Octave 3.8, jcobi/3

Percentage Accurate: 94.5% → 99.6%

Time: 17.4s

Precision: binary64

Cost: 1732

• 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

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

↓

\[\begin{array}{l} t_0 := \left(\beta + 2\right) + \alpha\\ \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+54}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{\frac{1 + \beta}{t_0}}{t_0 \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 - \alpha}{t_0}}{\beta + \left(\alpha + 3\right)} \cdot \left(-1 + \frac{1 + \alpha}{\beta}\right)\\ \end{array} \]

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 (+ (+ beta 2.0) alpha)))
   (if (<= beta 6.5e+54)
     (* (+ 1.0 alpha) (/ (/ (+ 1.0 beta) t_0) (* t_0 (+ alpha (+ beta 3.0)))))
     (*
      (/ (/ (- -1.0 alpha) t_0) (+ beta (+ alpha 3.0)))
      (+ -1.0 (/ (+ 1.0 alpha) beta))))))

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 = (beta + 2.0) + alpha;
	double tmp;
	if (beta <= 6.5e+54) {
		tmp = (1.0 + alpha) * (((1.0 + beta) / t_0) / (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = (((-1.0 - alpha) / t_0) / (beta + (alpha + 3.0))) * (-1.0 + ((1.0 + alpha) / beta));
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = (beta + 2.0d0) + alpha
    if (beta <= 6.5d+54) then
        tmp = (1.0d0 + alpha) * (((1.0d0 + beta) / t_0) / (t_0 * (alpha + (beta + 3.0d0))))
    else
        tmp = ((((-1.0d0) - alpha) / t_0) / (beta + (alpha + 3.0d0))) * ((-1.0d0) + ((1.0d0 + alpha) / beta))
    end if
    code = tmp
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 = (beta + 2.0) + alpha;
	double tmp;
	if (beta <= 6.5e+54) {
		tmp = (1.0 + alpha) * (((1.0 + beta) / t_0) / (t_0 * (alpha + (beta + 3.0))));
	} else {
		tmp = (((-1.0 - alpha) / t_0) / (beta + (alpha + 3.0))) * (-1.0 + ((1.0 + alpha) / beta));
	}
	return tmp;
}

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 = (beta + 2.0) + alpha
	tmp = 0
	if beta <= 6.5e+54:
		tmp = (1.0 + alpha) * (((1.0 + beta) / t_0) / (t_0 * (alpha + (beta + 3.0))))
	else:
		tmp = (((-1.0 - alpha) / t_0) / (beta + (alpha + 3.0))) * (-1.0 + ((1.0 + alpha) / beta))
	return tmp

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(beta + 2.0) + alpha)
	tmp = 0.0
	if (beta <= 6.5e+54)
		tmp = Float64(Float64(1.0 + alpha) * Float64(Float64(Float64(1.0 + beta) / t_0) / Float64(t_0 * Float64(alpha + Float64(beta + 3.0)))));
	else
		tmp = Float64(Float64(Float64(Float64(-1.0 - alpha) / t_0) / Float64(beta + Float64(alpha + 3.0))) * Float64(-1.0 + Float64(Float64(1.0 + alpha) / beta)));
	end
	return tmp
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_2 = code(alpha, beta)
	t_0 = (beta + 2.0) + alpha;
	tmp = 0.0;
	if (beta <= 6.5e+54)
		tmp = (1.0 + alpha) * (((1.0 + beta) / t_0) / (t_0 * (alpha + (beta + 3.0))));
	else
		tmp = (((-1.0 - alpha) / t_0) / (beta + (alpha + 3.0))) * (-1.0 + ((1.0 + alpha) / beta));
	end
	tmp_2 = tmp;
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[(beta + 2.0), $MachinePrecision] + alpha), $MachinePrecision]}, If[LessEqual[beta, 6.5e+54], N[(N[(1.0 + alpha), $MachinePrecision] * N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 * N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-1.0 - alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.5e54

