Average Error: 29.4 → 0.5
Time: 8.1s
Precision: 64
Internal precision: 1408
\[\log \left(N + 1\right) - \log N\]
⬇
\[\begin{array}{l} \mathbf{if}\;N \le 591618366256079.6:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{3}}{N} - \frac{1}{2}}{{N}^2} + \frac{1}{N}\\ \end{array}\]

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes.

  2. if N < 591618366256079.6

    1. Initial program 1.1

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

    3. Applied diff-log 0.9

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

    if 591618366256079.6 < N

    1. Initial program 60.5

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

    2. Applied taylor 0.0

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

    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 simplify 0.0

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

    5. Applied simplify 0.0

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{3}}{N} - \frac{1}{2}}{{N}^2}} + \frac{1}{N}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 8.1s) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(3052192724 3812927732 3686175817 630908657 2373248591 511094450)'
(FPCore (N)
  :name "NMSE problem 3.3.6"
  :pre (> N 0)
  (- (log (+ N 1)) (log N)))