Octave 3.8, jcobi/3

Average Accuracy: 94.3% → 99.7%

Time: 23.4s

Precision: binary64

Cost: 1732

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \alpha + \left(2 + \beta\right)\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+138}:\\ \;\;\;\;\left(\alpha + 1\right) \cdot \frac{\frac{1 + \beta}{t_0}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\alpha + 2\right) + \beta}}{\left(\beta + 4\right) + \alpha \cdot 2}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ alpha (+ 2.0 beta))))
   (if (<= beta 5e+138)
     (* (+ alpha 1.0) (/ (/ (+ 1.0 beta) t_0) (* (+ alpha (+ beta 3.0)) t_0)))
     (/
      (/ (+ alpha 1.0) (+ (+ alpha 2.0) beta))
      (+ (+ beta 4.0) (* alpha 2.0))))))

C

double code(double alpha, double beta) {
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0);
}

↓

double code(double alpha, double beta) {
	double t_0 = alpha + (2.0 + beta);
	double tmp;
	if (beta <= 5e+138) {
		tmp = (alpha + 1.0) * (((1.0 + beta) / t_0) / ((alpha + (beta + 3.0)) * t_0));
	} else {
		tmp = ((alpha + 1.0) / ((alpha + 2.0) + beta)) / ((beta + 4.0) + (alpha * 2.0));
	}
	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 = alpha + (2.0d0 + beta)
    if (beta <= 5d+138) then
        tmp = (alpha + 1.0d0) * (((1.0d0 + beta) / t_0) / ((alpha + (beta + 3.0d0)) * t_0))
    else
        tmp = ((alpha + 1.0d0) / ((alpha + 2.0d0) + beta)) / ((beta + 4.0d0) + (alpha * 2.0d0))
    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 + (2.0 + beta);
	double tmp;
	if (beta <= 5e+138) {
		tmp = (alpha + 1.0) * (((1.0 + beta) / t_0) / ((alpha + (beta + 3.0)) * t_0));
	} else {
		tmp = ((alpha + 1.0) / ((alpha + 2.0) + beta)) / ((beta + 4.0) + (alpha * 2.0));
	}
	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 = alpha + (2.0 + beta)
	tmp = 0
	if beta <= 5e+138:
		tmp = (alpha + 1.0) * (((1.0 + beta) / t_0) / ((alpha + (beta + 3.0)) * t_0))
	else:
		tmp = ((alpha + 1.0) / ((alpha + 2.0) + beta)) / ((beta + 4.0) + (alpha * 2.0))
	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(alpha + Float64(2.0 + beta))
	tmp = 0.0
	if (beta <= 5e+138)
		tmp = Float64(Float64(alpha + 1.0) * Float64(Float64(Float64(1.0 + beta) / t_0) / Float64(Float64(alpha + Float64(beta + 3.0)) * t_0)));
	else
		tmp = Float64(Float64(Float64(alpha + 1.0) / Float64(Float64(alpha + 2.0) + beta)) / Float64(Float64(beta + 4.0) + Float64(alpha * 2.0)));
	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 = alpha + (2.0 + beta);
	tmp = 0.0;
	if (beta <= 5e+138)
		tmp = (alpha + 1.0) * (((1.0 + beta) / t_0) / ((alpha + (beta + 3.0)) * t_0));
	else
		tmp = ((alpha + 1.0) / ((alpha + 2.0) + beta)) / ((beta + 4.0) + (alpha * 2.0));
	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[(alpha + N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5e+138], N[(N[(alpha + 1.0), $MachinePrecision] * N[(N[(N[(1.0 + beta), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(alpha + N[(beta + 3.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + 1.0), $MachinePrecision] / N[(N[(alpha + 2.0), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(beta + 4.0), $MachinePrecision] + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.00000000000000016e138

