Average Error: 29.1 → 0.1
Time: 29.7s
Precision: 64
Internal Precision: 1344
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \le 1.7643540376255373 \cdot 10^{-05}:\\ \;\;\;\;\frac{\frac{\frac{1}{3}}{N} - \frac{1}{2}}{N \cdot N} + \frac{1}{N}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array}\]

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if (- (log (+ N 1)) (log N)) < 1.7643540376255373e-05

    1. Initial program 59.6

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

    3. Applied diff-log59.4

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

    5. Applied add-exp-log59.4

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

    6. Taylor expanded around inf 5.7

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

    7. Applied simplify0.0

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

    if 1.7643540376255373e-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: 29.7s)Debug logProfile

herbie shell --seed '#(1071852389 864846987 1238109217 3425890003 4124793586 650694553)' 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))