Octave 3.8, jcobi/3

Percentage Accurate: 94.2% → 99.5%

Time: 23.5s

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

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

↓

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+22}:\\ \;\;\;\;\left(\beta + 1\right) \cdot \frac{-1 - \alpha}{t_0 \cdot \left(\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(-2 - \left(\beta + \alpha\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t_0}}{\frac{t_0}{1 + \alpha}}\\ \end{array} \]

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))))
   (if (<= beta 5e+22)
     (*
      (+ beta 1.0)
      (/
       (- -1.0 alpha)
       (* t_0 (* (+ beta (+ alpha 3.0)) (- -2.0 (+ beta alpha))))))
     (/ (/ 1.0 t_0) (/ t_0 (+ 1.0 alpha))))))

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);
	double tmp;
	if (beta <= 5e+22) {
		tmp = (beta + 1.0) * ((-1.0 - alpha) / (t_0 * ((beta + (alpha + 3.0)) * (-2.0 - (beta + alpha)))));
	} else {
		tmp = (1.0 / t_0) / (t_0 / (1.0 + alpha));
	}
	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 = alpha + (beta + 2.0d0)
    if (beta <= 5d+22) then
        tmp = (beta + 1.0d0) * (((-1.0d0) - alpha) / (t_0 * ((beta + (alpha + 3.0d0)) * ((-2.0d0) - (beta + alpha)))))
    else
        tmp = (1.0d0 / t_0) / (t_0 / (1.0d0 + alpha))
    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 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 5e+22) {
		tmp = (beta + 1.0) * ((-1.0 - alpha) / (t_0 * ((beta + (alpha + 3.0)) * (-2.0 - (beta + alpha)))));
	} else {
		tmp = (1.0 / t_0) / (t_0 / (1.0 + alpha));
	}
	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 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 5e+22:
		tmp = (beta + 1.0) * ((-1.0 - alpha) / (t_0 * ((beta + (alpha + 3.0)) * (-2.0 - (beta + alpha)))))
	else:
		tmp = (1.0 / t_0) / (t_0 / (1.0 + alpha))
	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(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 5e+22)
		tmp = Float64(Float64(beta + 1.0) * Float64(Float64(-1.0 - alpha) / Float64(t_0 * Float64(Float64(beta + Float64(alpha + 3.0)) * Float64(-2.0 - Float64(beta + alpha))))));
	else
		tmp = Float64(Float64(1.0 / t_0) / Float64(t_0 / Float64(1.0 + alpha)));
	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 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 5e+22)
		tmp = (beta + 1.0) * ((-1.0 - alpha) / (t_0 * ((beta + (alpha + 3.0)) * (-2.0 - (beta + alpha)))));
	else
		tmp = (1.0 / t_0) / (t_0 / (1.0 + alpha));
	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[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5e+22], N[(N[(beta + 1.0), $MachinePrecision] * N[(N[(-1.0 - alpha), $MachinePrecision] / N[(t$95$0 * N[(N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision] * N[(-2.0 - N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / N[(t$95$0 / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.9999999999999996e22

   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 99.4%

      \[\leadsto \color{blue}{\left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
      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/ [=>] 99.4	\[ \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]
      associate-+l+ [=>] 99.4	\[ \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      +-commutative [=>] 99.4	\[ \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      associate-+r+ [=>] 99.4	\[ \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      associate-+l+ [=>] 99.4	\[ \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      distribute-rgt1-in [=>] 99.4	\[ \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      *-rgt-identity [<=] 99.4	\[ \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      distribute-lft-out [=>] 99.4	\[ \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      +-commutative [=>] 99.4	\[ \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      associate-*r/ [<=] 99.4	\[ \frac{\color{blue}{\left(\beta + 1\right) \cdot \frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      associate-*r/ [<=] 99.4	\[ \color{blue}{\left(\beta + 1\right) \cdot \frac{\frac{1 + \alpha}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Applied egg-rr 99.4%

