Octave 3.8, jcobi/3

Percentage Accurate: 94.2% → 99.8%

Time: 25.6s

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

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

↓

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 3\right)\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 3 \cdot 10^{+18}:\\ \;\;\;\;\left(\alpha + 1\right) \cdot \frac{\frac{1 + \beta}{t_1}}{t_1 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\alpha + 1}{1 + \frac{\alpha - -1}{\beta}}}{t_0}}{t_1}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (/
  (/
   (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0)))
   (+ (+ alpha beta) (* 2.0 1.0)))
  (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ beta 3.0))) (t_1 (+ alpha (+ beta 2.0))))
   (if (<= beta 3e+18)
     (* (+ alpha 1.0) (/ (/ (+ 1.0 beta) t_1) (* t_1 t_0)))
     (/ (/ (/ (+ alpha 1.0) (+ 1.0 (/ (- alpha -1.0) beta))) t_0) t_1))))

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 + 3.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 3e+18) {
		tmp = (alpha + 1.0) * (((1.0 + beta) / t_1) / (t_1 * t_0));
	} else {
		tmp = (((alpha + 1.0) / (1.0 + ((alpha - -1.0) / beta))) / t_0) / t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = alpha + (beta + 3.0d0)
    t_1 = alpha + (beta + 2.0d0)
    if (beta <= 3d+18) then
        tmp = (alpha + 1.0d0) * (((1.0d0 + beta) / t_1) / (t_1 * t_0))
    else
        tmp = (((alpha + 1.0d0) / (1.0d0 + ((alpha - (-1.0d0)) / beta))) / t_0) / t_1
    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 + 3.0);
	double t_1 = alpha + (beta + 2.0);
	double tmp;
	if (beta <= 3e+18) {
		tmp = (alpha + 1.0) * (((1.0 + beta) / t_1) / (t_1 * t_0));
	} else {
		tmp = (((alpha + 1.0) / (1.0 + ((alpha - -1.0) / beta))) / t_0) / t_1;
	}
	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 + 3.0)
	t_1 = alpha + (beta + 2.0)
	tmp = 0
	if beta <= 3e+18:
		tmp = (alpha + 1.0) * (((1.0 + beta) / t_1) / (t_1 * t_0))
	else:
		tmp = (((alpha + 1.0) / (1.0 + ((alpha - -1.0) / beta))) / t_0) / t_1
	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 + 3.0))
	t_1 = Float64(alpha + Float64(beta + 2.0))
	tmp = 0.0
	if (beta <= 3e+18)
		tmp = Float64(Float64(alpha + 1.0) * Float64(Float64(Float64(1.0 + beta) / t_1) / Float64(t_1 * t_0)));
	else
		tmp = Float64(Float64(Float64(Float64(alpha + 1.0) / Float64(1.0 + Float64(Float64(alpha - -1.0) / beta))) / t_0) / t_1);
	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 + 3.0);
	t_1 = alpha + (beta + 2.0);
	tmp = 0.0;
	if (beta <= 3e+18)
		tmp = (alpha + 1.0) * (((1.0 + beta) / t_1) / (t_1 * t_0));
	else
		tmp = (((alpha + 1.0) / (1.0 + ((alpha - -1.0) / beta))) / t_0) / t_1;
	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 + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3e+18], N[(N[(alpha + 1.0), $MachinePrecision] * N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(1.0 + N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3e18

   1. Initial program 99.9%

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

   2. Simplified 94.6%

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

      [Start] 99.9	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      associate-/l/ [=>] 99.9	\[ \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]
      associate-+l+ [=>] 99.9	\[ \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      +-commutative [=>] 99.9	\[ \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      associate-+r+ [=>] 99.9	\[ \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      associate-+l+ [=>] 99.9	\[ \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      distribute-rgt1-in [=>] 99.9	\[ \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      *-rgt-identity [<=] 99.9	\[ \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      distribute-lft-out [=>] 99.9	\[ \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      +-commutative [=>] 99.9	\[ \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      associate-*l/ [<=] 99.9	\[ \frac{\color{blue}{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      *-commutative [=>] 99.9	\[ \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      associate-*r/ [<=] 94.6	\[ \color{blue}{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   if 3e18 < beta

   1. Initial program 88.9%

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

   2. Simplified 90.9%

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

      [Start] 88.9	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      associate-/l/ [=>] 83.5	\[ \color{blue}{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]
      associate-+l+ [=>] 83.5	\[ \frac{\frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      +-commutative [=>] 83.5	\[ \frac{\frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      associate-+r+ [=>] 83.5	\[ \frac{\frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      associate-+l+ [=>] 83.5	\[ \frac{\frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      distribute-rgt1-in [=>] 83.5	\[ \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      *-rgt-identity [<=] 83.5	\[ \frac{\frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      distribute-lft-out [=>] 83.5	\[ \frac{\frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      +-commutative [=>] 83.5	\[ \frac{\frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      associate-*l/ [<=] 91.1	\[ \frac{\color{blue}{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      *-commutative [=>] 91.1	\[ \frac{\color{blue}{\left(1 + \alpha\right) \cdot \frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)} \]
      associate-*r/ [<=] 90.9	\[ \color{blue}{\left(1 + \alpha\right) \cdot \frac{\frac{\beta + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Applied egg-rr 99.7%