   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%

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

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

   if 5.00000000000000016e138 < beta

   1. Initial program 83.5%

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

   2. Simplified 90.7%

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

      [Start] 83.5	\[ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      associate-/l/ [=>] 79.7	\[ \frac{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \]
      associate-/l/ [=>] 72.6	\[ \color{blue}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)}} \]
      associate-+l+ [=>] 72.6	\[ \frac{\color{blue}{\left(\alpha + \left(\beta + \beta \cdot \alpha\right)\right)} + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]
      +-commutative [=>] 72.6	\[ \frac{\left(\alpha + \color{blue}{\left(\beta \cdot \alpha + \beta\right)}\right) + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]
      associate-+r+ [=>] 72.6	\[ \frac{\color{blue}{\left(\left(\alpha + \beta \cdot \alpha\right) + \beta\right)} + 1}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]
      associate-+l+ [=>] 72.6	\[ \frac{\color{blue}{\left(\alpha + \beta \cdot \alpha\right) + \left(\beta + 1\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]
      distribute-rgt1-in [=>] 72.6	\[ \frac{\color{blue}{\left(\beta + 1\right) \cdot \alpha} + \left(\beta + 1\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]
      *-rgt-identity [<=] 72.6	\[ \frac{\left(\beta + 1\right) \cdot \alpha + \color{blue}{\left(\beta + 1\right) \cdot 1}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]
      distribute-lft-out [=>] 72.6	\[ \frac{\color{blue}{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]
      +-commutative [=>] 72.6	\[ \frac{\left(\beta + 1\right) \cdot \color{blue}{\left(1 + \alpha\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)\right)} \]
      times-frac [=>] 90.7	\[ \color{blue}{\frac{\beta + 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1} \cdot \frac{1 + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}} \]

   3. Applied egg-rr 99.9%

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

      [Start] 90.7	\[ \frac{\beta + 1}{\alpha + \left(\beta + 3\right)} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
      clear-num [=>] 90.7	\[ \color{blue}{\frac{1}{\frac{\alpha + \left(\beta + 3\right)}{\beta + 1}}} \cdot \frac{\alpha + 1}{\left(\alpha + \left(\beta + 2\right)\right) \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
      associate-/r* [=>] 99.9	\[ \frac{1}{\frac{\alpha + \left(\beta + 3\right)}{\beta + 1}} \cdot \color{blue}{\frac{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\alpha + \left(\beta + 2\right)}} \]
      frac-times [=>] 99.9	\[ \color{blue}{\frac{1 \cdot \frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}{\frac{\alpha + \left(\beta + 3\right)}{\beta + 1} \cdot \left(\alpha + \left(\beta + 2\right)\right)}} \]
      *-un-lft-identity [<=] 99.9	\[ \frac{\color{blue}{\frac{\alpha + 1}{\alpha + \left(\beta + 2\right)}}}{\frac{\alpha + \left(\beta + 3\right)}{\beta + 1} \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
      +-commutative [=>] 99.9	\[ \frac{\frac{\alpha + 1}{\alpha + \color{blue}{\left(2 + \beta\right)}}}{\frac{\alpha + \left(\beta + 3\right)}{\beta + 1} \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
      associate-+r+ [=>] 99.9	\[ \frac{\frac{\alpha + 1}{\color{blue}{\left(\alpha + 2\right) + \beta}}}{\frac{\alpha + \left(\beta + 3\right)}{\beta + 1} \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
      +-commutative [=>] 99.9	\[ \frac{\frac{\alpha + 1}{\left(\alpha + 2\right) + \beta}}{\frac{\alpha + \color{blue}{\left(3 + \beta\right)}}{\beta + 1} \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
      associate-+r+ [=>] 99.9	\[ \frac{\frac{\alpha + 1}{\left(\alpha + 2\right) + \beta}}{\frac{\color{blue}{\left(\alpha + 3\right) + \beta}}{\beta + 1} \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
      +-commutative [=>] 99.9	\[ \frac{\frac{\alpha + 1}{\left(\alpha + 2\right) + \beta}}{\frac{\left(\alpha + 3\right) + \beta}{\color{blue}{1 + \beta}} \cdot \left(\alpha + \left(\beta + 2\right)\right)} \]
      +-commutative [=>] 99.9	\[ \frac{\frac{\alpha + 1}{\left(\alpha + 2\right) + \beta}}{\frac{\left(\alpha + 3\right) + \beta}{1 + \beta} \cdot \left(\alpha + \color{blue}{\left(2 + \beta\right)}\right)} \]
      associate-+r+ [=>] 99.9	\[ \frac{\frac{\alpha + 1}{\left(\alpha + 2\right) + \beta}}{\frac{\left(\alpha + 3\right) + \beta}{1 + \beta} \cdot \color{blue}{\left(\left(\alpha + 2\right) + \beta\right)}} \]