   1. Initial program 99.8%

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

   2. Simplified 89.5%

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

      [Start] 99.8%	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      associate-/l/ [=>] 99.0%	\[ \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]
      associate-+l+ [=>] 99.0%	\[ \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      +-commutative [=>] 99.0%	\[ \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      associate-+r+ [=>] 99.0%	\[ \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      associate-+l+ [=>] 99.0%	\[ \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      distribute-rgt1-in [=>] 99.0%	\[ \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      *-rgt-identity [<=] 99.0%	\[ \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      distribute-lft-out [=>] 99.0%	\[ \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      +-commutative [=>] 99.0%	\[ \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      associate-*l/ [<=] 99.0%	\[ \frac{\color{blue}{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      *-commutative [=>] 99.0%	\[ \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      associate-*r/ [<=] 89.5%	\[ \color{blue}{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   if 6.5e54 < beta

   1. Initial program 78.6%

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

   2. Simplified 89.3%

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

      [Start] 78.6%	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      associate-/l/ [=>] 74.5%	\[ \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]
      associate-+l+ [=>] 74.5%	\[ \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      +-commutative [=>] 74.5%	\[ \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      associate-+r+ [=>] 74.5%	\[ \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      associate-+l+ [=>] 74.5%	\[ \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      distribute-rgt1-in [=>] 74.5%	\[ \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      *-rgt-identity [<=] 74.5%	\[ \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      distribute-lft-out [=>] 74.5%	\[ \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      +-commutative [=>] 74.5%	\[ \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      associate-*l/ [<=] 89.3%	\[ \frac{\color{blue}{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      *-commutative [=>] 89.3%	\[ \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      associate-*r/ [<=] 89.3%	\[ \color{blue}{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Applied egg-rr 89.3%

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

      [Start] 89.3%	\[ \left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
      associate-*r/ [=>] 89.3%	\[ \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
      +-commutative [=>] 89.3%	\[ \frac{\left(\alpha + 1\right) \cdot \frac{\color{blue}{1 + \beta}}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]

   4. Simplified 99.9%

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

      [Start] 89.3%	\[ \frac{\left(\alpha + 1\right) \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
      +-commutative [=>] 89.3%	\[ \frac{\color{blue}{\left(1 + \alpha\right)} \cdot \frac{1 + \beta}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
      *-commutative [=>] 89.3%	\[ \frac{\color{blue}{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \left(1 + \alpha\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
      +-commutative [<=] 89.3%	\[ \frac{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \color{blue}{\left(\alpha + 1\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
      associate-*r/ [<=] 89.3%	\[ \color{blue}{\frac{1 + \beta}{\alpha + \left(\beta + 2\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
      +-commutative [=>] 89.3%	\[ \frac{1 + \beta}{\color{blue}{\left(\beta + 2\right) + \alpha}} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
      associate-/r* [=>] 99.9%	\[ \frac{1 + \beta}{\left(\beta + 2\right) + \alpha} \cdot \color{blue}{\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}} \]
      +-commutative [=>] 99.9%	\[ \frac{1 + \beta}{\left(\beta + 2\right) + \alpha} \cdot \frac{\frac{\color{blue}{1 + \alpha}}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)} \]
      +-commutative [=>] 99.9%	\[ \frac{1 + \beta}{\left(\beta + 2\right) + \alpha} \cdot \frac{\frac{1 + \alpha}{\color{blue}{\left(\beta + 2\right) + \alpha}}}{\alpha + \left(\beta + 3\right)} \]
      +-commutative [=>] 99.9%	\[ \frac{1 + \beta}{\left(\beta + 2\right) + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(\beta + 2\right) + \alpha}}{\color{blue}{\left(\beta + 3\right) + \alpha}} \]
      associate-+r+ [<=] 99.9%	\[ \frac{1 + \beta}{\left(\beta + 2\right) + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(\beta + 2\right) + \alpha}}{\color{blue}{\beta + \left(3 + \alpha\right)}} \]
      +-commutative [=>] 99.9%	\[ \frac{1 + \beta}{\left(\beta + 2\right) + \alpha} \cdot \frac{\frac{1 + \alpha}{\left(\beta + 2\right) + \alpha}}{\beta + \color{blue}{\left(\alpha + 3\right)}} \]

