Octave 3.8, jcobi/3

Percentage Accurate: 94.5% → 99.7%

Time: 23.6s

Precision: binary64

Cost: 2372

• 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

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

↓

\[\begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 10^{+114}:\\ \;\;\;\;\frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \alpha}{\beta \cdot \beta + \left(\alpha + 2\right) \cdot \left(\left(\alpha + 2\right) + \beta \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\beta} + \left(\left(\left(1 + \alpha\right) + \frac{\alpha}{\beta}\right) + \frac{-1 - \alpha}{\frac{\beta}{\alpha + 2}}\right)}{t_0}}{1 + t_0}\\ \end{array} \]

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 (+ 2.0 (+ beta alpha))))
   (if (<= beta 1e+114)
     (*
      (/ (+ beta 1.0) (+ alpha (+ beta 3.0)))
      (/
       (+ 1.0 alpha)
       (+ (* beta beta) (* (+ alpha 2.0) (+ (+ alpha 2.0) (* beta 2.0))))))
     (/
      (/
       (+
        (/ 1.0 beta)
        (+
         (+ (+ 1.0 alpha) (/ alpha beta))
         (/ (- -1.0 alpha) (/ beta (+ alpha 2.0)))))
       t_0)
      (+ 1.0 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 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 1e+114) {
		tmp = ((beta + 1.0) / (alpha + (beta + 3.0))) * ((1.0 + alpha) / ((beta * beta) + ((alpha + 2.0) * ((alpha + 2.0) + (beta * 2.0)))));
	} else {
		tmp = (((1.0 / beta) + (((1.0 + alpha) + (alpha / beta)) + ((-1.0 - alpha) / (beta / (alpha + 2.0))))) / t_0) / (1.0 + t_0);
	}
	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 = 2.0d0 + (beta + alpha)
    if (beta <= 1d+114) then
        tmp = ((beta + 1.0d0) / (alpha + (beta + 3.0d0))) * ((1.0d0 + alpha) / ((beta * beta) + ((alpha + 2.0d0) * ((alpha + 2.0d0) + (beta * 2.0d0)))))
    else
        tmp = (((1.0d0 / beta) + (((1.0d0 + alpha) + (alpha / beta)) + (((-1.0d0) - alpha) / (beta / (alpha + 2.0d0))))) / t_0) / (1.0d0 + t_0)
    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 = 2.0 + (beta + alpha);
	double tmp;
	if (beta <= 1e+114) {
		tmp = ((beta + 1.0) / (alpha + (beta + 3.0))) * ((1.0 + alpha) / ((beta * beta) + ((alpha + 2.0) * ((alpha + 2.0) + (beta * 2.0)))));
	} else {
		tmp = (((1.0 / beta) + (((1.0 + alpha) + (alpha / beta)) + ((-1.0 - alpha) / (beta / (alpha + 2.0))))) / t_0) / (1.0 + t_0);
	}
	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 = 2.0 + (beta + alpha)
	tmp = 0
	if beta <= 1e+114:
		tmp = ((beta + 1.0) / (alpha + (beta + 3.0))) * ((1.0 + alpha) / ((beta * beta) + ((alpha + 2.0) * ((alpha + 2.0) + (beta * 2.0)))))
	else:
		tmp = (((1.0 / beta) + (((1.0 + alpha) + (alpha / beta)) + ((-1.0 - alpha) / (beta / (alpha + 2.0))))) / t_0) / (1.0 + t_0)
	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(2.0 + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 1e+114)
		tmp = Float64(Float64(Float64(beta + 1.0) / Float64(alpha + Float64(beta + 3.0))) * Float64(Float64(1.0 + alpha) / Float64(Float64(beta * beta) + Float64(Float64(alpha + 2.0) * Float64(Float64(alpha + 2.0) + Float64(beta * 2.0))))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / beta) + Float64(Float64(Float64(1.0 + alpha) + Float64(alpha / beta)) + Float64(Float64(-1.0 - alpha) / Float64(beta / Float64(alpha + 2.0))))) / t_0) / Float64(1.0 + t_0));
	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 = 2.0 + (beta + alpha);
	tmp = 0.0;
	if (beta <= 1e+114)
		tmp = ((beta + 1.0) / (alpha + (beta + 3.0))) * ((1.0 + alpha) / ((beta * beta) + ((alpha + 2.0) * ((alpha + 2.0) + (beta * 2.0)))));
	else
		tmp = (((1.0 / beta) + (((1.0 + alpha) + (alpha / beta)) + ((-1.0 - alpha) / (beta / (alpha + 2.0))))) / t_0) / (1.0 + t_0);
	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[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1e+114], N[(N[(N[(beta + 1.0), $MachinePrecision] / N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + alpha), $MachinePrecision] / N[(N[(beta * beta), $MachinePrecision] + N[(N[(alpha + 2.0), $MachinePrecision] * N[(N[(alpha + 2.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 / beta), $MachinePrecision] + N[(N[(N[(1.0 + alpha), $MachinePrecision] + N[(alpha / beta), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - alpha), $MachinePrecision] / N[(beta / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1e114

