Average Error: 3.8 → 1.2
Time: 15.7s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 1.1125280590322559 \cdot 10^{161}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\left(\alpha + \beta\right) - 2 \cdot 1}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) - 2 \cdot 1\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{\left(\frac{1}{\alpha} + \frac{1}{\beta}\right) - \frac{1}{{\alpha}^{2}}}}{\left(\alpha + \beta\right) - 2 \cdot 1}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) - 2 \cdot 1\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]
\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 1.1125280590322559 \cdot 10^{161}:\\
\;\;\;\;\frac{\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\left(\alpha + \beta\right) - 2 \cdot 1}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) - 2 \cdot 1\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{\frac{1}{\left(\frac{1}{\alpha} + \frac{1}{\beta}\right) - \frac{1}{{\alpha}^{2}}}}{\left(\alpha + \beta\right) - 2 \cdot 1}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) - 2 \cdot 1\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\

\end{array}
double f(double alpha, double beta) {
        double r176803 = alpha;
        double r176804 = beta;
        double r176805 = r176803 + r176804;
        double r176806 = r176804 * r176803;
        double r176807 = r176805 + r176806;
        double r176808 = 1.0;
        double r176809 = r176807 + r176808;
        double r176810 = 2.0;
        double r176811 = r176810 * r176808;
        double r176812 = r176805 + r176811;
        double r176813 = r176809 / r176812;
        double r176814 = r176813 / r176812;
        double r176815 = r176812 + r176808;
        double r176816 = r176814 / r176815;
        return r176816;
}
↓
double f(double alpha, double beta) {
        double r176817 = beta;
        double r176818 = 1.1125280590322559e+161;
        bool r176819 = r176817 <= r176818;
        double r176820 = alpha;
        double r176821 = r176820 + r176817;
        double r176822 = r176817 * r176820;
        double r176823 = r176821 + r176822;
        double r176824 = 1.0;
        double r176825 = r176823 + r176824;
        double r176826 = 2.0;
        double r176827 = fma(r176824, r176826, r176821);
        double r176828 = r176825 / r176827;
        double r176829 = r176826 * r176824;
        double r176830 = r176821 - r176829;
        double r176831 = r176828 / r176830;
        double r176832 = r176831 / r176827;
        double r176833 = r176832 * r176830;
        double r176834 = r176821 + r176829;
        double r176835 = r176834 + r176824;
        double r176836 = r176833 / r176835;
        double r176837 = 1.0;
        double r176838 = r176837 / r176820;
        double r176839 = r176837 / r176817;
        double r176840 = r176838 + r176839;
        double r176841 = 2.0;
        double r176842 = pow(r176820, r176841);
        double r176843 = r176837 / r176842;
        double r176844 = r176840 - r176843;
        double r176845 = r176837 / r176844;
        double r176846 = r176845 / r176830;
        double r176847 = r176846 / r176827;
        double r176848 = r176847 * r176830;
        double r176849 = r176848 / r176835;
        double r176850 = r176819 ? r176836 : r176849;
        return r176850;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if beta < 1.1125280590322559e+161

    1. Initial program 1.4

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

    3. Applied flip-+2.2

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

    4. Applied associate-/r/2.2

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

    5. Simplified1.4

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

    if 1.1125280590322559e+161 < 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 flip-+18.0

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

    4. Applied associate-/r/18.0

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

    5. Simplified16.2

      \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\left(\alpha + \beta\right) - 2 \cdot 1}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}} \cdot \left(\left(\alpha + \beta\right) - 2 \cdot 1\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
    6. Using strategy rm

    7. Applied clear-num16.2

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

    8. Taylor expanded around inf 0.1

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.1125280590322559 \cdot 10^{161}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\left(\alpha + \beta\right) - 2 \cdot 1}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) - 2 \cdot 1\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{\left(\frac{1}{\alpha} + \frac{1}{\beta}\right) - \frac{1}{{\alpha}^{2}}}}{\left(\alpha + \beta\right) - 2 \cdot 1}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)} \cdot \left(\left(\alpha + \beta\right) - 2 \cdot 1\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(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)))