Average Error: 29.3 → 0.1
Time: 3.7s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7930.6783706997021:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N}, 1 - \frac{0.5}{N}, \frac{0.333333333333333315}{{N}^{3}}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7930.6783706997021:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{N}, 1 - \frac{0.5}{N}, \frac{0.333333333333333315}{{N}^{3}}\right)\\

\end{array}
double f(double N) {
        double r36618 = N;
        double r36619 = 1.0;
        double r36620 = r36618 + r36619;
        double r36621 = log(r36620);
        double r36622 = log(r36618);
        double r36623 = r36621 - r36622;
        return r36623;
}
↓
double f(double N) {
        double r36624 = N;
        double r36625 = 7930.678370699702;
        bool r36626 = r36624 <= r36625;
        double r36627 = 1.0;
        double r36628 = r36624 + r36627;
        double r36629 = r36628 / r36624;
        double r36630 = log(r36629);
        double r36631 = 1.0;
        double r36632 = r36631 / r36624;
        double r36633 = 0.5;
        double r36634 = r36633 / r36624;
        double r36635 = r36627 - r36634;
        double r36636 = 0.3333333333333333;
        double r36637 = 3.0;
        double r36638 = pow(r36624, r36637);
        double r36639 = r36636 / r36638;
        double r36640 = fma(r36632, r36635, r36639);
        double r36641 = r36626 ? r36630 : r36640;
        return r36641;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 7930.678370699702

    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 7930.678370699702 < N

    1. Initial program 59.5

      \[\log \left(N + 1\right) - \log N\]
    2. Using strategy rm

    3. Applied diff-log59.3

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

    4. Taylor expanded around inf 0.0

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

    5. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7930.6783706997021:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N}, 1 - \frac{0.5}{N}, \frac{0.333333333333333315}{{N}^{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))