Average Error: 29.1 → 0.1
Time: 29.4s
Precision: 64
Internal Precision: 1600
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \le 4.1292561903105707 \cdot 10^{-07}:\\ \;\;\;\;\frac{\frac{\frac{1}{3}}{N \cdot N} + \left(1 - \frac{\frac{1}{2}}{N}\right)}{N}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\log \left(1 + N\right)\right)}^{3}} - \log N\\ \end{array}\]

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if (- (log (+ N 1)) (log N)) < 4.1292561903105707e-07

    1. Initial program 59.8

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

    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 4.1292561903105707e-07 < (- (log (+ N 1)) (log N))

    1. Initial program 0.2

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

    3. Applied add-cbrt-cube0.2

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

    4. Applied simplify0.2

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

Runtime

Time bar (total: 29.4s)Debug logProfile

herbie shell --seed '#(1071821486 549052472 3784827256 1559736200 3548510075 881134285)' 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))