Average Error: 28.8 → 0.1
Time: 5.1s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \le 2.35842836585 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N}, 1 - \frac{0.5}{N}, \frac{0.333333333333333315}{{N}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;\log \left(N + 1\right) - \log N \le 2.35842836585 \cdot 10^{-5}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{N}, 1 - \frac{0.5}{N}, \frac{0.333333333333333315}{{N}^{3}}\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

\end{array}
double f(double N) {
        double r36707 = N;
        double r36708 = 1.0;
        double r36709 = r36707 + r36708;
        double r36710 = log(r36709);
        double r36711 = log(r36707);
        double r36712 = r36710 - r36711;
        return r36712;
}
↓
double f(double N) {
        double r36713 = N;
        double r36714 = 1.0;
        double r36715 = r36713 + r36714;
        double r36716 = log(r36715);
        double r36717 = log(r36713);
        double r36718 = r36716 - r36717;
        double r36719 = 2.3584283658451e-05;
        bool r36720 = r36718 <= r36719;
        double r36721 = 1.0;
        double r36722 = r36721 / r36713;
        double r36723 = 0.5;
        double r36724 = r36723 / r36713;
        double r36725 = r36714 - r36724;
        double r36726 = 0.3333333333333333;
        double r36727 = 3.0;
        double r36728 = pow(r36713, r36727);
        double r36729 = r36726 / r36728;
        double r36730 = fma(r36722, r36725, r36729);
        double r36731 = r36715 / r36713;
        double r36732 = log(r36731);
        double r36733 = r36720 ? r36730 : r36732;
        return r36733;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if (- (log (+ N 1.0)) (log N)) < 2.3584283658451e-05

    1. Initial program 59.6

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

    3. Applied diff-log59.4

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

    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

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{N}, 1 - \frac{0.5}{N}, \frac{0.333333333333333315}{{N}^{3}}\right)}\]

    if 2.3584283658451e-05 < (- (log (+ N 1.0)) (log N))

    1. Initial program 0.1

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

    3. Applied diff-log0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \le 2.35842836585 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N}, 1 - \frac{0.5}{N}, \frac{0.333333333333333315}{{N}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))