Average Error: 29.2 → 0.1
Time: 13.6s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 10223.09734470723924459889531135559082031:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.3333333333333333148296162562473909929395}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 10223.09734470723924459889531135559082031:\\
\;\;\;\;\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 r55719 = N;
        double r55720 = 1.0;
        double r55721 = r55719 + r55720;
        double r55722 = log(r55721);
        double r55723 = log(r55719);
        double r55724 = r55722 - r55723;
        return r55724;
}
↓
double f(double N) {
        double r55725 = N;
        double r55726 = 10223.09734470724;
        bool r55727 = r55725 <= r55726;
        double r55728 = 1.0;
        double r55729 = r55725 + r55728;
        double r55730 = r55729 / r55725;
        double r55731 = log(r55730);
        double r55732 = 1.0;
        double r55733 = 2.0;
        double r55734 = pow(r55725, r55733);
        double r55735 = r55732 / r55734;
        double r55736 = 0.3333333333333333;
        double r55737 = r55736 / r55725;
        double r55738 = 0.5;
        double r55739 = r55737 - r55738;
        double r55740 = r55728 / r55725;
        double r55741 = fma(r55735, r55739, r55740);
        double r55742 = r55727 ? r55731 : r55741;
        return r55742;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 10223.09734470724

    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 10223.09734470724 < 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

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 10223.09734470723924459889531135559082031:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.3333333333333333148296162562473909929395}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))