Average Error: 29.4 → 0.1
Time: 35.8s
Precision: 64
Internal Precision: 1344
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(1 + N\right) - \log N \le 1.2266119196421323 \cdot 10^{-07}:\\ \;\;\;\;\frac{1}{N} - \frac{\frac{\frac{1}{2}}{N}}{N}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \end{array}\]

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)) < 1.2266119196421323e-07

    1. Initial program 60.0

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

    2. Taylor expanded around inf 60.0

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

    3. Applied simplify0.0

      \[\leadsto \color{blue}{\left(0 + \frac{1}{N}\right) - \frac{\frac{\frac{1}{2}}{N}}{N}}\]

    if 1.2266119196421323e-07 < (- (log (+ N 1)) (log N))

    1. Initial program 0.3

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

    3. Applied diff-log0.2

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

  4. Applied simplify0.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\log \left(1 + N\right) - \log N \le 1.2266119196421323 \cdot 10^{-07}:\\ \;\;\;\;\frac{1}{N} - \frac{\frac{\frac{1}{2}}{N}}{N}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \end{array}}\]

Runtime

Time bar (total: 35.8s)Debug logProfile

herbie shell --seed 2018201 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))