\frac{\frac{\left(\beta + \alpha\right) + 2}{\beta}}{\alpha}}} + \frac{\left(\beta + \alpha\right) - -1}{\beta + \left(2 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   metadata-eval [<=] 0.1	\[ \frac{\frac{\frac{1}{\frac{\frac{\left(\beta + \alpha\right) + \color{blue}{\left(-2 - -4\right)}}{\beta}}{\alpha}} + \frac{\left(\beta + \alpha\right) - -1}{\beta + \left(2 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   rational.json-simplify-13 [<=] 0.1	\[ \frac{\frac{\frac{1}{\frac{\frac{\color{blue}{\left(\left(\beta + \alpha\right) + -2\right) - -4}}{\beta}}{\alpha}} + \frac{\left(\beta + \alpha\right) - -1}{\beta + \left(2 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   rational.json-simplify-41 [<=] 0.1	\[ \frac{\frac{\frac{1}{\frac{\frac{\color{blue}{\left(-2 + \left(\beta + \alpha\right)\right)} - -4}{\beta}}{\alpha}} + \frac{\left(\beta + \alpha\right) - -1}{\beta + \left(2 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   metadata-eval [<=] 0.1	\[ \frac{\frac{\frac{1}{\frac{\frac{\left(-2 + \left(\beta + \alpha\right)\right) - \color{blue}{\left(-4 - 0\right)}}{\beta}}{\alpha}} + \frac{\left(\beta + \alpha\right) - -1}{\beta + \left(2 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   rational.json-simplify-5 [<=] 0.1	\[ \frac{\frac{\frac{1}{\frac{\frac{\color{blue}{0 - \left(-4 - \left(-2 + \left(\beta + \alpha\right)\right)\right)}}{\beta}}{\alpha}} + \frac{\left(\beta + \alpha\right) - -1}{\beta + \left(2 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   metadata-eval [<=] 0.1	\[ \frac{\frac{\frac{1}{\frac{\frac{0 - \left(\color{blue}{\left(-2 + -2\right)} - \left(-2 + \left(\beta + \alpha\right)\right)\right)}{\beta}}{\alpha}} + \frac{\left(\beta + \alpha\right) - -1}{\beta + \left(2 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   rational.json-simplify-51 [<=] 0.1	\[ \frac{\frac{\frac{1}{\frac{\frac{0 - \color{blue}{\left(-2 - \left(\beta + \alpha\right)\right)}}{\beta}}{\alpha}} + \frac{\left(\beta + \alpha\right) - -1}{\beta + \left(2 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   rational.json-simplify-5 [=>] 0.1	\[ \frac{\frac{\frac{1}{\frac{\frac{\color{blue}{\left(\beta + \alpha\right) - \left(-2 - 0\right)}}{\beta}}{\alpha}} + \frac{\left(\beta + \alpha\right) - -1}{\beta + \left(2 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   metadata-eval [=>] 0.1	\[ \frac{\frac{\frac{1}{\frac{\frac{\left(\beta + \alpha\right) - \color{blue}{-2}}{\beta}}{\alpha}} + \frac{\left(\beta + \alpha\right) - -1}{\beta + \left(2 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   rational.json-simplify-13 [=>] 0.1	\[ \frac{\frac{\frac{1}{\frac{\frac{\color{blue}{\beta + \left(\alpha - -2\right)}}{\beta}}{\alpha}} + \frac{\left(\beta + \alpha\right) - -1}{\beta + \left(2 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

9. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.3
Cost	2244

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 10^{+139}:\\ \;\;\;\;\frac{\frac{\frac{1}{\frac{1}{\alpha + \beta \cdot \left(\alpha + 1\right)}} + 1}{t_0 \cdot t_0}}{1 + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(2 + \alpha\right)}}{\beta + \left(3 + \alpha\right)}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	2240

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

Alternative 3
Error	0.1
Cost	2240

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

Alternative 4
Error	0.1
Cost	2240

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ t_1 := \beta + \left(2 + \alpha\right)\\ \frac{\frac{\alpha \cdot \frac{\beta}{t_1} + \frac{\left(\beta + \alpha\right) - -1}{t_1}}{t_0}}{t_0 + 1} \end{array} \]

Alternative 5
Error	0.3
Cost	1988

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\beta \leq 10^{+139}:\\ \;\;\;\;\frac{\frac{\left(\left(\alpha + \beta\right) + \alpha \cdot \beta\right) + 1}{t_0 \cdot t_0}}{1 + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(2 + \alpha\right)}}{\beta + \left(3 + \alpha\right)}\\ \end{array} \]

Alternative 6
Error	0.3
Cost	1860

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 9.8 \cdot 10^{+138}:\\ \;\;\;\;\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{t_0 \cdot t_0} \cdot \frac{1}{\alpha + \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(2 + \alpha\right)}}{\beta + \left(3 + \alpha\right)}\\ \end{array} \]

Alternative 7
Error	0.3
Cost	1860

\[\begin{array}{l} t_0 := \beta + \left(2 + \alpha\right)\\ \mathbf{if}\;\beta \leq 1.08 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{1 + \left(\alpha + \beta \cdot \left(\alpha - -1\right)\right)}{t_0}}{t_0 \cdot \left(3 + \left(\beta + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t_0}}{\beta + \left(3 + \alpha\right)}\\ \end{array} \]

Alternative 8
Error	0.7
Cost	1348

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{\frac{1 + \beta}{\beta + 2}}{\beta + 2}}{\left(\left(\alpha + \beta\right) + 2\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(2 + \alpha\right)}}{\beta + \left(3 + \alpha\right)}\\ \end{array} \]

Alternative 9
Error	1.7
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 80:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \alpha}}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\beta}\\ \end{array} \]

Alternative 10
Error	1.6
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.4:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \alpha}}{\left(2 + \alpha\right) \cdot \left(3 + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(2 + \alpha\right)}}{\beta + \left(3 + \alpha\right)}\\ \end{array} \]

Alternative 11
Error	1.0
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.4 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{1 + \beta}{\beta + 2}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(2 + \alpha\right)}}{\beta + \left(3 + \alpha\right)}\\ \end{array} \]

Alternative 12
Error	1.0
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{\frac{\beta + 1}{\beta + 3}}{\beta + 2}}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(2 + \alpha\right)}}{\beta + \left(3 + \alpha\right)}\\ \end{array} \]

Alternative 13
Error	24.2
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\beta}\\ \end{array} \]

Alternative 14
Error	24.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.32 \cdot 10^{+22}:\\ \;\;\;\;\frac{1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 15
Error	24.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.1 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{1}{\beta + 2}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 16
Error	28.9
Cost	576

\[\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta} \]

Alternative 17
Error	32.1
Cost	448

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

Alternative 18
Error	31.9
Cost	448

\[\frac{\frac{1}{\beta}}{\beta + 3} \]

Alternative 19
Error	62.4
Cost	192

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

Alternative 20
Error	60.1
Cost	192

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

Reproduce

herbie shell --seed 2023066 
(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)))