Average Error: 29.9 → 0.1
Time: 5.0s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 9607.783785361794798518531024456024169922:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 9607.783785361794798518531024456024169922:\\
\;\;\;\;\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 r38607 = N;
        double r38608 = 1.0;
        double r38609 = r38607 + r38608;
        double r38610 = log(r38609);
        double r38611 = log(r38607);
        double r38612 = r38610 - r38611;
        return r38612;
}
↓
double f(double N) {
        double r38613 = N;
        double r38614 = 9607.783785361795;
        bool r38615 = r38613 <= r38614;
        double r38616 = 1.0;
        double r38617 = r38613 + r38616;
        double r38618 = r38617 / r38613;
        double r38619 = log(r38618);
        double r38620 = 1.0;
        double r38621 = 2.0;
        double r38622 = pow(r38613, r38621);
        double r38623 = r38620 / r38622;
        double r38624 = 0.3333333333333333;
        double r38625 = r38624 / r38613;
        double r38626 = 0.5;
        double r38627 = r38625 - r38626;
        double r38628 = r38623 * r38627;
        double r38629 = r38616 / r38613;
        double r38630 = r38628 + r38629;
        double r38631 = r38615 ? r38619 : r38630;
        return r38631;
}

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 < 9607.783785361795

    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 9607.783785361795 < 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

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9607.783785361794798518531024456024169922:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))