Average Error: 30.3 → 0.1
Time: 3.9s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 20452.1348824906744994223117828369140625:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 20452.1348824906744994223117828369140625:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\

\end{array}
double f(double N) {
        double r38613 = N;
        double r38614 = 1.0;
        double r38615 = r38613 + r38614;
        double r38616 = log(r38615);
        double r38617 = log(r38613);
        double r38618 = r38616 - r38617;
        return r38618;
}
↓
double f(double N) {
        double r38619 = N;
        double r38620 = 20452.134882490674;
        bool r38621 = r38619 <= r38620;
        double r38622 = 1.0;
        double r38623 = r38619 + r38622;
        double r38624 = r38623 / r38619;
        double r38625 = log(r38624);
        double r38626 = 1.0;
        double r38627 = 2.0;
        double r38628 = pow(r38619, r38627);
        double r38629 = r38626 / r38628;
        double r38630 = 0.3333333333333333;
        double r38631 = r38630 / r38619;
        double r38632 = 0.5;
        double r38633 = r38631 - r38632;
        double r38634 = r38629 * r38633;
        double r38635 = r38622 / r38619;
        double r38636 = r38634 + r38635;
        double r38637 = r38621 ? r38625 : r38636;
        return r38637;
}

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 < 20452.134882490674

    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 20452.134882490674 < N

    1. Initial program 59.5

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

    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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 20452.1348824906744994223117828369140625:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))