Average Error: 29.0 → 0.2
Time: 42.0s
Precision: 64
Internal Precision: 1600
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \le 3.158716666007422 \cdot 10^{-08}:\\ \;\;\;\;\frac{\frac{\frac{1}{3}}{N \cdot N} + \left(1 - \frac{\frac{1}{2}}{N}\right)}{N}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\log \left(N + 1\right)} \cdot \sqrt[3]{\log \left(N + 1\right)}\right) \cdot \sqrt[3]{\log \left(N + 1\right)} - \log N\\ \end{array}\]

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if (- (log (+ N 1)) (log N)) < 3.158716666007422e-08

    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

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

    if 3.158716666007422e-08 < (- (log (+ N 1)) (log N))

    1. Initial program 0.3

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

    3. Applied add-cube-cbrt0.4

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

Runtime

Time bar (total: 42.0s)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))