Average Error: 29.4 → 0.1
Time: 13.5s
Precision: 64
Internal Precision: 128
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 6703.766594082657:\\ \;\;\;\;\left(\log \left(\sqrt{\sqrt{\frac{1 + N}{N}}}\right) + \log \left(\sqrt{\sqrt{\frac{1 + N}{N}}}\right)\right) + \left(\log \left(\sqrt{1 + N}\right) - \log \left(\sqrt{N}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \left(\frac{1}{2} - \frac{\frac{1}{3}}{N}\right) \cdot \frac{1}{N \cdot N}\\ \end{array}\]

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if N < 6703.766594082657

    1. Initial program 0.1

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

    3. Applied diff-log0.1

      \[\leadsto \color{blue}{\log \left(\frac{N + 1}{N}\right)}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt0.1

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

    6. Applied log-prod0.1

      \[\leadsto \color{blue}{\log \left(\sqrt{\frac{N + 1}{N}}\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)}\]
    7. Using strategy rm

    8. Applied sqrt-div0.1

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

    9. Applied log-div0.1

      \[\leadsto \color{blue}{\left(\log \left(\sqrt{N + 1}\right) - \log \left(\sqrt{N}\right)\right)} + \log \left(\sqrt{\frac{N + 1}{N}}\right)\]
    10. Using strategy rm

    11. Applied add-sqr-sqrt0.1

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

    12. Applied log-prod0.1

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

    if 6703.766594082657 < N

    1. Initial program 59.6

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

    3. Applied diff-log59.3

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

    4. Taylor expanded around -inf 0.0

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

    5. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 6703.766594082657:\\ \;\;\;\;\left(\log \left(\sqrt{\sqrt{\frac{1 + N}{N}}}\right) + \log \left(\sqrt{\sqrt{\frac{1 + N}{N}}}\right)\right) + \left(\log \left(\sqrt{1 + N}\right) - \log \left(\sqrt{N}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \left(\frac{1}{2} - \frac{\frac{1}{3}}{N}\right) \cdot \frac{1}{N \cdot N}\\ \end{array}\]

Runtime

Time bar (total: 13.5s)Debug logProfile

herbie shell --seed 2018258 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))