Average Error: 29.1 → 0.2
Time: 27.0s
Precision: 64
Internal Precision: 1344
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(1 + N\right) - \log N \le 9.441470229268852 \cdot 10^{-09}:\\ \;\;\;\;\frac{1}{N} - \frac{\frac{1}{2} - \frac{\frac{1}{3}}{N}}{N \cdot N}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \end{array}\]

Error

Bits error versus N

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Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (- (log (+ N 1)) (log N)) < 9.441470229268852e-09

    1. Initial program 60.1

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

    2. Initial simplification60.1

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

    3. Taylor expanded around inf 0.0

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

    4. Simplified0.0

      \[\leadsto \color{blue}{\frac{1}{N} - \frac{\frac{1}{2} - \frac{\frac{1}{3}}{N}}{N \cdot N}}\]

    if 9.441470229268852e-09 < (- (log (+ N 1)) (log N))

    1. Initial program 0.4

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

    2. Initial simplification0.4

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

    4. Applied diff-log0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 + N\right) - \log N \le 9.441470229268852 \cdot 10^{-09}:\\ \;\;\;\;\frac{1}{N} - \frac{\frac{1}{2} - \frac{\frac{1}{3}}{N}}{N \cdot N}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \end{array}\]

Runtime

Time bar (total: 27.0s)Debug logProfile

herbie shell --seed 2018225 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))