Average Error: 29.9 → 0.1
Time: 3.4s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 9607.783785361794798518531024456024169922:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.3333333333333333148296162562473909929395}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]
\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}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.3333333333333333148296162562473909929395}{N} - 0.5, \frac{1}{N}\right)\\

\end{array}
double f(double N) {
        double r39707 = N;
        double r39708 = 1.0;
        double r39709 = r39707 + r39708;
        double r39710 = log(r39709);
        double r39711 = log(r39707);
        double r39712 = r39710 - r39711;
        return r39712;
}
↓
double f(double N) {
        double r39713 = N;
        double r39714 = 9607.783785361795;
        bool r39715 = r39713 <= r39714;
        double r39716 = 1.0;
        double r39717 = r39713 + r39716;
        double r39718 = r39717 / r39713;
        double r39719 = log(r39718);
        double r39720 = 1.0;
        double r39721 = 2.0;
        double r39722 = pow(r39713, r39721);
        double r39723 = r39720 / r39722;
        double r39724 = 0.3333333333333333;
        double r39725 = r39724 / r39713;
        double r39726 = 0.5;
        double r39727 = r39725 - r39726;
        double r39728 = r39716 / r39713;
        double r39729 = fma(r39723, r39727, r39728);
        double r39730 = r39715 ? r39719 : r39729;
        return r39730;
}

Error

Bits error versus N

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

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

  4. Final simplification0.1

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

Reproduce

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