Average Error: 29.8 → 0.1
Time: 3.8s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 6381.7936198495444841682910919189453125:\\ \;\;\;\;\frac{1}{2} \cdot \log \left(\frac{N + 1}{N}\right) + \left(\log \left(\sqrt{\frac{\sqrt{N + 1}}{1}}\right) + \log \left(\sqrt{\frac{\sqrt{N + 1}}{N}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 6381.7936198495444841682910919189453125:\\
\;\;\;\;\frac{1}{2} \cdot \log \left(\frac{N + 1}{N}\right) + \left(\log \left(\sqrt{\frac{\sqrt{N + 1}}{1}}\right) + \log \left(\sqrt{\frac{\sqrt{N + 1}}{N}}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\

\end{array}
double f(double N) {
        double r41397 = N;
        double r41398 = 1.0;
        double r41399 = r41397 + r41398;
        double r41400 = log(r41399);
        double r41401 = log(r41397);
        double r41402 = r41400 - r41401;
        return r41402;
}
↓
double f(double N) {
        double r41403 = N;
        double r41404 = 6381.7936198495445;
        bool r41405 = r41403 <= r41404;
        double r41406 = 0.5;
        double r41407 = 1.0;
        double r41408 = r41403 + r41407;
        double r41409 = r41408 / r41403;
        double r41410 = log(r41409);
        double r41411 = r41406 * r41410;
        double r41412 = sqrt(r41408);
        double r41413 = 1.0;
        double r41414 = r41412 / r41413;
        double r41415 = sqrt(r41414);
        double r41416 = log(r41415);
        double r41417 = r41412 / r41403;
        double r41418 = sqrt(r41417);
        double r41419 = log(r41418);
        double r41420 = r41416 + r41419;
        double r41421 = r41411 + r41420;
        double r41422 = 2.0;
        double r41423 = pow(r41403, r41422);
        double r41424 = r41413 / r41423;
        double r41425 = 0.3333333333333333;
        double r41426 = r41425 / r41403;
        double r41427 = 0.5;
        double r41428 = r41426 - r41427;
        double r41429 = r41424 * r41428;
        double r41430 = r41407 / r41403;
        double r41431 = r41429 + r41430;
        double r41432 = r41405 ? r41421 : r41431;
        return r41432;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if N < 6381.7936198495445

    1. Initial program 0.1

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

    3. Applied diff-log0.1

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

    5. Applied add-sqr-sqrt0.1

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

    6. Applied log-prod0.1

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

    8. Applied pow1/20.1

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

    9. Applied log-pow0.1

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \log \left(\frac{N + 1}{N}\right)} + \log \left(\sqrt{\frac{N + 1}{N}}\right)\]
    10. Using strategy rm

    11. Applied *-un-lft-identity0.1

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

    12. Applied add-sqr-sqrt0.1

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

    13. Applied times-frac0.1

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

    14. Applied sqrt-prod0.1

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

    15. Applied log-prod0.1

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

    if 6381.7936198495445 < N

    1. Initial program 59.4

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

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.3333333333333333148296162562473909929395 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 6381.7936198495444841682910919189453125:\\ \;\;\;\;\frac{1}{2} \cdot \log \left(\frac{N + 1}{N}\right) + \left(\log \left(\sqrt{\frac{\sqrt{N + 1}}{1}}\right) + \log \left(\sqrt{\frac{\sqrt{N + 1}}{N}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))