Average Error: 29.1 → 0.1
Time: 1.5m
Precision: 64
Internal Precision: 1600
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7931.132604025559:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\frac{\frac{1}{3}}{N} + \frac{1}{2}\right) - \frac{\frac{1}{4}}{N}\right) - \frac{\frac{\frac{-1}{9}}{N} \cdot \frac{1}{N}}{N}}{\frac{1}{2} \cdot N + \frac{1}{3}}\\ \end{array}\]

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 N < 7931.132604025559

    1. Initial program 0.1

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

    2. Initial simplification0.1

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

    4. Applied diff-log0.1

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

    if 7931.132604025559 < N

    1. Initial program 59.5

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

    2. Initial simplification59.5

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

    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

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

    6. Applied flip--0.0

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

    7. Applied associate-*r/0.0

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

    8. Applied frac-sub0.0

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

    9. Simplified0.0

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{\frac{1}{3}}{N} + \frac{1}{2}\right) - \frac{\frac{1}{4}}{N}\right) - \frac{\frac{\frac{-1}{9}}{N} \cdot \frac{1}{N}}{N}}}{N \cdot \left(\frac{1}{2} + \frac{\frac{1}{3}}{N}\right)}\]

    10. Simplified0.0

      \[\leadsto \frac{\left(\left(\frac{\frac{1}{3}}{N} + \frac{1}{2}\right) - \frac{\frac{1}{4}}{N}\right) - \frac{\frac{\frac{-1}{9}}{N} \cdot \frac{1}{N}}{N}}{\color{blue}{\frac{1}{3} + N \cdot \frac{1}{2}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7931.132604025559:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\frac{\frac{1}{3}}{N} + \frac{1}{2}\right) - \frac{\frac{1}{4}}{N}\right) - \frac{\frac{\frac{-1}{9}}{N} \cdot \frac{1}{N}}{N}}{\frac{1}{2} \cdot N + \frac{1}{3}}\\ \end{array}\]

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed 2018251 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))