Average Error: 29.6 → 0.2
Time: 22.3s
Precision: 64
Internal Precision: 1408
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 5887.230341303003:\\ \;\;\;\;\frac{{\left(\log \left(N + 1\right)\right)}^{3} - {\left(\log N\right)}^{3}}{\log \left(1 + N\right) \cdot \log \left(1 + N\right) + \left(\log N + \log \left(1 + N\right)\right) \cdot \log N}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \left(\frac{\frac{1}{3}}{N} - \frac{1}{2}\right) \cdot \frac{1}{N \cdot N}\\ \end{array}\]

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 5887.230341303003

    1. Initial program 0.1

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

    3. Applied flip3--0.3

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

    4. Applied simplify0.3

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

    if 5887.230341303003 < N

    1. Initial program 59.5

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

    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

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

Runtime

Time bar (total: 22.3s)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))