Average Error: 29.6 → 0.2
Time: 59.6s
Precision: 64
Internal Precision: 1344
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(1 + N\right) - \log N \le 2.656015105277077 \cdot 10^{-09}:\\ \;\;\;\;\frac{1 - \frac{\frac{1}{2}}{N}}{N}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \end{array}\]

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 (- (log (+ N 1)) (log N)) < 2.656015105277077e-09

    1. Initial program 60.1

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

    2. Taylor expanded around inf 60.1

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

    3. Applied simplify0.0

      \[\leadsto \color{blue}{\frac{1}{N} - (\left(\frac{1}{N}\right) \cdot \left(\frac{\frac{1}{2}}{N}\right) + 0)_*}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt0.0

      \[\leadsto \frac{1}{N} - \color{blue}{\left(\sqrt[3]{(\left(\frac{1}{N}\right) \cdot \left(\frac{\frac{1}{2}}{N}\right) + 0)_*} \cdot \sqrt[3]{(\left(\frac{1}{N}\right) \cdot \left(\frac{\frac{1}{2}}{N}\right) + 0)_*}\right) \cdot \sqrt[3]{(\left(\frac{1}{N}\right) \cdot \left(\frac{\frac{1}{2}}{N}\right) + 0)_*}}\]

    6. Applied *-un-lft-identity0.0

      \[\leadsto \color{blue}{1 \cdot \frac{1}{N}} - \left(\sqrt[3]{(\left(\frac{1}{N}\right) \cdot \left(\frac{\frac{1}{2}}{N}\right) + 0)_*} \cdot \sqrt[3]{(\left(\frac{1}{N}\right) \cdot \left(\frac{\frac{1}{2}}{N}\right) + 0)_*}\right) \cdot \sqrt[3]{(\left(\frac{1}{N}\right) \cdot \left(\frac{\frac{1}{2}}{N}\right) + 0)_*}\]

    7. Applied prod-diff0.0

      \[\leadsto \color{blue}{(1 \cdot \left(\frac{1}{N}\right) + \left(-\sqrt[3]{(\left(\frac{1}{N}\right) \cdot \left(\frac{\frac{1}{2}}{N}\right) + 0)_*} \cdot \left(\sqrt[3]{(\left(\frac{1}{N}\right) \cdot \left(\frac{\frac{1}{2}}{N}\right) + 0)_*} \cdot \sqrt[3]{(\left(\frac{1}{N}\right) \cdot \left(\frac{\frac{1}{2}}{N}\right) + 0)_*}\right)\right))_* + (\left(-\sqrt[3]{(\left(\frac{1}{N}\right) \cdot \left(\frac{\frac{1}{2}}{N}\right) + 0)_*}\right) \cdot \left(\sqrt[3]{(\left(\frac{1}{N}\right) \cdot \left(\frac{\frac{1}{2}}{N}\right) + 0)_*} \cdot \sqrt[3]{(\left(\frac{1}{N}\right) \cdot \left(\frac{\frac{1}{2}}{N}\right) + 0)_*}\right) + \left(\sqrt[3]{(\left(\frac{1}{N}\right) \cdot \left(\frac{\frac{1}{2}}{N}\right) + 0)_*} \cdot \left(\sqrt[3]{(\left(\frac{1}{N}\right) \cdot \left(\frac{\frac{1}{2}}{N}\right) + 0)_*} \cdot \sqrt[3]{(\left(\frac{1}{N}\right) \cdot \left(\frac{\frac{1}{2}}{N}\right) + 0)_*}\right)\right))_*}\]

    8. Applied simplify0.0

      \[\leadsto \color{blue}{\frac{1 - \frac{\frac{1}{2}}{N}}{N}} + (\left(-\sqrt[3]{(\left(\frac{1}{N}\right) \cdot \left(\frac{\frac{1}{2}}{N}\right) + 0)_*}\right) \cdot \left(\sqrt[3]{(\left(\frac{1}{N}\right) \cdot \left(\frac{\frac{1}{2}}{N}\right) + 0)_*} \cdot \sqrt[3]{(\left(\frac{1}{N}\right) \cdot \left(\frac{\frac{1}{2}}{N}\right) + 0)_*}\right) + \left(\sqrt[3]{(\left(\frac{1}{N}\right) \cdot \left(\frac{\frac{1}{2}}{N}\right) + 0)_*} \cdot \left(\sqrt[3]{(\left(\frac{1}{N}\right) \cdot \left(\frac{\frac{1}{2}}{N}\right) + 0)_*} \cdot \sqrt[3]{(\left(\frac{1}{N}\right) \cdot \left(\frac{\frac{1}{2}}{N}\right) + 0)_*}\right)\right))_*\]

    9. Applied simplify0.0

      \[\leadsto \frac{1 - \frac{\frac{1}{2}}{N}}{N} + \color{blue}{0}\]

    if 2.656015105277077e-09 < (- (log (+ N 1)) (log N))

    1. Initial program 0.4

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

    3. Applied diff-log0.4

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

  4. Applied simplify0.2

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\log \left(1 + N\right) - \log N \le 2.656015105277077 \cdot 10^{-09}:\\ \;\;\;\;\frac{1 - \frac{\frac{1}{2}}{N}}{N}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \end{array}}\]

Runtime

Time bar (total: 59.6s)Debug logProfile

herbie shell --seed 2018170 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))