Average Error: 29.9 → 0.0
Time: 16.0s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7536.677708381748743704520165920257568359:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{0.3333333333333333148296162562473909929395}{N}}{N \cdot N} - \frac{0.5}{N \cdot N}\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7536.677708381748743704520165920257568359:\\
\;\;\;\;\log \left(\frac{1 + N}{N}\right)\\

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

\end{array}
double f(double N) {
        double r4623738 = N;
        double r4623739 = 1.0;
        double r4623740 = r4623738 + r4623739;
        double r4623741 = log(r4623740);
        double r4623742 = log(r4623738);
        double r4623743 = r4623741 - r4623742;
        return r4623743;
}
↓
double f(double N) {
        double r4623744 = N;
        double r4623745 = 7536.677708381749;
        bool r4623746 = r4623744 <= r4623745;
        double r4623747 = 1.0;
        double r4623748 = r4623747 + r4623744;
        double r4623749 = r4623748 / r4623744;
        double r4623750 = log(r4623749);
        double r4623751 = 0.3333333333333333;
        double r4623752 = r4623751 / r4623744;
        double r4623753 = r4623744 * r4623744;
        double r4623754 = r4623752 / r4623753;
        double r4623755 = 0.5;
        double r4623756 = r4623755 / r4623753;
        double r4623757 = r4623754 - r4623756;
        double r4623758 = r4623747 / r4623744;
        double r4623759 = r4623757 + r4623758;
        double r4623760 = r4623746 ? r4623750 : r4623759;
        return r4623760;
}

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 < 7536.677708381749

    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 7536.677708381749 < 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

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7536.677708381748743704520165920257568359:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{0.3333333333333333148296162562473909929395}{N}}{N \cdot N} - \frac{0.5}{N \cdot N}\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1.0)) (log N)))