Average Error: 3.9 → 2.3
Time: 25.1s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 3.78966527199751605985093206761738747102 \cdot 10^{160}:\\ \;\;\;\;\frac{\frac{\frac{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{1 \cdot 2 + \left(\left(\alpha + \beta\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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}\;\alpha \le 3.78966527199751605985093206761738747102 \cdot 10^{160}:\\
\;\;\;\;\frac{\frac{\frac{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{1 \cdot 2 + \left(\left(\alpha + \beta\right) + 1\right)}\\

\mathbf{else}:\\
\;\;\;\;0\\

\end{array}
double f(double alpha, double beta) {
        double r68873 = alpha;
        double r68874 = beta;
        double r68875 = r68873 + r68874;
        double r68876 = r68874 * r68873;
        double r68877 = r68875 + r68876;
        double r68878 = 1.0;
        double r68879 = r68877 + r68878;
        double r68880 = 2.0;
        double r68881 = r68880 * r68878;
        double r68882 = r68875 + r68881;
        double r68883 = r68879 / r68882;
        double r68884 = r68883 / r68882;
        double r68885 = r68882 + r68878;
        double r68886 = r68884 / r68885;
        return r68886;
}
↓
double f(double alpha, double beta) {
        double r68887 = alpha;
        double r68888 = 3.789665271997516e+160;
        bool r68889 = r68887 <= r68888;
        double r68890 = 1.0;
        double r68891 = beta;
        double r68892 = r68887 + r68891;
        double r68893 = fma(r68887, r68891, r68892);
        double r68894 = r68890 + r68893;
        double r68895 = 2.0;
        double r68896 = fma(r68890, r68895, r68892);
        double r68897 = r68894 / r68896;
        double r68898 = r68897 / r68896;
        double r68899 = r68890 * r68895;
        double r68900 = r68892 + r68890;
        double r68901 = r68899 + r68900;
        double r68902 = r68898 / r68901;
        double r68903 = 0.0;
        double r68904 = r68889 ? r68902 : r68903;
        return r68904;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 3.789665271997516e+160

    1. Initial program 1.3

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

    2. Simplified1.3

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

    4. Applied fma-udef1.3

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

    5. Applied associate-+l+1.3

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

    if 3.789665271997516e+160 < alpha

    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. Simplified16.2

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

    3. Taylor expanded around inf 7.2

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3.78966527199751605985093206761738747102 \cdot 10^{160}:\\ \;\;\;\;\frac{\frac{\frac{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}{1 \cdot 2 + \left(\left(\alpha + \beta\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +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)))