Average Error: 29.5 → 0.2
Time: 34.6s
Precision: 64
Ground Truth: 128
\[\log \left(N + 1\right) - \log N\]
⬇
\[\begin{array}{l} \mathbf{if}\;N \le 1438490.0801097094:\\ \;\;\;\;\frac{{\left(\log \left(N + 1\right)\right)}^3 - {\left(\log N\right)}^3}{{\left(\log \left(N + 1\right)\right)}^2 + \left({\left(\log N\right)}^2 + \log \left(N + 1\right) \cdot \log N\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{3}}{N} - \frac{1}{2}}{{N}^2} + \frac{1}{N}\\ \end{array}\]

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes.

  2. if N < 1438490.0801097094

    1. Initial program 0.2

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

    3. Applied flip3-- 0.5

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

    4. Applied simplify 0.3

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

    if 1438490.0801097094 < N

    1. Initial program 59.8

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

    2. Applied taylor 0.0

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

    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 simplify 0.0

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

    5. Applied simplify 0.0

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{3}}{N} - \frac{1}{2}}{{N}^2}} + \frac{1}{N}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions

Runtime

Total time: 34.6s Debug log

Please include this information when filing a bug report:

herbie --seed '#(2915042545 3591512314 4229004281 3744598893 166345636 3506032321)'
(FPCore (N)
  :name "NMSE problem 3.3.6"
  (- (log (+ N 1)) (log N)))