   5. Taylor expanded in beta around inf 90.5%

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

   6. Simplified 90.5%

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

      [Start] 90.5%	\[ \left(1 + -1 \cdot \frac{1 + \alpha}{\beta}\right) \cdot \frac{\frac{1 + \alpha}{\left(\beta + 2\right) + \alpha}}{\beta + \left(\alpha + 3\right)} \]
      associate-*r/ [=>] 90.5%	\[ \left(1 + \color{blue}{\frac{-1 \cdot \left(1 + \alpha\right)}{\beta}}\right) \cdot \frac{\frac{1 + \alpha}{\left(\beta + 2\right) + \alpha}}{\beta + \left(\alpha + 3\right)} \]
      neg-mul-1 [<=] 90.5%	\[ \left(1 + \frac{\color{blue}{-\left(1 + \alpha\right)}}{\beta}\right) \cdot \frac{\frac{1 + \alpha}{\left(\beta + 2\right) + \alpha}}{\beta + \left(\alpha + 3\right)} \]
      distribute-neg-in [=>] 90.5%	\[ \left(1 + \frac{\color{blue}{\left(-1\right) + \left(-\alpha\right)}}{\beta}\right) \cdot \frac{\frac{1 + \alpha}{\left(\beta + 2\right) + \alpha}}{\beta + \left(\alpha + 3\right)} \]
      metadata-eval [=>] 90.5%	\[ \left(1 + \frac{\color{blue}{-1} + \left(-\alpha\right)}{\beta}\right) \cdot \frac{\frac{1 + \alpha}{\left(\beta + 2\right) + \alpha}}{\beta + \left(\alpha + 3\right)} \]
      unsub-neg [=>] 90.5%	\[ \left(1 + \frac{\color{blue}{-1 - \alpha}}{\beta}\right) \cdot \frac{\frac{1 + \alpha}{\left(\beta + 2\right) + \alpha}}{\beta + \left(\alpha + 3\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+54}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{\frac{1 + \beta}{\left(\beta + 2\right) + \alpha}}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 - \alpha}{\left(\beta + 2\right) + \alpha}}{\beta + \left(\alpha + 3\right)} \cdot \left(-1 + \frac{1 + \alpha}{\beta}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	1732

\[\begin{array}{l} t_0 := \left(\beta + 2\right) + \alpha\\ \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+54}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{\frac{1 + \beta}{t_0}}{t_0 \cdot \left(\alpha + \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 - \alpha}{t_0}}{\beta + \left(\alpha + 3\right)} \cdot \left(-1 + \frac{1 + \alpha}{\beta}\right)\\ \end{array} \]

Alternative 2
Accuracy	99.2%
Cost	1608

\[\begin{array}{l} t_0 := \left(\beta + 2\right) + \alpha\\ \mathbf{if}\;\beta \leq 3.2 \cdot 10^{-94}:\\ \;\;\;\;\frac{1 + \alpha}{t_0 \cdot \left(6 + \alpha \cdot \left(\alpha + 5\right)\right)}\\ \mathbf{elif}\;\beta \leq 5 \cdot 10^{+49}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{\frac{1 + \beta}{t_0}}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]

Alternative 3
Accuracy	99.2%
Cost	1608

\[\begin{array}{l} t_0 := \left(\beta + 2\right) + \alpha\\ t_1 := \frac{1 + \beta}{t_0}\\ \mathbf{if}\;\beta \leq 3.2 \cdot 10^{-94}:\\ \;\;\;\;\frac{1 + \alpha}{t_0 \cdot \left(6 + \alpha \cdot \left(\alpha + 5\right)\right)}\\ \mathbf{elif}\;\beta \leq 7 \cdot 10^{+49}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \frac{t_1}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 4
Accuracy	98.2%
Cost	1604

\[\begin{array}{l} t_0 := \left(\beta + 2\right) + \alpha\\ \mathbf{if}\;\beta \leq 34:\\ \;\;\;\;\frac{1 + \alpha}{t_0 \cdot \left(6 + \alpha \cdot \left(\alpha + 5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 - \alpha}{t_0}}{\beta + \left(\alpha + 3\right)} \cdot \left(-1 + \frac{1 + \alpha}{\beta}\right)\\ \end{array} \]

