Average Error: 29.3 → 0.1
Time: 3.0s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 9398.8604648901419:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.333333333333333315}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 9398.8604648901419:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.333333333333333315}{N} - 0.5, \frac{1}{N}\right)\\

\end{array}
double f(double N) {
        double r40883 = N;
        double r40884 = 1.0;
        double r40885 = r40883 + r40884;
        double r40886 = log(r40885);
        double r40887 = log(r40883);
        double r40888 = r40886 - r40887;
        return r40888;
}
↓
double f(double N) {
        double r40889 = N;
        double r40890 = 9398.860464890142;
        bool r40891 = r40889 <= r40890;
        double r40892 = 1.0;
        double r40893 = r40889 + r40892;
        double r40894 = r40893 / r40889;
        double r40895 = log(r40894);
        double r40896 = 1.0;
        double r40897 = 2.0;
        double r40898 = pow(r40889, r40897);
        double r40899 = r40896 / r40898;
        double r40900 = 0.3333333333333333;
        double r40901 = r40900 / r40889;
        double r40902 = 0.5;
        double r40903 = r40901 - r40902;
        double r40904 = r40892 / r40889;
        double r40905 = fma(r40899, r40903, r40904);
        double r40906 = r40891 ? r40895 : r40905;
        return r40906;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 9398.860464890142

    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 9398.860464890142 < N

    1. Initial program 59.4

      \[\log \left(N + 1\right) - \log N\]

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9398.8604648901419:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.333333333333333315}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))