Average Error: 29.1 → 0.1
Time: 3.6s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 6058.78819213588122:\\ \;\;\;\;\log \left(\sqrt{N + 1}\right) + \log \left(\frac{\sqrt{N + 1}}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 6058.78819213588122:\\
\;\;\;\;\log \left(\sqrt{N + 1}\right) + \log \left(\frac{\sqrt{N + 1}}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\

\end{array}
double f(double N) {
        double r63371 = N;
        double r63372 = 1.0;
        double r63373 = r63371 + r63372;
        double r63374 = log(r63373);
        double r63375 = log(r63371);
        double r63376 = r63374 - r63375;
        return r63376;
}
↓
double f(double N) {
        double r63377 = N;
        double r63378 = 6058.788192135881;
        bool r63379 = r63377 <= r63378;
        double r63380 = 1.0;
        double r63381 = r63377 + r63380;
        double r63382 = sqrt(r63381);
        double r63383 = log(r63382);
        double r63384 = r63382 / r63377;
        double r63385 = log(r63384);
        double r63386 = r63383 + r63385;
        double r63387 = 1.0;
        double r63388 = 2.0;
        double r63389 = pow(r63377, r63388);
        double r63390 = r63387 / r63389;
        double r63391 = 0.3333333333333333;
        double r63392 = r63391 / r63377;
        double r63393 = 0.5;
        double r63394 = r63392 - r63393;
        double r63395 = r63390 * r63394;
        double r63396 = r63380 / r63377;
        double r63397 = r63395 + r63396;
        double r63398 = r63379 ? r63386 : r63397;
        return r63398;
}

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 < 6058.788192135881

    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 *-un-lft-identity0.1

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

    6. Applied add-sqr-sqrt0.1

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

    7. Applied times-frac0.1

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

    8. Applied log-prod0.1

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

    9. Simplified0.1

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

    if 6058.788192135881 < N

    1. Initial program 59.5

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

    2. Taylor expanded around inf 0.1

      \[\leadsto \color{blue}{\left(0.333333333333333315 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

    3. Simplified0.1

      \[\leadsto \color{blue}{\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 6058.78819213588122:\\ \;\;\;\;\log \left(\sqrt{N + 1}\right) + \log \left(\frac{\sqrt{N + 1}}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))