Average Error: 29.4 → 0.1
Time: 16.0s
Precision: 64
Internal Precision: 576
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8142.718291694942:\\ \;\;\;\;\frac{1}{2} \cdot \log \left(\frac{N + 1}{N}\right) + \frac{1}{2} \cdot \log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{1}{N \cdot N}\right) \cdot \left(\frac{\frac{1}{3}}{N} - \frac{1}{2}\right) + \left(\frac{1}{N}\right))_*\\ \end{array}\]

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 8142.718291694942

    1. Initial program 0.1

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

    3. Applied diff-log0.1

      \[\leadsto \color{blue}{\log \left(\frac{N + 1}{N}\right)}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt0.1

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

    6. Applied log-prod0.1

      \[\leadsto \color{blue}{\log \left(\sqrt{\frac{N + 1}{N}}\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)}\]
    7. Using strategy rm

    8. Applied pow1/20.1

      \[\leadsto \log \color{blue}{\left({\left(\frac{N + 1}{N}\right)}^{\frac{1}{2}}\right)} + \log \left(\sqrt{\frac{N + 1}{N}}\right)\]

    9. Applied log-pow0.1

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \log \left(\frac{N + 1}{N}\right)} + \log \left(\sqrt{\frac{N + 1}{N}}\right)\]
    10. Using strategy rm

    11. Applied pow1/20.1

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

    12. Applied log-pow0.1

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

    if 8142.718291694942 < N

    1. Initial program 59.6

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

    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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8142.718291694942:\\ \;\;\;\;\frac{1}{2} \cdot \log \left(\frac{N + 1}{N}\right) + \frac{1}{2} \cdot \log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{1}{N \cdot N}\right) \cdot \left(\frac{\frac{1}{3}}{N} - \frac{1}{2}\right) + \left(\frac{1}{N}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 16.0s)Debug logProfile

herbie shell --seed 2018256 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))