Average Error: 29.8 → 0.2
Time: 33.1s
Precision: 64
Internal Precision: 1344
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \le 1.0007174098329816 \cdot 10^{-09}:\\ \;\;\;\;(\left(\frac{1}{N \cdot N}\right) \cdot \left(\frac{\frac{1}{3}}{N} - \frac{1}{2}\right) + \left(\frac{1}{N}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\log_* (1 + N)} \cdot \sqrt[3]{\log_* (1 + N)}\right) \cdot \sqrt[3]{\log_* (1 + N)} - \log N\\ \end{array}\]

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if (- (log (+ N 1)) (log N)) < 1.0007174098329816e-09

    1. Initial program 60.1

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

    2. Applied simplify60.1

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

    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

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

    if 1.0007174098329816e-09 < (- (log (+ N 1)) (log N))

    1. Initial program 0.4

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

    2. Applied simplify0.4

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

    4. Applied add-cube-cbrt0.5

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

Runtime

Time bar (total: 33.1s)Debug logProfile

herbie shell --seed 2018193 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))