Average Error: 3.6 → 1.5
Time: 32.8s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 2.458067370716520590176800600215761955635 \cdot 10^{153}:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - \left(2 \cdot 1\right) \cdot \left(2 \cdot 1\right)}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(\left(\alpha + \beta\right) - 2 \cdot 1\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]
\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}
\mathbf{if}\;\beta \le 2.458067370716520590176800600215761955635 \cdot 10^{153}:\\
\;\;\;\;\frac{\frac{1}{\frac{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - \left(2 \cdot 1\right) \cdot \left(2 \cdot 1\right)}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(\left(\alpha + \beta\right) - 2 \cdot 1\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\

\end{array}
double f(double alpha, double beta) {
        double r148693 = alpha;
        double r148694 = beta;
        double r148695 = r148693 + r148694;
        double r148696 = r148694 * r148693;
        double r148697 = r148695 + r148696;
        double r148698 = 1.0;
        double r148699 = r148697 + r148698;
        double r148700 = 2.0;
        double r148701 = r148700 * r148698;
        double r148702 = r148695 + r148701;
        double r148703 = r148699 / r148702;
        double r148704 = r148703 / r148702;
        double r148705 = r148702 + r148698;
        double r148706 = r148704 / r148705;
        return r148706;
}
↓
double f(double alpha, double beta) {
        double r148707 = beta;
        double r148708 = 2.4580673707165206e+153;
        bool r148709 = r148707 <= r148708;
        double r148710 = 1.0;
        double r148711 = alpha;
        double r148712 = r148711 + r148707;
        double r148713 = r148712 * r148712;
        double r148714 = 2.0;
        double r148715 = 1.0;
        double r148716 = r148714 * r148715;
        double r148717 = r148716 * r148716;
        double r148718 = r148713 - r148717;
        double r148719 = r148707 * r148711;
        double r148720 = r148712 + r148719;
        double r148721 = r148720 + r148715;
        double r148722 = r148712 + r148716;
        double r148723 = r148721 / r148722;
        double r148724 = r148712 - r148716;
        double r148725 = r148723 * r148724;
        double r148726 = r148718 / r148725;
        double r148727 = r148710 / r148726;
        double r148728 = r148722 + r148715;
        double r148729 = r148727 / r148728;
        double r148730 = 2.0;
        double r148731 = r148707 / r148711;
        double r148732 = r148711 / r148707;
        double r148733 = r148731 + r148732;
        double r148734 = r148730 + r148733;
        double r148735 = r148710 / r148734;
        double r148736 = r148735 / r148728;
        double r148737 = r148709 ? r148729 : r148736;
        return r148737;
}

Error

Bits error versus alpha

Bits error versus beta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if beta < 2.4580673707165206e+153

    1. Initial program 1.0

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

    3. Applied clear-num1.0

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

    5. Applied flip-+1.6

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

    6. Applied associate-/l/1.6

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

    if 2.4580673707165206e+153 < beta

    1. Initial program 16.2

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

    3. Applied clear-num16.2

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

    4. Taylor expanded around inf 1.1

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 2.458067370716520590176800600215761955635 \cdot 10^{153}:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - \left(2 \cdot 1\right) \cdot \left(2 \cdot 1\right)}{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(\left(\alpha + \beta\right) - 2 \cdot 1\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2 + \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1)))