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

      [Start] 99.4	\[ \left(\beta + 1\right) \cdot \frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
      frac-2neg [=>] 99.4	\[ \left(\beta + 1\right) \cdot \color{blue}{\frac{-\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{-\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
      div-inv [=>] 99.4	\[ \left(\beta + 1\right) \cdot \color{blue}{\left(\left(-\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}\right) \cdot \frac{1}{-\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\right)} \]
      distribute-neg-frac [=>] 99.4	\[ \left(\beta + 1\right) \cdot \left(\color{blue}{\frac{-\left(\alpha + 1\right)}{\alpha + \left(\beta + 2\right)}} \cdot \frac{1}{-\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\right) \]
      +-commutative [=>] 99.4	\[ \left(\beta + 1\right) \cdot \left(\frac{-\color{blue}{\left(1 + \alpha\right)}}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{-\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\right) \]
      distribute-neg-in [=>] 99.4	\[ \left(\beta + 1\right) \cdot \left(\frac{\color{blue}{\left(-1\right) + \left(-\alpha\right)}}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{-\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\right) \]
      metadata-eval [=>] 99.4	\[ \left(\beta + 1\right) \cdot \left(\frac{\color{blue}{-1} + \left(-\alpha\right)}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{-\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\right) \]
      +-commutative [=>] 99.4	\[ \left(\beta + 1\right) \cdot \left(\frac{-1 + \left(-\alpha\right)}{\color{blue}{\left(\beta + 2\right) + \alpha}} \cdot \frac{1}{-\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\right) \]
      associate-+l+ [=>] 99.4	\[ \left(\beta + 1\right) \cdot \left(\frac{-1 + \left(-\alpha\right)}{\color{blue}{\beta + \left(2 + \alpha\right)}} \cdot \frac{1}{-\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}\right) \]
      distribute-rgt-neg-in [=>] 99.4	\[ \left(\beta + 1\right) \cdot \left(\frac{-1 + \left(-\alpha\right)}{\beta + \left(2 + \alpha\right)} \cdot \frac{1}{\color{blue}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(-\left(\alpha + \left(\beta + 3\right)\right)\right)}}\right) \]
      +-commutative [=>] 99.4	\[ \left(\beta + 1\right) \cdot \left(\frac{-1 + \left(-\alpha\right)}{\beta + \left(2 + \alpha\right)} \cdot \frac{1}{\color{blue}{\left(\left(\beta + 2\right) + \alpha\right)} \cdot \left(-\left(\alpha + \left(\beta + 3\right)\right)\right)}\right) \]
      associate-+l+ [=>] 99.4	\[ \left(\beta + 1\right) \cdot \left(\frac{-1 + \left(-\alpha\right)}{\beta + \left(2 + \alpha\right)} \cdot \frac{1}{\color{blue}{\left(\beta + \left(2 + \alpha\right)\right)} \cdot \left(-\left(\alpha + \left(\beta + 3\right)\right)\right)}\right) \]
      +-commutative [=>] 99.4	\[ \left(\beta + 1\right) \cdot \left(\frac{-1 + \left(-\alpha\right)}{\beta + \left(2 + \alpha\right)} \cdot \frac{1}{\left(\beta + \left(2 + \alpha\right)\right) \cdot \left(-\color{blue}{\left(\left(\beta + 3\right) + \alpha\right)}\right)}\right) \]
      associate-+l+ [=>] 99.4	\[ \left(\beta + 1\right) \cdot \left(\frac{-1 + \left(-\alpha\right)}{\beta + \left(2 + \alpha\right)} \cdot \frac{1}{\left(\beta + \left(2 + \alpha\right)\right) \cdot \left(-\color{blue}{\left(\beta + \left(3 + \alpha\right)\right)}\right)}\right) \]