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

      [Start] 90.9	\[ \left(\alpha + 1\right) \cdot \frac{\frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)} \]
      associate-*r/ [=>] 91.1	\[ \color{blue}{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 3\right)\right)}} \]
      *-commutative [=>] 91.1	\[ \frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\color{blue}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]
      associate-/r* [=>] 99.7	\[ \color{blue}{\frac{\frac{\left(\alpha + 1\right) \cdot \frac{\beta + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}} \]
      clear-num [=>] 99.7	\[ \frac{\frac{\left(\alpha + 1\right) \cdot \color{blue}{\frac{1}{\frac{\alpha + \left(\beta + 2\right)}{\beta + 1}}}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)} \]
      un-div-inv [=>] 99.7	\[ \frac{\frac{\color{blue}{\frac{\alpha + 1}{\frac{\alpha + \left(\beta + 2\right)}{\beta + 1}}}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)} \]
      +-commutative [=>] 99.7	\[ \frac{\frac{\frac{\alpha + 1}{\frac{\alpha + \left(\beta + 2\right)}{\color{blue}{1 + \beta}}}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

   4. Taylor expanded in beta around -inf 99.7%

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

   5. Simplified 99.7%

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

      [Start] 99.7	\[ \frac{\frac{\frac{\alpha + 1}{1 + -1 \cdot \frac{1 + -1 \cdot \left(2 + \alpha\right)}{\beta}}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)} \]
      mul-1-neg [=>] 99.7	\[ \frac{\frac{\frac{\alpha + 1}{1 + \color{blue}{\left(-\frac{1 + -1 \cdot \left(2 + \alpha\right)}{\beta}\right)}}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)} \]
      unsub-neg [=>] 99.7	\[ \frac{\frac{\frac{\alpha + 1}{\color{blue}{1 - \frac{1 + -1 \cdot \left(2 + \alpha\right)}{\beta}}}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)} \]
      distribute-lft-in [=>] 99.7	\[ \frac{\frac{\frac{\alpha + 1}{1 - \frac{1 + \color{blue}{\left(-1 \cdot 2 + -1 \cdot \alpha\right)}}{\beta}}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)} \]
      metadata-eval [=>] 99.7	\[ \frac{\frac{\frac{\alpha + 1}{1 - \frac{1 + \left(\color{blue}{-2} + -1 \cdot \alpha\right)}{\beta}}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)} \]
      neg-mul-1 [<=] 99.7	\[ \frac{\frac{\frac{\alpha + 1}{1 - \frac{1 + \left(-2 + \color{blue}{\left(-\alpha\right)}\right)}{\beta}}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)} \]
      associate-+r+ [=>] 99.7	\[ \frac{\frac{\frac{\alpha + 1}{1 - \frac{\color{blue}{\left(1 + -2\right) + \left(-\alpha\right)}}{\beta}}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)} \]
      metadata-eval [=>] 99.7	\[ \frac{\frac{\frac{\alpha + 1}{1 - \frac{\color{blue}{-1} + \left(-\alpha\right)}{\beta}}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)} \]
      unsub-neg [=>] 99.7	\[ \frac{\frac{\frac{\alpha + 1}{1 - \frac{\color{blue}{-1 - \alpha}}{\beta}}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 96.2%

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

Alternatives

Alternative 1
Accuracy	98.9%
Cost	1604

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1520000000:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + \left(\alpha + 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\alpha + 1}{1 + \frac{\alpha - -1}{\beta}}}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]

Alternative 2
Accuracy	99.8%
Cost	1600

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

Alternative 3
Accuracy	99.7%
Cost	1600

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

Alternative 4
Accuracy	98.5%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.06 \cdot 10^{+17}:\\ \;\;\;\;\frac{1 + \beta}{\left(\beta + 3\right) \cdot \left(\left(\beta + 2\right) \cdot \left(\beta + \left(\alpha + 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{\alpha + \left(\beta + 2\right)}\\ \end{array} \]

Alternative 5
Accuracy	97.2%
Cost	1092

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 1.66:\\ \;\;\;\;\frac{0.16666666666666666 + \beta \cdot 0.027777777777777776}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{t_0}\\ \end{array} \]

Alternative 6
Accuracy	97.2%
Cost	1092

\[\begin{array}{l} t_0 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\frac{\frac{1 + \beta}{6 + \beta \cdot 5}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 3\right)}}{t_0}\\ \end{array} \]

Alternative 7
Accuracy	97.0%
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.8:\\ \;\;\;\;\frac{0.16666666666666666 + \beta \cdot 0.027777777777777776}{\alpha + \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 8
Accuracy	97.1%
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.4:\\ \;\;\;\;\frac{0.16666666666666666 + \beta \cdot 0.027777777777777776}{\alpha + \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\alpha + \left(\beta + 3\right)}\\ \end{array} \]

Alternative 9
Accuracy	93.6%
Cost	712

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.4:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + \left(\beta + 2\right)}\\ \mathbf{elif}\;\beta \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 10
Accuracy	96.3%
Cost	712

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.5:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + \left(\beta + 2\right)}\\ \mathbf{elif}\;\beta \leq 1.6 \cdot 10^{+154}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 11
Accuracy	96.7%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.5:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + \left(\beta + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta} \cdot \frac{1}{\beta}\\ \end{array} \]

Alternative 12
Accuracy	93.6%
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.7:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{elif}\;\beta \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 13
Accuracy	93.6%
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.8:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + \left(\beta + 2\right)}\\ \mathbf{elif}\;\beta \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 14
Accuracy	91.0%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.7:\\ \;\;\;\;\frac{0.16666666666666666}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 15
Accuracy	45.5%
Cost	320

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

Alternative 16
Accuracy	3.7%
Cost	192

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

Alternative 17
Accuracy	6.0%
Cost	192

\[\frac{0.3333333333333333}{\beta} \]

Reproduce

herbie shell --seed 2023160 
(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)))