   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 99.8%

      \[\leadsto \color{blue}{\frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]
      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/ [=>] 99.8%	\[ \frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      associate-/l/ [=>] 95.4%	\[ \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]
      associate-+l+ [=>] 95.4%	\[ \frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]
      +-commutative [=>] 95.4%	\[ \frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]
      associate-+r+ [=>] 95.4%	\[ \frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]
      associate-+l+ [=>] 95.4%	\[ \frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]
      distribute-rgt1-in [=>] 95.4%	\[ \frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]
      *-rgt-identity [<=] 95.4%	\[ \frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]
      distribute-lft-out [=>] 95.4%	\[ \frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]
      +-commutative [=>] 95.4%	\[ \frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]
      times-frac [=>] 99.8%	\[ \color{blue}{\frac{\beta + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \cdot \frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Taylor expanded in beta around -inf 99.8%

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

   4. Simplified 99.8%

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

      [Start] 99.8%	\[ \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{{\beta}^{2} + \left(2 \cdot \left(\beta \cdot \left(2 + \alpha\right)\right) + {\left(2 + \alpha\right)}^{2}\right)} \]
      unpow2 [=>] 99.8%	\[ \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\color{blue}{\beta \cdot \beta} + \left(2 \cdot \left(\beta \cdot \left(2 + \alpha\right)\right) + {\left(2 + \alpha\right)}^{2}\right)} \]
      +-commutative [=>] 99.8%	\[ \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\beta \cdot \beta + \color{blue}{\left({\left(2 + \alpha\right)}^{2} + 2 \cdot \left(\beta \cdot \left(2 + \alpha\right)\right)\right)}} \]
      unpow2 [=>] 99.8%	\[ \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\beta \cdot \beta + \left(\color{blue}{\left(2 + \alpha\right) \cdot \left(2 + \alpha\right)} + 2 \cdot \left(\beta \cdot \left(2 + \alpha\right)\right)\right)} \]
      associate-*r* [=>] 99.8%	\[ \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\beta \cdot \beta + \left(\left(2 + \alpha\right) \cdot \left(2 + \alpha\right) + \color{blue}{\left(2 \cdot \beta\right) \cdot \left(2 + \alpha\right)}\right)} \]
      distribute-rgt-out [=>] 99.8%	\[ \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\beta \cdot \beta + \color{blue}{\left(2 + \alpha\right) \cdot \left(\left(2 + \alpha\right) + 2 \cdot \beta\right)}} \]
      +-commutative [=>] 99.8%	\[ \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\beta \cdot \beta + \color{blue}{\left(\alpha + 2\right)} \cdot \left(\left(2 + \alpha\right) + 2 \cdot \beta\right)} \]
      +-commutative [=>] 99.8%	\[ \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\beta \cdot \beta + \left(\alpha + 2\right) \cdot \left(\color{blue}{\left(\alpha + 2\right)} + 2 \cdot \beta\right)} \]

   if 1e114 < beta

   1. Initial program 79.5%

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

   2. Taylor expanded in beta around inf 81.5%

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

   3. Simplified 86.0%

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

      [Start] 81.5%	\[ \frac{\frac{\left(\frac{1}{\beta} + \left(\frac{\alpha}{\beta} + \left(1 + \alpha\right)\right)\right) - \frac{\left(1 + \alpha\right) \cdot \left(2 + \alpha\right)}{\beta}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      associate--l+ [=>] 81.5%	\[ \frac{\frac{\color{blue}{\frac{1}{\beta} + \left(\left(\frac{\alpha}{\beta} + \left(1 + \alpha\right)\right) - \frac{\left(1 + \alpha\right) \cdot \left(2 + \alpha\right)}{\beta}\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      +-commutative [=>] 81.5%	\[ \frac{\frac{\frac{1}{\beta} + \left(\color{blue}{\left(\left(1 + \alpha\right) + \frac{\alpha}{\beta}\right)} - \frac{\left(1 + \alpha\right) \cdot \left(2 + \alpha\right)}{\beta}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      associate-/l* [=>] 86.0%	\[ \frac{\frac{\frac{1}{\beta} + \left(\left(\left(1 + \alpha\right) + \frac{\alpha}{\beta}\right) - \color{blue}{\frac{1 + \alpha}{\frac{\beta}{2 + \alpha}}}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      +-commutative [=>] 86.0%	\[ \frac{\frac{\frac{1}{\beta} + \left(\left(\left(1 + \alpha\right) + \frac{\alpha}{\beta}\right) - \frac{1 + \alpha}{\frac{\beta}{\color{blue}{\alpha + 2}}}\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]