   4. Simplified 95.2%

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

      [Start] 99.4	\[ \left(\beta + 1\right) \cdot \left(\frac{-1 + \left(-\alpha\right)}{\beta + \left(2 + \alpha\right)} \cdot \frac{1}{\left(\beta + \left(2 + \alpha\right)\right) \cdot \left(-\left(\beta + \left(3 + \alpha\right)\right)\right)}\right) \]
      associate-*r/ [=>] 99.4	\[ \left(\beta + 1\right) \cdot \color{blue}{\frac{\frac{-1 + \left(-\alpha\right)}{\beta + \left(2 + \alpha\right)} \cdot 1}{\left(\beta + \left(2 + \alpha\right)\right) \cdot \left(-\left(\beta + \left(3 + \alpha\right)\right)\right)}} \]
      *-rgt-identity [=>] 99.4	\[ \left(\beta + 1\right) \cdot \frac{\color{blue}{\frac{-1 + \left(-\alpha\right)}{\beta + \left(2 + \alpha\right)}}}{\left(\beta + \left(2 + \alpha\right)\right) \cdot \left(-\left(\beta + \left(3 + \alpha\right)\right)\right)} \]
      metadata-eval [<=] 99.4	\[ \left(\beta + 1\right) \cdot \frac{\frac{\color{blue}{\left(-1\right)} + \left(-\alpha\right)}{\beta + \left(2 + \alpha\right)}}{\left(\beta + \left(2 + \alpha\right)\right) \cdot \left(-\left(\beta + \left(3 + \alpha\right)\right)\right)} \]
      distribute-neg-in [<=] 99.4	\[ \left(\beta + 1\right) \cdot \frac{\frac{\color{blue}{-\left(1 + \alpha\right)}}{\beta + \left(2 + \alpha\right)}}{\left(\beta + \left(2 + \alpha\right)\right) \cdot \left(-\left(\beta + \left(3 + \alpha\right)\right)\right)} \]
      distribute-frac-neg [=>] 99.4	\[ \left(\beta + 1\right) \cdot \frac{\color{blue}{-\frac{1 + \alpha}{\beta + \left(2 + \alpha\right)}}}{\left(\beta + \left(2 + \alpha\right)\right) \cdot \left(-\left(\beta + \left(3 + \alpha\right)\right)\right)} \]
      distribute-neg-frac [<=] 99.4	\[ \left(\beta + 1\right) \cdot \color{blue}{\left(-\frac{\frac{1 + \alpha}{\beta + \left(2 + \alpha\right)}}{\left(\beta + \left(2 + \alpha\right)\right) \cdot \left(-\left(\beta + \left(3 + \alpha\right)\right)\right)}\right)} \]
      associate-/l/ [=>] 95.2	\[ \left(\beta + 1\right) \cdot \left(-\color{blue}{\frac{1 + \alpha}{\left(\left(\beta + \left(2 + \alpha\right)\right) \cdot \left(-\left(\beta + \left(3 + \alpha\right)\right)\right)\right) \cdot \left(\beta + \left(2 + \alpha\right)\right)}}\right) \]
      distribute-neg-frac [=>] 95.2	\[ \left(\beta + 1\right) \cdot \color{blue}{\frac{-\left(1 + \alpha\right)}{\left(\left(\beta + \left(2 + \alpha\right)\right) \cdot \left(-\left(\beta + \left(3 + \alpha\right)\right)\right)\right) \cdot \left(\beta + \left(2 + \alpha\right)\right)}} \]
      distribute-neg-in [=>] 95.2	\[ \left(\beta + 1\right) \cdot \frac{\color{blue}{\left(-1\right) + \left(-\alpha\right)}}{\left(\left(\beta + \left(2 + \alpha\right)\right) \cdot \left(-\left(\beta + \left(3 + \alpha\right)\right)\right)\right) \cdot \left(\beta + \left(2 + \alpha\right)\right)} \]
      metadata-eval [=>] 95.2	\[ \left(\beta + 1\right) \cdot \frac{\color{blue}{-1} + \left(-\alpha\right)}{\left(\left(\beta + \left(2 + \alpha\right)\right) \cdot \left(-\left(\beta + \left(3 + \alpha\right)\right)\right)\right) \cdot \left(\beta + \left(2 + \alpha\right)\right)} \]
      unsub-neg [=>] 95.2	\[ \left(\beta + 1\right) \cdot \frac{\color{blue}{-1 - \alpha}}{\left(\left(\beta + \left(2 + \alpha\right)\right) \cdot \left(-\left(\beta + \left(3 + \alpha\right)\right)\right)\right) \cdot \left(\beta + \left(2 + \alpha\right)\right)} \]

   if 4.9999999999999996e22 < beta

   1. Initial program 82.0%

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

   2. Simplified 93.0%

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

      [Start] 82.0	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      associate-/l/ [=>] 79.3	\[ \frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      associate-/l/ [=>] 66.1	\[ \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]
      associate-+l+ [=>] 66.1	\[ \frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]
      +-commutative [=>] 66.1	\[ \frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]
      associate-+r+ [=>] 66.1	\[ \frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]
      associate-+l+ [=>] 66.1	\[ \frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]
      distribute-rgt1-in [=>] 66.1	\[ \frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]
      *-rgt-identity [<=] 66.1	\[ \frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]
      distribute-lft-out [=>] 66.1	\[ \frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]
      +-commutative [=>] 66.1	\[ \frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]
      times-frac [=>] 93.0	\[ \color{blue}{\frac{\beta + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \cdot \frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Applied egg-rr 99.6%

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

      [Start] 93.0	\[ \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
      associate-/r* [=>] 99.8	\[ \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \color{blue}{\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}} \]
      div-inv [=>] 99.6	\[ \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \color{blue}{\left(\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)} \cdot \frac{1}{\alpha + \left(\beta + 2\right)}\right)} \]
      +-commutative [=>] 99.6	\[ \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \left(\frac{\alpha + 1}{\alpha + \color{blue}{\left(2 + \beta\right)}} \cdot \frac{1}{\alpha + \left(\beta + 2\right)}\right) \]
      associate-+r+ [=>] 99.6	\[ \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \left(\frac{\alpha + 1}{\color{blue}{\left(\alpha + 2\right) + \beta}} \cdot \frac{1}{\alpha + \left(\beta + 2\right)}\right) \]
      +-commutative [=>] 99.6	\[ \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \left(\frac{\alpha + 1}{\left(\alpha + 2\right) + \beta} \cdot \frac{1}{\alpha + \color{blue}{\left(2 + \beta\right)}}\right) \]
      associate-+r+ [=>] 99.6	\[ \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \left(\frac{\alpha + 1}{\left(\alpha + 2\right) + \beta} \cdot \frac{1}{\color{blue}{\left(\alpha + 2\right) + \beta}}\right) \]