   4. Taylor expanded in beta around inf 99.5%

      \[\leadsto \frac{\frac{\alpha + 1}{\left(\alpha + 2\right) + \beta}}{\color{blue}{\beta + \left(4 + 2 \cdot \alpha\right)}} \]

   5. Simplified 99.5%

      \[\leadsto \frac{\frac{\alpha + 1}{\left(\alpha + 2\right) + \beta}}{\color{blue}{\left(\beta + 4\right) + 2 \cdot \alpha}} \]
      Proof

      [Start] 99.5	\[ \frac{\frac{\alpha + 1}{\left(\alpha + 2\right) + \beta}}{\beta + \left(4 + 2 \cdot \alpha\right)} \]
      associate-+r+ [=>] 99.5	\[ \frac{\frac{\alpha + 1}{\left(\alpha + 2\right) + \beta}}{\color{blue}{\left(\beta + 4\right) + 2 \cdot \alpha}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+138}:\\ \;\;\;\;\left(\alpha + 1\right) \cdot \frac{\frac{1 + \beta}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + \left(\beta + 3\right)\right) \cdot \left(\alpha + \left(2 + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\alpha + 2\right) + \beta}}{\left(\beta + 4\right) + \alpha \cdot 2}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	1600

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

Alternative 2
Accuracy	99.7%
Cost	1600

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

Alternative 3
Accuracy	99.7%
Cost	1600

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

Alternative 4
Accuracy	99.8%
Cost	1600

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

Alternative 5
Accuracy	98.0%
Cost	1220

\[\begin{array}{l} t_0 := \frac{\alpha + 1}{\left(\alpha + 2\right) + \beta}\\ \mathbf{if}\;\beta \leq 7.6:\\ \;\;\;\;\frac{t_0}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\beta + 4}\\ \end{array} \]

Alternative 6
Accuracy	98.0%
Cost	1220

\[\begin{array}{l} t_0 := \frac{\alpha + 1}{\left(\alpha + 2\right) + \beta}\\ \mathbf{if}\;\beta \leq 3.05:\\ \;\;\;\;\frac{t_0}{6 + \alpha \cdot \left(\alpha + 5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\beta + 4}\\ \end{array} \]

Alternative 7
Accuracy	98.3%
Cost	1220

\[\begin{array}{l} t_0 := \frac{\alpha + 1}{\left(\alpha + 2\right) + \beta}\\ \mathbf{if}\;\beta \leq 2.8:\\ \;\;\;\;\frac{t_0}{6 + \alpha \cdot \left(\alpha + 5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\left(\beta + 4\right) + \alpha \cdot 2}\\ \end{array} \]

Alternative 8
Accuracy	98.0%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.45:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\alpha + 2\right) + \beta}}{\beta + 4}\\ \end{array} \]

Alternative 9
Accuracy	97.0%
Cost	964

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.5:\\ \;\;\;\;\frac{\alpha + 1}{\left(\alpha + 2\right) \cdot \left(6 + \alpha \cdot 5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\alpha + 2\right) + \beta}}{\beta}\\ \end{array} \]

Alternative 10
Accuracy	97.6%
Cost	964

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.62:\\ \;\;\;\;\frac{\alpha + 1}{\left(\alpha + 2\right) \cdot \left(6 + \alpha \cdot 5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\alpha + 2\right) + \beta}}{\beta + 4}\\ \end{array} \]

Alternative 11
Accuracy	96.6%
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\left(\alpha + 2\right) + \beta}}{\beta}\\ \end{array} \]

Alternative 12
Accuracy	96.0%
Cost	712

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.6:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \beta}\\ \mathbf{elif}\;\beta \leq 1.95 \cdot 10^{+160}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 13
Accuracy	93.6%
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.6:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \beta}\\ \mathbf{elif}\;\beta \leq 2.6 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 14
Accuracy	96.6%
Cost	580

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

Alternative 15
Accuracy	91.0%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.6:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 16
Accuracy	91.4%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.6:\\ \;\;\;\;\frac{0.16666666666666666}{2 + \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \end{array} \]

Alternative 17
Accuracy	47.1%
Cost	320

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

Alternative 18
Accuracy	2.5%
Cost	192

\[\frac{0.2}{\alpha} \]

Reproduce

herbie shell --seed 2023153 
(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)))