Average Error: 29.7 → 0.1

Time: 35.9s

Precision: 64

Internal Precision: 1600

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

↓

\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \le 8.011007441346735 \cdot 10^{-06}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\frac{1}{3}}{N \cdot N}}\right) + \left(1 - \frac{\frac{1}{2}}{N}\right)}{N}\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(1 + {N}^{3}\right) - \log \left(N \cdot N + \left(1 - N\right)\right)\right) - \log N\\ \end{array}\]

Error

The X axis uses an exponential scale — Bits error versus N

The X axis uses an exponential scale — Bits error versus N

Try it out

In
Out

Enter valid numbers for all inputs

Derivation

Split input into 2 regimes
if (- (log (+ N 1)) (log N)) < 8.011007441346735e-06
1. Initial program 59.6
  \[\log \left(N + 1\right) - \log N\]
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
  \[\leadsto \color{blue}{\frac{\frac{\frac{1}{3}}{N \cdot N} + \left(1 - \frac{\frac{1}{2}}{N}\right)}{N}}\]
4. Using strategy rm
5. Applied add-log-exp0.0
  \[\leadsto \frac{\color{blue}{\log \left(e^{\frac{\frac{1}{3}}{N \cdot N}}\right)} + \left(1 - \frac{\frac{1}{2}}{N}\right)}{N}\]
if 8.011007441346735e-06 < (- (log (+ N 1)) (log N))
1. Initial program 0.1
  \[\log \left(N + 1\right) - \log N\]
2. Using strategy rm
3. Applied flip3-+0.1
  \[\leadsto \log \color{blue}{\left(\frac{{N}^{3} + {1}^{3}}{N \cdot N + \left(1 \cdot 1 - N \cdot 1\right)}\right)} - \log N\]
4. Applied log-div0.2
  \[\leadsto \color{blue}{\left(\log \left({N}^{3} + {1}^{3}\right) - \log \left(N \cdot N + \left(1 \cdot 1 - N \cdot 1\right)\right)\right)} - \log N\]
5. Applied simplify0.2
  \[\leadsto \left(\color{blue}{\log \left(1 + {N}^{3}\right)} - \log \left(N \cdot N + \left(1 \cdot 1 - N \cdot 1\right)\right)\right) - \log N\]
6. Applied simplify0.2
  \[\leadsto \left(\log \left(1 + {N}^{3}\right) - \color{blue}{\log \left(N \cdot N + \left(1 - N\right)\right)}\right) - \log N\]
Recombined 2 regimes into one program.

Runtime

herbie shell --seed 2018199 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))