Average Error: 29.8 → 0.2
Time: 27.5s
Precision: 64
Internal Precision: 1408
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \le 3.0668800832245324 \cdot 10^{-09}:\\ \;\;\;\;\frac{1}{N} + \left(\frac{\frac{1}{3}}{N} - \frac{1}{2}\right) \cdot \frac{1}{N \cdot N}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{\frac{N + 1}{N}} \cdot \sqrt{\frac{N + 1}{N}}\right)\\ \end{array}\]

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if (- (log (+ N 1)) (log N)) < 3.0668800832245324e-09

    1. Initial program 60.0

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

    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. Applied simplify0.0

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

    if 3.0668800832245324e-09 < (- (log (+ N 1)) (log N))

    1. Initial program 0.5

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

    3. Applied diff-log0.4

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

    5. Applied add-sqr-sqrt0.4

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

Runtime

Time bar (total: 27.5s)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))