Average Error: 30.5 → 0.1
Time: 53.0s
Precision: 64
Internal Precision: 1344
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \le 0.0029377285253137393:\\ \;\;\;\;(\left(\frac{1}{N \cdot N}\right) \cdot \left(\frac{\frac{1}{3}}{N} - \frac{1}{2}\right) + \left(\frac{1}{N}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array}\]

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if (- (log (+ N 1)) (log N)) < 0.0029377285253137393

    1. Initial program 59.4

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

    2. Taylor expanded around inf 0.1

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

    3. Applied simplify0.1

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

    if 0.0029377285253137393 < (- (log (+ N 1)) (log N))

    1. Initial program 0.0

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

    3. Applied diff-log0.0

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

Runtime

Time bar (total: 53.0s)Debug logProfile

herbie shell --seed 2018178 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))