Average Error: 29.5 → 0.2
Time: 37.5s
Precision: 64
Internal Precision: 1600
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(1 + N\right) - \log N \le 4.0142178601380495 \cdot 10^{-08}:\\ \;\;\;\;\frac{1}{N} - \frac{\frac{\frac{1}{2}}{N}}{N}\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(1 + {N}^{3}\right) - \log \left(\left(1 - N\right) + N \cdot N\right)\right) - \log N\\ \end{array}\]

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)) < 4.0142178601380495e-08

    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 4.0142178601380495e-08 < (- (log (+ N 1)) (log N))

    1. Initial program 0.4

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

    3. Applied flip3-+0.4

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

    4. Applied log-div0.4

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

    5. Applied simplify0.4

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

    6. Applied simplify0.4

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

  4. Applied simplify0.2

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\log \left(1 + N\right) - \log N \le 4.0142178601380495 \cdot 10^{-08}:\\ \;\;\;\;\frac{1}{N} - \frac{\frac{\frac{1}{2}}{N}}{N}\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(1 + {N}^{3}\right) - \log \left(\left(1 - N\right) + N \cdot N\right)\right) - \log N\\ \end{array}}\]

Runtime

Time bar (total: 37.5s)Debug logProfile

herbie shell --seed 2018208 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))