Average Error: 29.0 → 0.1
Time: 14.0s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7726.51655081009:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{3}}{N \cdot N}, \frac{1}{N}, \frac{1}{N} - \frac{\frac{1}{2}}{N \cdot N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7726.51655081009:\\
\;\;\;\;\log \left(\frac{1 + N}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{3}}{N \cdot N}, \frac{1}{N}, \frac{1}{N} - \frac{\frac{1}{2}}{N \cdot N}\right)\\

\end{array}
double f(double N) {
        double r1095713 = N;
        double r1095714 = 1.0;
        double r1095715 = r1095713 + r1095714;
        double r1095716 = log(r1095715);
        double r1095717 = log(r1095713);
        double r1095718 = r1095716 - r1095717;
        return r1095718;
}
↓
double f(double N) {
        double r1095719 = N;
        double r1095720 = 7726.51655081009;
        bool r1095721 = r1095719 <= r1095720;
        double r1095722 = 1.0;
        double r1095723 = r1095722 + r1095719;
        double r1095724 = r1095723 / r1095719;
        double r1095725 = log(r1095724);
        double r1095726 = 0.3333333333333333;
        double r1095727 = r1095719 * r1095719;
        double r1095728 = r1095726 / r1095727;
        double r1095729 = r1095722 / r1095719;
        double r1095730 = 0.5;
        double r1095731 = r1095730 / r1095727;
        double r1095732 = r1095729 - r1095731;
        double r1095733 = fma(r1095728, r1095729, r1095732);
        double r1095734 = r1095721 ? r1095725 : r1095733;
        return r1095734;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 7726.51655081009

    1. Initial program 0.1

      \[\log \left(N + 1\right) - \log N\]
    2. Using strategy rm

    3. Applied add-log-exp0.1

      \[\leadsto \color{blue}{\log \left(e^{\log \left(N + 1\right) - \log N}\right)}\]

    4. Simplified0.1

      \[\leadsto \log \color{blue}{\left(\frac{N + 1}{N}\right)}\]

    if 7726.51655081009 < N

    1. Initial program 59.5

      \[\log \left(N + 1\right) - \log N\]

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7726.51655081009:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{3}}{N \cdot N}, \frac{1}{N}, \frac{1}{N} - \frac{\frac{1}{2}}{N \cdot N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))