Alternative 5
Accuracy	99.8%
Cost	1600

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

Alternative 6
Accuracy	96.4%
Cost	1480

\[\begin{array}{l} t_0 := \left(\beta + 2\right) + \alpha\\ \mathbf{if}\;\beta \leq 3 \cdot 10^{-94}:\\ \;\;\;\;\frac{1 + \alpha}{t_0 \cdot \left(6 + \alpha \cdot \left(\alpha + 5\right)\right)}\\ \mathbf{elif}\;\beta \leq 1.52 \cdot 10^{+38}:\\ \;\;\;\;\frac{1 + \beta}{t_0} \cdot \frac{1}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \left(\frac{-1}{\beta} \cdot \frac{-1}{\beta}\right)\\ \end{array} \]

Alternative 7
Accuracy	99.0%
Cost	1480

\[\begin{array}{l} t_0 := \left(\beta + 2\right) + \alpha\\ \mathbf{if}\;\beta \leq 10^{-94}:\\ \;\;\;\;\frac{1 + \alpha}{t_0 \cdot \left(6 + \alpha \cdot \left(\alpha + 5\right)\right)}\\ \mathbf{elif}\;\beta \leq 1.52 \cdot 10^{+38}:\\ \;\;\;\;\frac{1 + \beta}{t_0} \cdot \frac{1}{\left(\beta + 2\right) \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]

Alternative 8
Accuracy	96.4%
Cost	1352

\[\begin{array}{l} t_0 := \left(\beta + 2\right) + \alpha\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{-94}:\\ \;\;\;\;\frac{1 + \alpha}{t_0 \cdot \left(6 + \alpha \cdot \left(\alpha + 5\right)\right)}\\ \mathbf{elif}\;\beta \leq 2.5 \cdot 10^{+36}:\\ \;\;\;\;\frac{1 + \beta}{t_0 \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \left(\frac{-1}{\beta} \cdot \frac{-1}{\beta}\right)\\ \end{array} \]

Alternative 9
Accuracy	94.8%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 36:\\ \;\;\;\;\frac{1 + \alpha}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \left(\left(\alpha + 3\right) \cdot \left(2 + \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \left(\frac{-1}{\beta} \cdot \frac{-1}{\beta}\right)\\ \end{array} \]

Alternative 10
Accuracy	94.8%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.6:\\ \;\;\;\;\frac{1 + \alpha}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \left(6 + \alpha \cdot \left(\alpha + 5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \left(\frac{-1}{\beta} \cdot \frac{-1}{\beta}\right)\\ \end{array} \]

Alternative 11
Accuracy	94.4%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 59:\\ \;\;\;\;\frac{1 + \alpha}{\left(\left(\beta + 2\right) + \alpha\right) \cdot \left(6 + \alpha \cdot 5\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \left(\frac{-1}{\beta} \cdot \frac{-1}{\beta}\right)\\ \end{array} \]

Alternative 12
Accuracy	93.9%
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.8:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \alpha\right) \cdot \left(\frac{-1}{\beta} \cdot \frac{-1}{\beta}\right)\\ \end{array} \]

Alternative 13
Accuracy	93.4%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.8:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 14
Accuracy	73.9%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7 \cdot 10^{+38}:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 15
Accuracy	47.1%
Cost	320

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

Alternative 16
Accuracy	2.5%
Cost	192

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

Alternative 17
Accuracy	6.0%
Cost	192

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

Reproduce

herbie shell --seed 2023178 
(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)))