Average Error: 28.9 → 0.1
Time: 16.1s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7287.699358996014780132099986076354980469:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333148296162562473909929395}{{N}^{3}} + \frac{1 - \frac{0.5}{N}}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7287.699358996014780132099986076354980469:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

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

\end{array}
double f(double N) {
        double r47432 = N;
        double r47433 = 1.0;
        double r47434 = r47432 + r47433;
        double r47435 = log(r47434);
        double r47436 = log(r47432);
        double r47437 = r47435 - r47436;
        return r47437;
}
↓
double f(double N) {
        double r47438 = N;
        double r47439 = 7287.699358996015;
        bool r47440 = r47438 <= r47439;
        double r47441 = 1.0;
        double r47442 = r47438 + r47441;
        double r47443 = r47442 / r47438;
        double r47444 = log(r47443);
        double r47445 = 0.3333333333333333;
        double r47446 = 3.0;
        double r47447 = pow(r47438, r47446);
        double r47448 = r47445 / r47447;
        double r47449 = 0.5;
        double r47450 = r47449 / r47438;
        double r47451 = r47441 - r47450;
        double r47452 = r47451 / r47438;
        double r47453 = r47448 + r47452;
        double r47454 = r47440 ? r47444 : r47453;
        return r47454;
}

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 < 7287.699358996015

    1. Initial program 0.1

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

    3. Applied diff-log0.1

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

    if 7287.699358996015 < 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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7287.699358996014780132099986076354980469:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333148296162562473909929395}{{N}^{3}} + \frac{1 - \frac{0.5}{N}}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))