Average Error: 29.2 → 0.1
Time: 1.2m
Precision: 64
Internal Precision: 128
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 5819.417453615807:\\ \;\;\;\;\left(\log_* (1 + N) - \log \left(\sqrt{N}\right)\right) - \log \left(\sqrt{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\left(1 + \frac{\frac{\frac{1}{3}}{N}}{N}\right) \cdot \left(1 + \frac{\frac{\frac{1}{3}}{N}}{N}\right)\right) \cdot \left(1 + \frac{\frac{\frac{1}{3}}{N}}{N}\right) + \left(\frac{\frac{-1}{8}}{\left(N \cdot N\right) \cdot N}\right))_*}{(\left(\left(1 + \frac{\frac{\frac{1}{3}}{N}}{N}\right) \cdot \left(1 + \frac{\frac{\frac{1}{3}}{N}}{N}\right)\right) \cdot N + \left((\frac{1}{2} \cdot \left(1 + \frac{\frac{\frac{1}{3}}{N}}{N}\right) + \left(\frac{\frac{1}{4}}{N}\right))_*\right))_*}\\ \end{array}\]

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 5819.417453615807

    1. Initial program 0.1

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

    2. Simplified0.1

      \[\leadsto \color{blue}{\log_* (1 + N) - \log N}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt0.1

      \[\leadsto \log_* (1 + N) - \log \color{blue}{\left(\sqrt{N} \cdot \sqrt{N}\right)}\]

    5. Applied log-prod0.1

      \[\leadsto \log_* (1 + N) - \color{blue}{\left(\log \left(\sqrt{N}\right) + \log \left(\sqrt{N}\right)\right)}\]

    6. Applied associate--r+0.1

      \[\leadsto \color{blue}{\left(\log_* (1 + N) - \log \left(\sqrt{N}\right)\right) - \log \left(\sqrt{N}\right)}\]

    if 5819.417453615807 < N

    1. Initial program 59.5

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

    2. Simplified59.5

      \[\leadsto \color{blue}{\log_* (1 + N) - \log N}\]

    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

      \[\leadsto \color{blue}{\left(\frac{\frac{-1}{2}}{N \cdot N} + \frac{1}{N}\right) - \frac{\frac{-1}{3}}{\left(N \cdot N\right) \cdot N}}\]
    5. Using strategy rm

    6. Applied add-log-exp0.5

      \[\leadsto \left(\color{blue}{\log \left(e^{\frac{\frac{-1}{2}}{N \cdot N}}\right)} + \frac{1}{N}\right) - \frac{\frac{-1}{3}}{\left(N \cdot N\right) \cdot N}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity0.5

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

    9. Applied times-frac0.5

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

    10. Applied add-sqr-sqrt0.9

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

    11. Applied prod-diff0.9

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

    12. Simplified0.0

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

    13. Simplified0.0

      \[\leadsto \frac{1}{N} \cdot \left(\frac{\frac{-1}{2}}{N} + (\left(\frac{1}{N}\right) \cdot \left(\frac{\frac{1}{3}}{N}\right) + 1)_*\right) + \color{blue}{0}\]
    14. Using strategy rm

    15. Applied flip3-+0.0

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

    16. Applied frac-times0.0

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

    17. Simplified0.0

      \[\leadsto \frac{\color{blue}{(\left(\left(\frac{\frac{\frac{1}{3}}{N}}{N} + 1\right) \cdot \left(\frac{\frac{\frac{1}{3}}{N}}{N} + 1\right)\right) \cdot \left(\frac{\frac{\frac{1}{3}}{N}}{N} + 1\right) + \left(\frac{\frac{-1}{8}}{\left(N \cdot N\right) \cdot N}\right))_*}}{N \cdot \left(\frac{\frac{-1}{2}}{N} \cdot \frac{\frac{-1}{2}}{N} + \left((\left(\frac{1}{N}\right) \cdot \left(\frac{\frac{1}{3}}{N}\right) + 1)_* \cdot (\left(\frac{1}{N}\right) \cdot \left(\frac{\frac{1}{3}}{N}\right) + 1)_* - \frac{\frac{-1}{2}}{N} \cdot (\left(\frac{1}{N}\right) \cdot \left(\frac{\frac{1}{3}}{N}\right) + 1)_*\right)\right)} + 0\]

    18. Simplified0.0

      \[\leadsto \frac{(\left(\left(\frac{\frac{\frac{1}{3}}{N}}{N} + 1\right) \cdot \left(\frac{\frac{\frac{1}{3}}{N}}{N} + 1\right)\right) \cdot \left(\frac{\frac{\frac{1}{3}}{N}}{N} + 1\right) + \left(\frac{\frac{-1}{8}}{\left(N \cdot N\right) \cdot N}\right))_*}{\color{blue}{(\left(\left(\frac{\frac{\frac{1}{3}}{N}}{N} + 1\right) \cdot \left(\frac{\frac{\frac{1}{3}}{N}}{N} + 1\right)\right) \cdot N + \left((\frac{1}{2} \cdot \left(\frac{\frac{\frac{1}{3}}{N}}{N} + 1\right) + \left(\frac{\frac{1}{4}}{N}\right))_*\right))_*}} + 0\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 5819.417453615807:\\ \;\;\;\;\left(\log_* (1 + N) - \log \left(\sqrt{N}\right)\right) - \log \left(\sqrt{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\left(1 + \frac{\frac{\frac{1}{3}}{N}}{N}\right) \cdot \left(1 + \frac{\frac{\frac{1}{3}}{N}}{N}\right)\right) \cdot \left(1 + \frac{\frac{\frac{1}{3}}{N}}{N}\right) + \left(\frac{\frac{-1}{8}}{\left(N \cdot N\right) \cdot N}\right))_*}{(\left(\left(1 + \frac{\frac{\frac{1}{3}}{N}}{N}\right) \cdot \left(1 + \frac{\frac{\frac{1}{3}}{N}}{N}\right)\right) \cdot N + \left((\frac{1}{2} \cdot \left(1 + \frac{\frac{\frac{1}{3}}{N}}{N}\right) + \left(\frac{\frac{1}{4}}{N}\right))_*\right))_*}\\ \end{array}\]

Reproduce

herbie shell --seed 2019094 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))