Average Error: 29.3 → 0.1
Time: 5.3s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 9218.49831353843729:\\ \;\;\;\;\log \left(\frac{N \cdot N - 1 \cdot 1}{N \cdot \left(N - 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 9218.49831353843729:\\
\;\;\;\;\log \left(\frac{N \cdot N - 1 \cdot 1}{N \cdot \left(N - 1\right)}\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 r47632 = N;
        double r47633 = 1.0;
        double r47634 = r47632 + r47633;
        double r47635 = log(r47634);
        double r47636 = log(r47632);
        double r47637 = r47635 - r47636;
        return r47637;
}
↓
double f(double N) {
        double r47638 = N;
        double r47639 = 9218.498313538437;
        bool r47640 = r47638 <= r47639;
        double r47641 = r47638 * r47638;
        double r47642 = 1.0;
        double r47643 = r47642 * r47642;
        double r47644 = r47641 - r47643;
        double r47645 = r47638 - r47642;
        double r47646 = r47638 * r47645;
        double r47647 = r47644 / r47646;
        double r47648 = log(r47647);
        double r47649 = 1.0;
        double r47650 = 2.0;
        double r47651 = pow(r47638, r47650);
        double r47652 = r47649 / r47651;
        double r47653 = 0.3333333333333333;
        double r47654 = r47653 / r47638;
        double r47655 = 0.5;
        double r47656 = r47654 - r47655;
        double r47657 = r47652 * r47656;
        double r47658 = r47642 / r47638;
        double r47659 = r47657 + r47658;
        double r47660 = r47640 ? r47648 : r47659;
        return r47660;
}

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 < 9218.498313538437

    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 flip-+0.1

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

    6. Applied associate-/l/0.1

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

    if 9218.498313538437 < N

    1. Initial program 59.4

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9218.49831353843729:\\ \;\;\;\;\log \left(\frac{N \cdot N - 1 \cdot 1}{N \cdot \left(N - 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))