Average Error: 29.3 → 0.3
Time: 40.0s
Precision: 64
Internal Precision: 1408
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \le 1.2434497875801753 \cdot 10^{-13}:\\ \;\;\;\;\left(\frac{1}{N} - \log 1\right) - \frac{\frac{\frac{1}{2}}{N}}{N}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array}\]

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if (- (log (+ N 1)) (log N)) < 1.2434497875801753e-13

    1. Initial program 60.5

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

    2. Taylor expanded around inf 60.5

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

    3. Applied simplify0.0

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

    if 1.2434497875801753e-13 < (- (log (+ N 1)) (log N))

    1. Initial program 0.8

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

    3. Applied diff-log0.6

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

Runtime

Time bar (total: 40.0s)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))