Average Error: 29.3 → 1.1
Time: 11.7s
Precision: 64
Internal precision: 1408
\[\log \left(N + 1\right) - \log N\]
⬇
\[\begin{array}{l} \mathbf{if}\;N \le 2.400759143774319 \cdot 10^{+19}:\\ \;\;\;\;\log \left(N + 1\right) - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{3}}{N} - \frac{1}{2}}{{N}^2} + \frac{1}{N}\\ \end{array}\]

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes.

  2. if N < 2.400759143774319e+19

    1. Initial program 2.1

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

    if 2.400759143774319e+19 < N

    1. Initial program 60.5

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

    2. Applied taylor 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

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

    4. Applied simplify 0

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

    5. Applied simplify 0

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

Runtime

Time bar (total: 11.7s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1065033997 2389885643 4100569014 2620012693 26800780 3144211646)'
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))