Average Error: 29.3 → 0.1
Time: 23.0s
Precision: 64
Internal Precision: 1408
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 4712.227220586418:\\ \;\;\;\;\log \left(\frac{e^{\log_* (1 + N)}}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\frac{1}{3}}{N} - \frac{1}{2}\right) \cdot \left(\frac{1}{N \cdot N}\right) + \left(\frac{1}{N}\right))_*\\ \end{array}\]

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 4712.227220586418

    1. Initial program 0.1

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

    2. Applied simplify0.1

      \[\leadsto \color{blue}{\log_* (1 + N) - \log N}\]
    3. Using strategy rm

    4. Applied add-log-exp0.1

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

    5. Applied diff-log0.1

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

    if 4712.227220586418 < N

    1. Initial program 59.7

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

    2. Applied simplify59.7

      \[\leadsto \color{blue}{\log_* (1 + N) - \log N}\]

    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. Applied simplify0.0

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

Runtime

Time bar (total: 23.0s)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))