Input Error: 40.8b
Output Error: 19.9b
Time: 19.8s
Precision: 64b
Ground Truth: 128b
\[\log \left(N + 1\right) - \log N\]
⬇
\[\begin{cases} \log \left(\frac{N + 1}{N}\right) & \text{when } N \le 2136887.5541925835 \\ \frac{1}{N} + \frac{\frac{\frac{1}{3}}{N} - \frac{1}{2}}{{N}^2} & \text{otherwise} \end{cases}\]

Error

Bits error versus N

Derivation

    if N < 2136887.5541925835

    1. Initial program 31.7b

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

    3. Applied diff-log 29.3b

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

    if 2136887.5541925835 < N

    1. Initial program 59.8b

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

    2. Applied taylor 0.0b

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

    3. Taylor expanded around inf 0.0b

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

    4. Applied simplify 0.0b

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

    5. Applied simplify 0.0b

      \[\leadsto \frac{1}{N} + \color{blue}{\frac{\frac{\frac{1}{3}}{N} - \frac{1}{2}}{{N}^2}}\]
  1. Removed slow pow expressions

Runtime

Total time: 19.8s Debug log

herbie --seed '#(1912305969 168510591 1604000121 706033541 4079405032 799739787)'
(FPCore (N)
  :name "NMSE problem 3.3.6"
  (- (log (+ N 1)) (log N)))