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

↓

\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \le 1.6787765542147781 \cdot 10^{-06}:\\ \;\;\;\;(\left(\frac{\frac{1}{3}}{N} - \frac{1}{2}\right) \cdot \left(\frac{1}{N \cdot N}\right) + \left(\frac{1}{N}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{\log_* (1 + N)}}{N}\right)\\ \end{array}\]

Derivation

Split input into 2 regimes
if (- (log (+ N 1)) (log N)) < 1.6787765542147781e-06
1. Initial program 59.7
  \[\log \left(N + 1\right) - \log N\]
2. Applied simplify59.7
  \[\leadsto \color{blue}{\log_* (1 + N) - \log N}\]
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. Applied simplify0.0
  \[\leadsto \color{blue}{(\left(\frac{\frac{1}{3}}{N} - \frac{1}{2}\right) \cdot \left(\frac{1}{N \cdot N}\right) + \left(\frac{1}{N}\right))_*}\]
if 1.6787765542147781e-06 < (- (log (+ N 1)) (log N))
1. Initial program 0.2
  \[\log \left(N + 1\right) - \log N\]
2. Applied simplify0.2
  \[\leadsto \color{blue}{\log_* (1 + N) - \log N}\]
3. Using strategy rm
4. Applied add-log-exp0.2
  \[\leadsto \color{blue}{\log \left(e^{\log_* (1 + N)}\right)} - \log N\]
5. Applied diff-log0.2
  \[\leadsto \color{blue}{\log \left(\frac{e^{\log_* (1 + N)}}{N}\right)}\]
Recombined 2 regimes into one program.

Runtime

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))

Error

Derivation

`if (- (log (+ N 1)) (log N)) < 1.6787765542147781e-06`

`if 1.6787765542147781e-06 < (- (log (+ N 1)) (log N))`

Runtime