Average Error: 29.4 → 0.1
Time: 37.1s
Precision: 64
Internal Precision: 1344
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \le 6.718382873077259 \cdot 10^{-05}:\\ \;\;\;\;\frac{\left(1 - \frac{\frac{1}{2}}{N}\right) + \log \left(e^{\frac{\frac{\frac{1}{3}}{N}}{N}}\right)}{N}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array}\]

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)) < 6.718382873077259e-05

    1. Initial program 59.5

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

    2. Taylor expanded around inf 0.0

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

    3. Applied simplify0.0

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

    5. Applied add-log-exp0.0

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

    if 6.718382873077259e-05 < (- (log (+ N 1)) (log N))

    1. Initial program 0.1

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

    3. Applied diff-log0.1

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

Runtime

Time bar (total: 37.1s)Debug logProfile

herbie shell --seed 2020178 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))