Average Error: 29.7 → 0.0
Time: 14.8s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8023.454188180856363032944500446319580078:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right) + \frac{\frac{0.3333333333333333148296162562473909929395}{N}}{N \cdot N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8023.454188180856363032944500446319580078:\\
\;\;\;\;\log \left(\frac{1 + N}{N}\right)\\

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

\end{array}
double f(double N) {
        double r3248419 = N;
        double r3248420 = 1.0;
        double r3248421 = r3248419 + r3248420;
        double r3248422 = log(r3248421);
        double r3248423 = log(r3248419);
        double r3248424 = r3248422 - r3248423;
        return r3248424;
}
↓
double f(double N) {
        double r3248425 = N;
        double r3248426 = 8023.454188180856;
        bool r3248427 = r3248425 <= r3248426;
        double r3248428 = 1.0;
        double r3248429 = r3248428 + r3248425;
        double r3248430 = r3248429 / r3248425;
        double r3248431 = log(r3248430);
        double r3248432 = r3248428 / r3248425;
        double r3248433 = 0.5;
        double r3248434 = r3248425 * r3248425;
        double r3248435 = r3248433 / r3248434;
        double r3248436 = r3248432 - r3248435;
        double r3248437 = 0.3333333333333333;
        double r3248438 = r3248437 / r3248425;
        double r3248439 = r3248438 / r3248434;
        double r3248440 = r3248436 + r3248439;
        double r3248441 = r3248427 ? r3248431 : r3248440;
        return r3248441;
}

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 < 8023.454188180856

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

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

    4. Simplified0.1

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

    if 8023.454188180856 < N

    1. Initial program 59.6

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

    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

      \[\leadsto \color{blue}{\frac{1}{N} - \left(\frac{0.5}{N \cdot N} - \frac{\frac{0.3333333333333333148296162562473909929395}{N}}{N \cdot N}\right)}\]
    4. Using strategy rm

    5. Applied associate--r-0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8023.454188180856363032944500446319580078:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right) + \frac{\frac{0.3333333333333333148296162562473909929395}{N}}{N \cdot N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1.0)) (log N)))