Average Error: 16.0 → 6.0
Time: 10.0s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1015227874.2528594:\\ \;\;\;\;\frac{e^{\log \left({\left(\frac{\beta}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}^{3}\right) - \log \left(\mathsf{fma}\left(\frac{\beta}{\left(\alpha + \beta\right) + 2}, \frac{\beta}{\left(\alpha + \beta\right) + 2}, \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right) \cdot \left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right) + \frac{\beta}{\left(\alpha + \beta\right) + 2}\right)\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1015227874.2528594:\\
\;\;\;\;\frac{e^{\log \left({\left(\frac{\beta}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}^{3}\right) - \log \left(\mathsf{fma}\left(\frac{\beta}{\left(\alpha + \beta\right) + 2}, \frac{\beta}{\left(\alpha + \beta\right) + 2}, \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right) \cdot \left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right) + \frac{\beta}{\left(\alpha + \beta\right) + 2}\right)\right)\right)}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r100819 = beta;
        double r100820 = alpha;
        double r100821 = r100819 - r100820;
        double r100822 = r100820 + r100819;
        double r100823 = 2.0;
        double r100824 = r100822 + r100823;
        double r100825 = r100821 / r100824;
        double r100826 = 1.0;
        double r100827 = r100825 + r100826;
        double r100828 = r100827 / r100823;
        return r100828;
}
↓
double f(double alpha, double beta) {
        double r100829 = alpha;
        double r100830 = 1015227874.2528594;
        bool r100831 = r100829 <= r100830;
        double r100832 = beta;
        double r100833 = r100829 + r100832;
        double r100834 = 2.0;
        double r100835 = r100833 + r100834;
        double r100836 = r100832 / r100835;
        double r100837 = 3.0;
        double r100838 = pow(r100836, r100837);
        double r100839 = r100829 / r100835;
        double r100840 = 1.0;
        double r100841 = r100839 - r100840;
        double r100842 = pow(r100841, r100837);
        double r100843 = r100838 - r100842;
        double r100844 = log(r100843);
        double r100845 = r100841 + r100836;
        double r100846 = r100841 * r100845;
        double r100847 = fma(r100836, r100836, r100846);
        double r100848 = log(r100847);
        double r100849 = r100844 - r100848;
        double r100850 = exp(r100849);
        double r100851 = r100850 / r100834;
        double r100852 = 4.0;
        double r100853 = r100829 * r100829;
        double r100854 = r100852 / r100853;
        double r100855 = 8.0;
        double r100856 = pow(r100829, r100837);
        double r100857 = r100855 / r100856;
        double r100858 = r100854 - r100857;
        double r100859 = r100834 / r100829;
        double r100860 = r100858 - r100859;
        double r100861 = r100836 - r100860;
        double r100862 = r100861 / r100834;
        double r100863 = r100831 ? r100851 : r100862;
        return r100863;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 1015227874.2528594

    1. Initial program 0.1

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

    3. Applied div-sub0.1

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

    4. Applied associate-+l-0.1

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

    6. Applied add-exp-log0.1

      \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}}}{2}\]
    7. Using strategy rm

    8. Applied flip3--0.1

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

    9. Applied log-div0.1

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

    10. Simplified0.1

      \[\leadsto \frac{e^{\log \left({\left(\frac{\beta}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}^{3}\right) - \color{blue}{\log \left(\mathsf{fma}\left(\frac{\beta}{\left(\alpha + \beta\right) + 2}, \frac{\beta}{\left(\alpha + \beta\right) + 2}, \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right) \cdot \left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right) + \frac{\beta}{\left(\alpha + \beta\right) + 2}\right)\right)\right)}}}{2}\]

    if 1015227874.2528594 < alpha

    1. Initial program 50.3

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

    3. Applied div-sub50.3

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

    4. Applied associate-+l-48.6

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

    5. Taylor expanded around inf 18.7

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2}\]

    6. Simplified18.7

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1015227874.2528594:\\ \;\;\;\;\frac{e^{\log \left({\left(\frac{\beta}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}^{3}\right) - \log \left(\mathsf{fma}\left(\frac{\beta}{\left(\alpha + \beta\right) + 2}, \frac{\beta}{\left(\alpha + \beta\right) + 2}, \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right) \cdot \left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right) + \frac{\beta}{\left(\alpha + \beta\right) + 2}\right)\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{8}{{\alpha}^{3}}\right) - \frac{2}{\alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))