Octave 3.8, jcobi/3

Percentage Accurate: 94.4% → 99.8%

Time: 16.4s

Precision: binary64

Cost: 1600

• Unsound rule application detected in e-graph. Results may not be sound.

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

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

Math

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

↓

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

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

Derivation

1. Initial program 94.0%

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

2. Simplified 94.1%

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

   [Start] 94.0%	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
   associate-/l/ [=>] 92.2%	\[ \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]
   associate-+l+ [=>] 92.2%	\[ \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
   +-commutative [=>] 92.2%	\[ \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
   associate-+r+ [=>] 92.2%	\[ \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
   associate-+l+ [=>] 92.2%	\[ \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
   distribute-rgt1-in [=>] 92.2%	\[ \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
   *-rgt-identity [<=] 92.2%	\[ \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
   distribute-lft-out [=>] 92.2%	\[ \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
   +-commutative [=>] 92.2%	\[ \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
   associate-*l/ [<=] 95.5%	\[ \frac{\color{blue}{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
   *-commutative [=>] 95.5%	\[ \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
   associate-*r/ [<=] 94.0%	\[ \color{blue}{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

3. Applied egg-rr 95.5%

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

   [Start] 94.1%	\[ \left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   associate-*r/ [=>] 95.5%	\[ \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
   +-commutative [=>] 95.5%	\[ \frac{\left(\alpha + 1\right) \cdot \frac{\color{blue}{1 + \beta}}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   associate-+r+ [=>] 95.5%	\[ \frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{\color{blue}{\left(\alpha + \beta\right) + 2}}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   +-commutative [=>] 95.5%	\[ \frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{\color{blue}{2 + \left(\alpha + \beta\right)}}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   associate-+r+ [=>] 95.5%	\[ \frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\color{blue}{\left(\left(\alpha + \beta\right) + 2\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   +-commutative [=>] 95.5%	\[ \frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\color{blue}{\left(2 + \left(\alpha + \beta\right)\right)} \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

4. Simplified 99.7%

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

   [Start] 95.5%	\[ \frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\left(2 + \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
   times-frac [=>] 99.7%	\[ \color{blue}{\frac{\alpha + 1}{2 + \left(\alpha + \beta\right)} \cdot \frac{\frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 3\right)}} \]
   +-commutative [=>] 99.7%	\[ \frac{\color{blue}{1 + \alpha}}{2 + \left(\alpha + \beta\right)} \cdot \frac{\frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 3\right)} \]
   associate-+r+ [=>] 99.7%	\[ \frac{1 + \alpha}{\color{blue}{\left(2 + \alpha\right) + \beta}} \cdot \frac{\frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 3\right)} \]
   +-commutative [<=] 99.7%	\[ \frac{1 + \alpha}{\left(2 + \alpha\right) + \beta} \cdot \frac{\frac{\color{blue}{\beta + 1}}{2 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 3\right)} \]
   associate-+r+ [=>] 99.7%	\[ \frac{1 + \alpha}{\left(2 + \alpha\right) + \beta} \cdot \frac{\frac{\beta + 1}{\color{blue}{\left(2 + \alpha\right) + \beta}}}{\alpha + \left(\beta + 3\right)} \]
   +-commutative [=>] 99.7%	\[ \frac{1 + \alpha}{\left(2 + \alpha\right) + \beta} \cdot \frac{\frac{\beta + 1}{\left(2 + \alpha\right) + \beta}}{\color{blue}{\left(\beta + 3\right) + \alpha}} \]

5. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}\right)\right)} \cdot \frac{\frac{\beta + 1}{\left(2 + \alpha\right) + \beta}}{\left(\beta + 3\right) + \alpha} \]
   Step-by-step derivation

   [Start] 99.7%	\[ \frac{1 + \alpha}{\left(2 + \alpha\right) + \beta} \cdot \frac{\frac{\beta + 1}{\left(2 + \alpha\right) + \beta}}{\left(\beta + 3\right) + \alpha} \]
   expm1-log1p-u [=>] 99.7%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \alpha}{\left(2 + \alpha\right) + \beta}\right)\right)} \cdot \frac{\frac{\beta + 1}{\left(2 + \alpha\right) + \beta}}{\left(\beta + 3\right) + \alpha} \]
   associate-+r+ [<=] 99.7%	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \alpha}{\color{blue}{2 + \left(\alpha + \beta\right)}}\right)\right) \cdot \frac{\frac{\beta + 1}{\left(2 + \alpha\right) + \beta}}{\left(\beta + 3\right) + \alpha} \]
   +-commutative [=>] 99.7%	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \alpha}{\color{blue}{\left(\alpha + \beta\right) + 2}}\right)\right) \cdot \frac{\frac{\beta + 1}{\left(2 + \alpha\right) + \beta}}{\left(\beta + 3\right) + \alpha} \]
   associate-+r+ [<=] 99.7%	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \alpha}{\color{blue}{\alpha + \left(\beta + 2\right)}}\right)\right) \cdot \frac{\frac{\beta + 1}{\left(2 + \alpha\right) + \beta}}{\left(\beta + 3\right) + \alpha} \]
   +-commutative [=>] 99.7%	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \alpha}{\alpha + \color{blue}{\left(2 + \beta\right)}}\right)\right) \cdot \frac{\frac{\beta + 1}{\left(2 + \alpha\right) + \beta}}{\left(\beta + 3\right) + \alpha} \]

