Average Error: 29.6 → 0.1
Time: 9.4s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8386.707428118045:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{N}}{N}, \frac{\frac{1}{3}}{N} - \frac{1}{2}, \frac{1}{N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8386.707428118045:\\
\;\;\;\;\log \left(\frac{1 + N}{N}\right)\\

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

\end{array}
double f(double N) {
        double r1048910 = N;
        double r1048911 = 1.0;
        double r1048912 = r1048910 + r1048911;
        double r1048913 = log(r1048912);
        double r1048914 = log(r1048910);
        double r1048915 = r1048913 - r1048914;
        return r1048915;
}
↓
double f(double N) {
        double r1048916 = N;
        double r1048917 = 8386.707428118045;
        bool r1048918 = r1048916 <= r1048917;
        double r1048919 = 1.0;
        double r1048920 = r1048919 + r1048916;
        double r1048921 = r1048920 / r1048916;
        double r1048922 = log(r1048921);
        double r1048923 = r1048919 / r1048916;
        double r1048924 = r1048923 / r1048916;
        double r1048925 = 0.3333333333333333;
        double r1048926 = r1048925 / r1048916;
        double r1048927 = 0.5;
        double r1048928 = r1048926 - r1048927;
        double r1048929 = fma(r1048924, r1048928, r1048923);
        double r1048930 = r1048918 ? r1048922 : r1048929;
        return r1048930;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 8386.707428118045

    1. Initial program 0.1

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

    2. Simplified0.1

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N}\]
    3. Using strategy rm

    4. Applied log1p-udef0.1

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

    5. Applied diff-log0.1

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

    if 8386.707428118045 < N

    1. Initial program 59.4

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

    2. Simplified59.4

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N}\]

    3. Taylor expanded around -inf 0.0

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

    4. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8386.707428118045:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{N}}{N}, \frac{\frac{1}{3}}{N} - \frac{1}{2}, \frac{1}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))