Average Error: 29.6 → 0.2
Time: 26.1s
Precision: 64
Internal Precision: 1344
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \le 2.656015105277077 \cdot 10^{-09}:\\ \;\;\;\;\frac{\frac{\frac{1}{3}}{N \cdot N} + \left(1 - \frac{\frac{1}{2}}{N}\right)}{N}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array}\]

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)) < 2.656015105277077e-09

    1. Initial program 60.1

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

    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

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

    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.

Runtime

Time bar (total: 26.1s)Debug logProfile

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