6. Applied egg-rr 99.7%

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

   [Start] 99.7%	\[ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}\right)\right) \cdot \frac{\frac{\beta + 1}{\left(2 + \alpha\right) + \beta}}{\left(\beta + 3\right) + \alpha} \]
   associate-*r/ [=>] 99.7%	\[ \color{blue}{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}\right)\right) \cdot \frac{\beta + 1}{\left(2 + \alpha\right) + \beta}}{\left(\beta + 3\right) + \alpha}} \]
   expm1-log1p-u [<=] 99.7%	\[ \frac{\color{blue}{\frac{1 + \alpha}{\alpha + \left(2 + \beta\right)}} \cdot \frac{\beta + 1}{\left(2 + \alpha\right) + \beta}}{\left(\beta + 3\right) + \alpha} \]
   associate-+r+ [=>] 99.7%	\[ \frac{\frac{1 + \alpha}{\color{blue}{\left(\alpha + 2\right) + \beta}} \cdot \frac{\beta + 1}{\left(2 + \alpha\right) + \beta}}{\left(\beta + 3\right) + \alpha} \]
   +-commutative [<=] 99.7%	\[ \frac{\frac{1 + \alpha}{\color{blue}{\left(2 + \alpha\right)} + \beta} \cdot \frac{\beta + 1}{\left(2 + \alpha\right) + \beta}}{\left(\beta + 3\right) + \alpha} \]
   associate-+l+ [=>] 99.7%	\[ \frac{\frac{1 + \alpha}{\color{blue}{2 + \left(\alpha + \beta\right)}} \cdot \frac{\beta + 1}{\left(2 + \alpha\right) + \beta}}{\left(\beta + 3\right) + \alpha} \]
   +-commutative [=>] 99.7%	\[ \frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)} \cdot \frac{\color{blue}{1 + \beta}}{\left(2 + \alpha\right) + \beta}}{\left(\beta + 3\right) + \alpha} \]
   associate-+l+ [=>] 99.7%	\[ \frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)} \cdot \frac{1 + \beta}{\color{blue}{2 + \left(\alpha + \beta\right)}}}{\left(\beta + 3\right) + \alpha} \]
   +-commutative [=>] 99.7%	\[ \frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)} \cdot \frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\color{blue}{\alpha + \left(\beta + 3\right)}} \]

7. Simplified 99.6%

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

   [Start] 99.7%	\[ \frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)} \cdot \frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}{\alpha + \left(\beta + 3\right)} \]
   associate-/l* [=>] 99.6%	\[ \color{blue}{\frac{\frac{1 + \alpha}{2 + \left(\alpha + \beta\right)}}{\frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}}} \]
   associate-+r+ [=>] 99.6%	\[ \frac{\frac{1 + \alpha}{\color{blue}{\left(2 + \alpha\right) + \beta}}}{\frac{\alpha + \left(\beta + 3\right)}{\frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}} \]
   +-commutative [=>] 99.6%	\[ \frac{\frac{1 + \alpha}{\left(2 + \alpha\right) + \beta}}{\frac{\color{blue}{\left(\beta + 3\right) + \alpha}}{\frac{1 + \beta}{2 + \left(\alpha + \beta\right)}}} \]
   +-commutative [<=] 99.6%	\[ \frac{\frac{1 + \alpha}{\left(2 + \alpha\right) + \beta}}{\frac{\left(\beta + 3\right) + \alpha}{\frac{\color{blue}{\beta + 1}}{2 + \left(\alpha + \beta\right)}}} \]
   associate-+r+ [=>] 99.6%	\[ \frac{\frac{1 + \alpha}{\left(2 + \alpha\right) + \beta}}{\frac{\left(\beta + 3\right) + \alpha}{\frac{\beta + 1}{\color{blue}{\left(2 + \alpha\right) + \beta}}}} \]

8. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	99.8%
Cost	1600

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

Alternative 2
Accuracy	99.7%
Cost	1732

\[\begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 1.28 \cdot 10^{+63}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{\frac{1 + \beta}{t_0}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + 2\right) + \beta}}{\beta + \left(4 + \alpha \cdot 2\right)}\\ \end{array} \]

Alternative 3
Accuracy	99.7%
Cost	1600

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

Alternative 4
Accuracy	96.7%
Cost	1220

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

Alternative 5
Accuracy	97.6%
Cost	1220

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

Alternative 6
Accuracy	98.5%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\left(\beta + 3\right) \cdot \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + 2\right) + \beta}}{\beta}\\ \end{array} \]

Alternative 7
Accuracy	98.8%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 880000000:\\ \;\;\;\;\frac{1 + \beta}{\left(\alpha + \left(2 + \beta\right)\right) \cdot \left(\left(\beta + 3\right) \cdot \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + 2\right) + \beta}}{\beta + \left(4 + \alpha \cdot 2\right)}\\ \end{array} \]

Alternative 8
Accuracy	93.9%
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.8:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{1}{\beta \cdot \left(\beta + 5\right)}\\ \end{array} \]

Alternative 9
Accuracy	96.7%
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.7:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\left(\alpha + 2\right) + \beta}}{\beta}\\ \end{array} \]

Alternative 10
Accuracy	93.8%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.5:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 11
Accuracy	93.8%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.8:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 12
Accuracy	52.0%
Cost	452

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

Alternative 13
Accuracy	52.7%
Cost	448

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

Alternative 14
Accuracy	33.7%
Cost	320

\[\frac{0.5}{\beta \cdot \beta} \]

Alternative 15
Accuracy	50.0%
Cost	320

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

Reproduce

herbie shell --seed 2023165 
(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)))