Average Error: 29.4 → 0.1
Time: 33.0s
Precision: 64
Internal Precision: 1600
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8431.959514385991:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \left(\frac{\frac{\frac{1}{2}}{N}}{N} - \frac{\frac{1}{3}}{{N}^{3}}\right)\\ \end{array}\]

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 < 8431.959514385991

    1. Initial program 0.1

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

    3. Applied diff-log0.1

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

    if 8431.959514385991 < N

    1. Initial program 59.6

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

    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. Simplified0.0

      \[\leadsto \color{blue}{\frac{1 - \frac{\frac{1}{2}}{N}}{N} + \frac{\frac{1}{3}}{{N}^{3}}}\]
    4. Using strategy rm

    5. Applied div-sub0.0

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

    6. Applied associate-+l-0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8431.959514385991:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \left(\frac{\frac{\frac{1}{2}}{N}}{N} - \frac{\frac{1}{3}}{{N}^{3}}\right)\\ \end{array}\]

Runtime

Time bar (total: 33.0s)Debug logProfile

herbie shell --seed 2018221 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))