3. Recombined 2 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+114}:\\ \;\;\;\;\frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \alpha}{\beta \cdot \beta + \left(\alpha + 2\right) \cdot \left(\left(\alpha + 2\right) + \beta \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\beta} + \left(\left(\left(1 + \alpha\right) + \frac{\alpha}{\beta}\right) + \frac{-1 - \alpha}{\frac{\beta}{\alpha + 2}}\right)}{2 + \left(\beta + \alpha\right)}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	2372

\[\begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 10^{+114}:\\ \;\;\;\;\frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \alpha}{\beta \cdot \beta + \left(\alpha + 2\right) \cdot \left(\left(\alpha + 2\right) + \beta \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\beta} + \left(\left(\left(1 + \alpha\right) + \frac{\alpha}{\beta}\right) + \frac{-1 - \alpha}{\frac{\beta}{\alpha + 2}}\right)}{t_0}}{1 + t_0}\\ \end{array} \]

Alternative 2
Accuracy	99.6%
Cost	1988

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+112}:\\ \;\;\;\;\frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \alpha}{\beta \cdot \beta + \left(\alpha + 2\right) \cdot \left(\left(\alpha + 2\right) + \beta \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 2\right)}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]

Alternative 3
Accuracy	99.6%
Cost	1988

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+112}:\\ \;\;\;\;\frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{1 + \alpha}{\beta \cdot \beta + \left(\alpha + 2\right) \cdot \left(\left(\alpha + 2\right) + \beta \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta} + \left(\frac{\alpha}{\beta} - \frac{1 + \alpha}{\beta} \cdot \frac{\alpha + 2}{\beta}\right)}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]

Alternative 4
Accuracy	99.5%
Cost	1732

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+90}:\\ \;\;\;\;\frac{\frac{-1 - \alpha}{t_0}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot t_0} \cdot \left(-1 - \beta\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 2\right)}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]

Alternative 5
Accuracy	99.6%
Cost	1732

\[\begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 10^{+114}:\\ \;\;\;\;\frac{\beta + 1}{t_0} \cdot \frac{1 + \alpha}{\left(\alpha + \left(\beta + 3\right)\right) \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 2\right)}}{1 + t_0}\\ \end{array} \]

Alternative 6
Accuracy	98.6%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.9 \cdot 10^{+15}:\\ \;\;\;\;\frac{\beta + 1}{\left(\beta + 2\right) \cdot \left(\beta \cdot \left(\beta + 5\right) + 6\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 2\right)}}{1 + \left(2 + \left(\beta + \alpha\right)\right)}\\ \end{array} \]

Alternative 7
Accuracy	62.8%
Cost	1092

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 3\right)\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+112}:\\ \;\;\;\;\frac{1 + \alpha}{\left(\beta + \left(\alpha + 2\right)\right) \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{t_0}\\ \end{array} \]

Alternative 8
Accuracy	98.6%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 10^{+16}:\\ \;\;\;\;\frac{\beta + 1}{\left(\beta + 2\right) \cdot \left(\beta \cdot \left(\beta + 5\right) + 6\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 9
Accuracy	98.6%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.4 \cdot 10^{+16}:\\ \;\;\;\;\frac{\beta + 1}{\left(\beta + 2\right) \cdot \left(\beta \cdot \left(\beta + 5\right) + 6\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{2 + \left(\beta + \alpha\right)}}{\beta + 1}\\ \end{array} \]

Alternative 10
Accuracy	62.6%
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 11
Accuracy	62.6%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta + 1}\\ \end{array} \]

Alternative 12
Accuracy	57.2%
Cost	576

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

Alternative 13
Accuracy	50.9%
Cost	448

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

Alternative 14
Accuracy	53.6%
Cost	448

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

Alternative 15
Accuracy	34.3%
Cost	320

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

Alternative 16
Accuracy	50.7%
Cost	320

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

Alternative 17
Accuracy	51.2%
Cost	320

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

Alternative 18
Accuracy	6.0%
Cost	192

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

Alternative 19
Accuracy	2.2%
Cost	64

\[-0.125 \]

Reproduce

herbie shell --seed 2023271 
(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)))