Average Error: 29.4 → 0.1
Time: 1.3m
Precision: 64
Internal Precision: 1344
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \le 0.0002516277025645313:\\ \;\;\;\;\frac{1}{N} - \frac{\frac{1}{2} - \frac{\frac{1}{3}}{N}}{N \cdot N}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(N + 1\right)} \cdot \sqrt{\log \left(N + 1\right)} - \log N\\ \end{array}\]

Error

Bits error versus N

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (- (log (+ N 1)) (log N)) < 0.0002516277025645313

    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

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

    if 0.0002516277025645313 < (- (log (+ N 1)) (log N))

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2019053 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))