Average Error: 3.9 → 3.1
Time: 15.1s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 1.3363296163984674 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - \left(2 \cdot 1\right) \cdot \left(2 \cdot 1\right)} \cdot \frac{\left(\alpha + \beta\right) - 2 \cdot 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot \alpha + \left(0.5 + 0.25 \cdot \beta\right)}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - \left(2 \cdot 1\right) \cdot \left(2 \cdot 1\right)} \cdot \frac{\left(\alpha + \beta\right) - 2 \cdot 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]
\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 1.3363296163984674 \cdot 10^{154}:\\
\;\;\;\;\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - \left(2 \cdot 1\right) \cdot \left(2 \cdot 1\right)} \cdot \frac{\left(\alpha + \beta\right) - 2 \cdot 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{0.25 \cdot \alpha + \left(0.5 + 0.25 \cdot \beta\right)}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - \left(2 \cdot 1\right) \cdot \left(2 \cdot 1\right)} \cdot \frac{\left(\alpha + \beta\right) - 2 \cdot 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\

\end{array}
double f(double alpha, double beta) {
        double r108747 = alpha;
        double r108748 = beta;
        double r108749 = r108747 + r108748;
        double r108750 = r108748 * r108747;
        double r108751 = r108749 + r108750;
        double r108752 = 1.0;
        double r108753 = r108751 + r108752;
        double r108754 = 2.0;
        double r108755 = r108754 * r108752;
        double r108756 = r108749 + r108755;
        double r108757 = r108753 / r108756;
        double r108758 = r108757 / r108756;
        double r108759 = r108756 + r108752;
        double r108760 = r108758 / r108759;
        return r108760;
}
↓
double f(double alpha, double beta) {
        double r108761 = beta;
        double r108762 = 1.3363296163984674e+154;
        bool r108763 = r108761 <= r108762;
        double r108764 = alpha;
        double r108765 = r108764 + r108761;
        double r108766 = r108761 * r108764;
        double r108767 = r108765 + r108766;
        double r108768 = 1.0;
        double r108769 = r108767 + r108768;
        double r108770 = 2.0;
        double r108771 = r108770 * r108768;
        double r108772 = r108765 + r108771;
        double r108773 = r108769 / r108772;
        double r108774 = r108765 * r108765;
        double r108775 = r108771 * r108771;
        double r108776 = r108774 - r108775;
        double r108777 = r108773 / r108776;
        double r108778 = r108765 - r108771;
        double r108779 = r108772 + r108768;
        double r108780 = r108778 / r108779;
        double r108781 = r108777 * r108780;
        double r108782 = 0.25;
        double r108783 = r108782 * r108764;
        double r108784 = 0.5;
        double r108785 = r108782 * r108761;
        double r108786 = r108784 + r108785;
        double r108787 = r108783 + r108786;
        double r108788 = r108787 / r108776;
        double r108789 = r108788 * r108780;
        double r108790 = r108763 ? r108781 : r108789;
        return r108790;
}

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 < 1.3363296163984674e+154

    1. Initial program 1.2

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

    3. Applied *-un-lft-identity1.2

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

    4. Applied flip-+1.8

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

    5. Applied associate-/r/1.8

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

    6. Applied times-frac1.8

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

    7. Simplified1.8

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

    if 1.3363296163984674e+154 < beta

    1. Initial program 17.4

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

    3. Applied *-un-lft-identity17.4

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

    4. Applied flip-+20.0

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

    5. Applied associate-/r/20.0

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

    6. Applied times-frac20.0

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

    7. Simplified20.0

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

    8. Taylor expanded around 0 9.7

      \[\leadsto \frac{\color{blue}{0.25 \cdot \alpha + \left(0.5 + 0.25 \cdot \beta\right)}}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - \left(2 \cdot 1\right) \cdot \left(2 \cdot 1\right)} \cdot \frac{\left(\alpha + \beta\right) - 2 \cdot 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.3363296163984674 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - \left(2 \cdot 1\right) \cdot \left(2 \cdot 1\right)} \cdot \frac{\left(\alpha + \beta\right) - 2 \cdot 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot \alpha + \left(0.5 + 0.25 \cdot \beta\right)}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - \left(2 \cdot 1\right) \cdot \left(2 \cdot 1\right)} \cdot \frac{\left(\alpha + \beta\right) - 2 \cdot 1}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 
(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)))