   4. Applied egg-rr 99.8%

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

      [Start] 99.6	\[ \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \left(\frac{\alpha + 1}{\left(\alpha + 2\right) + \beta} \cdot \frac{1}{\left(\alpha + 2\right) + \beta}\right) \]
      *-commutative [=>] 99.6	\[ \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \color{blue}{\left(\frac{1}{\left(\alpha + 2\right) + \beta} \cdot \frac{\alpha + 1}{\left(\alpha + 2\right) + \beta}\right)} \]
      clear-num [=>] 99.6	\[ \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \left(\frac{1}{\left(\alpha + 2\right) + \beta} \cdot \color{blue}{\frac{1}{\frac{\left(\alpha + 2\right) + \beta}{\alpha + 1}}}\right) \]
      un-div-inv [=>] 99.8	\[ \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \color{blue}{\frac{\frac{1}{\left(\alpha + 2\right) + \beta}}{\frac{\left(\alpha + 2\right) + \beta}{\alpha + 1}}} \]
      associate-+l+ [=>] 99.8	\[ \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\frac{1}{\color{blue}{\alpha + \left(2 + \beta\right)}}}{\frac{\left(\alpha + 2\right) + \beta}{\alpha + 1}} \]
      associate-+l+ [=>] 99.8	\[ \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\frac{1}{\alpha + \left(2 + \beta\right)}}{\frac{\color{blue}{\alpha + \left(2 + \beta\right)}}{\alpha + 1}} \]

   5. Taylor expanded in beta around inf 92.1%

      \[\leadsto \color{blue}{1} \cdot \frac{\frac{1}{\alpha + \left(2 + \beta\right)}}{\frac{\alpha + \left(2 + \beta\right)}{\alpha + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+22}:\\ \;\;\;\;\left(\beta + 1\right) \cdot \frac{-1 - \alpha}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + \left(\alpha + 3\right)\right) \cdot \left(-2 - \left(\beta + \alpha\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\alpha + \left(\beta + 2\right)}}{\frac{\alpha + \left(\beta + 2\right)}{1 + \alpha}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	1728

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

Alternative 2
Accuracy	99.8%
Cost	1600

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

Alternative 3
Accuracy	98.5%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{\beta + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\left(\beta + 3\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 2\right)}}{\beta + \left(\alpha + 3\right)}\\ \end{array} \]

Alternative 4
Accuracy	98.5%
Cost	1220

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\beta + 1}{t_0 \cdot \left(\left(\beta + 3\right) \cdot \left(\beta + 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t_0}}{\frac{t_0}{1 + \alpha}}\\ \end{array} \]

Alternative 5
Accuracy	97.4%
Cost	1092

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 3\right)\\ \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 2\right)}}{t_0}\\ \end{array} \]

Alternative 6
Accuracy	97.2%
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{\beta + \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 7
Accuracy	97.3%
Cost	836

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 3\right)\\ \mathbf{if}\;\beta \leq 4.8:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 2}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{t_0}\\ \end{array} \]

Alternative 8
Accuracy	96.4%
Cost	712

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;0.16666666666666666 \cdot \frac{\beta + 1}{\beta + 2}\\ \mathbf{elif}\;\beta \leq 4.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 9
Accuracy	96.4%
Cost	712

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.9:\\ \;\;\;\;\left(\beta + 1\right) \cdot \frac{0.16666666666666666}{\beta + 2}\\ \mathbf{elif}\;\beta \leq 4.8 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 10
Accuracy	96.7%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;\left(\beta + 1\right) \cdot \frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 11
Accuracy	96.7%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 9.5:\\ \;\;\;\;\frac{0.5}{\left(\beta + 3\right) \cdot \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 12
Accuracy	96.7%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 9:\\ \;\;\;\;\frac{\frac{0.5}{\beta + 3}}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \alpha}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 13
Accuracy	55.8%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 14
Accuracy	52.9%
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 15
Accuracy	53.3%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.05:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 16
Accuracy	55.7%
Cost	452

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

Alternative 17
Accuracy	51.0%
Cost	320

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

Alternative 18
Accuracy	6.0%
Cost	192

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

Reproduce

herbie shell --seed 2023159 
(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)))