Average Error: 29.5 → 0.1
Time: 1.9m
Precision: 64
Internal Precision: 1600
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \le 0.000159168316661983:\\ \;\;\;\;(\left(\frac{1}{N \cdot N}\right) \cdot \left(\frac{\frac{1}{3}}{N} - \frac{1}{2}\right) + \left(\frac{1}{N}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\sqrt{\log \left(N + 1\right)}\right) \cdot \left(\sqrt{\log \left(N + 1\right)}\right) + \left(-\log N\right))_*\\ \end{array}\]

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if (- (log (+ N 1)) (log N)) < 0.000159168316661983

    1. Initial program 59.5

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

    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

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

    if 0.000159168316661983 < (- (log (+ N 1)) (log N))

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied fma-neg0.1

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

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2018198 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))