Average Error: 40.4 → 19.6
Time: 5.4s
Precision: 64
Internal precision: 1408
\[\log \left(N + 1\right) - \log N\]
⬇
\[\begin{array}{l}
\mathbf{if}\;N \le 3659161.101236961:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\
\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{1}{3}}{N} - \frac{1}{2}}{{N}^2} + \frac{1}{N}\\
\end{array}\]
Derivation
- Split input into 2 regimes.
-
if N < 3659161.101236961
Initial program 31.1
\[\log \left(N + 1\right) - \log N\]
- Using strategy
rm
Applied diff-log 28.9
\[\leadsto \color{blue}{\log \left(\frac{N + 1}{N}\right)}\]
if 3659161.101236961 < N
Initial program 59.9
\[\log \left(N + 1\right) - \log N\]
Applied taylor 0.0
\[\leadsto \left(\frac{1}{3} \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \frac{1}{2} \cdot \frac{1}{{N}^2}\]
Taylor expanded around inf 0.0
\[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \frac{1}{2} \cdot \frac{1}{{N}^2}}\]
Applied simplify 0.0
\[\leadsto \color{blue}{\frac{1}{N \cdot N} \cdot \left(\frac{\frac{1}{3}}{N} - \frac{1}{2}\right) + \frac{1}{N}}\]
Applied simplify 0.0
\[\leadsto \color{blue}{\frac{\frac{\frac{1}{3}}{N} - \frac{1}{2}}{{N}^2}} + \frac{1}{N}\]
- Recombined 2 regimes into one program.
- Removed slow pow expressions
Runtime
Please include this information when filing a bug report:
herbie shell --seed '#(1064555532 179913862 452496668 2441903500 287849034 462453547)'
(FPCore (N)
:name "2log (problem 3.3.6)"
(- (log (+ N 1)) (log N)))
