Average Error: 29.2 → 0.2
Time: 33.8s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 4798.860931730865559075027704238891601562:\\ \;\;\;\;\left(\log \left(1 + N\right) - 2 \cdot \log \left(\sqrt[3]{N}\right)\right) - \log \left(\sqrt[3]{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.3333333333333333148296162562473909929395}{{N}^{3}} - \frac{\frac{0.5}{N}}{N}\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 4798.860931730865559075027704238891601562:\\
\;\;\;\;\left(\log \left(1 + N\right) - 2 \cdot \log \left(\sqrt[3]{N}\right)\right) - \log \left(\sqrt[3]{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{0.3333333333333333148296162562473909929395}{{N}^{3}} - \frac{\frac{0.5}{N}}{N}\right) + \frac{1}{N}\\

\end{array}
double f(double N) {
        double r104385 = N;
        double r104386 = 1.0;
        double r104387 = r104385 + r104386;
        double r104388 = log(r104387);
        double r104389 = log(r104385);
        double r104390 = r104388 - r104389;
        return r104390;
}
↓
double f(double N) {
        double r104391 = N;
        double r104392 = 4798.860931730866;
        bool r104393 = r104391 <= r104392;
        double r104394 = 1.0;
        double r104395 = r104394 + r104391;
        double r104396 = log(r104395);
        double r104397 = 2.0;
        double r104398 = cbrt(r104391);
        double r104399 = log(r104398);
        double r104400 = r104397 * r104399;
        double r104401 = r104396 - r104400;
        double r104402 = r104401 - r104399;
        double r104403 = 0.3333333333333333;
        double r104404 = 3.0;
        double r104405 = pow(r104391, r104404);
        double r104406 = r104403 / r104405;
        double r104407 = 0.5;
        double r104408 = r104407 / r104391;
        double r104409 = r104408 / r104391;
        double r104410 = r104406 - r104409;
        double r104411 = r104394 / r104391;
        double r104412 = r104410 + r104411;
        double r104413 = r104393 ? r104402 : r104412;
        return r104413;
}

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 < 4798.860931730866

    1. Initial program 0.1

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

    2. Simplified0.1

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

    4. Applied add-cube-cbrt0.1

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

    5. Applied log-prod0.4

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

    6. Applied associate--r+0.4

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

    7. Simplified0.4

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

    if 4798.860931730866 < N

    1. Initial program 59.7

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

    2. Simplified59.7

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

    3. Taylor expanded around inf 0.0

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

    4. Simplified0.0

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 4798.860931730865559075027704238891601562:\\ \;\;\;\;\left(\log \left(1 + N\right) - 2 \cdot \log \left(\sqrt[3]{N}\right)\right) - \log \left(\sqrt[3]{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.3333333333333333148296162562473909929395}{{N}^{3}} - \frac{\frac{0.5}{N}}{N}\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1.0)) (log N)))