Average Error: 29.8 → 0.1
Time: 6.4s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 3680.1889511455383:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(N + 1\right)\right)\right) - \log N\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{N}, \mathsf{fma}\left(0.333333333333333315, \frac{1}{\left(1 - \log \left(\frac{1}{N}\right)\right) \cdot {N}^{3}}, 5.55112 \cdot 10^{-17} \cdot \frac{\log \left(\frac{1}{N}\right)}{{\left(1 - \log \left(\frac{1}{N}\right)\right)}^{3} \cdot {N}^{3}}\right) - \mathsf{fma}\left(5.55112 \cdot 10^{-17}, \frac{1}{{\left(1 - \log \left(\frac{1}{N}\right)\right)}^{3} \cdot {N}^{3}}, \mathsf{fma}\left(0.333333333333333315, \frac{\log \left(\frac{1}{N}\right)}{\left(1 - \log \left(\frac{1}{N}\right)\right) \cdot {N}^{3}}, 0.5 \cdot \frac{1}{{N}^{2}}\right)\right)\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 3680.1889511455383:\\
\;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(N + 1\right)\right)\right) - \log N\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(1, \frac{1}{N}, \mathsf{fma}\left(0.333333333333333315, \frac{1}{\left(1 - \log \left(\frac{1}{N}\right)\right) \cdot {N}^{3}}, 5.55112 \cdot 10^{-17} \cdot \frac{\log \left(\frac{1}{N}\right)}{{\left(1 - \log \left(\frac{1}{N}\right)\right)}^{3} \cdot {N}^{3}}\right) - \mathsf{fma}\left(5.55112 \cdot 10^{-17}, \frac{1}{{\left(1 - \log \left(\frac{1}{N}\right)\right)}^{3} \cdot {N}^{3}}, \mathsf{fma}\left(0.333333333333333315, \frac{\log \left(\frac{1}{N}\right)}{\left(1 - \log \left(\frac{1}{N}\right)\right) \cdot {N}^{3}}, 0.5 \cdot \frac{1}{{N}^{2}}\right)\right)\right)\\

\end{array}
double f(double N) {
        double r70849 = N;
        double r70850 = 1.0;
        double r70851 = r70849 + r70850;
        double r70852 = log(r70851);
        double r70853 = log(r70849);
        double r70854 = r70852 - r70853;
        return r70854;
}
↓
double f(double N) {
        double r70855 = N;
        double r70856 = 3680.1889511455383;
        bool r70857 = r70855 <= r70856;
        double r70858 = 1.0;
        double r70859 = r70855 + r70858;
        double r70860 = log(r70859);
        double r70861 = log1p(r70860);
        double r70862 = expm1(r70861);
        double r70863 = log(r70855);
        double r70864 = r70862 - r70863;
        double r70865 = 1.0;
        double r70866 = r70865 / r70855;
        double r70867 = 0.3333333333333333;
        double r70868 = log(r70866);
        double r70869 = r70865 - r70868;
        double r70870 = 3.0;
        double r70871 = pow(r70855, r70870);
        double r70872 = r70869 * r70871;
        double r70873 = r70865 / r70872;
        double r70874 = 5.551115123125783e-17;
        double r70875 = pow(r70869, r70870);
        double r70876 = r70875 * r70871;
        double r70877 = r70868 / r70876;
        double r70878 = r70874 * r70877;
        double r70879 = fma(r70867, r70873, r70878);
        double r70880 = r70865 / r70876;
        double r70881 = r70868 / r70872;
        double r70882 = 0.5;
        double r70883 = 2.0;
        double r70884 = pow(r70855, r70883);
        double r70885 = r70865 / r70884;
        double r70886 = r70882 * r70885;
        double r70887 = fma(r70867, r70881, r70886);
        double r70888 = fma(r70874, r70880, r70887);
        double r70889 = r70879 - r70888;
        double r70890 = fma(r70858, r70866, r70889);
        double r70891 = r70857 ? r70864 : r70890;
        return r70891;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 3680.1889511455383

    1. Initial program 0.1

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

    3. Applied expm1-log1p-u0.1

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(N + 1\right)\right)\right)} - \log N\]

    if 3680.1889511455383 < N

    1. Initial program 59.3

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

    3. Applied expm1-log1p-u60.2

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(N + 1\right)\right)\right)} - \log N\]

    4. Taylor expanded around inf 0.1

      \[\leadsto \color{blue}{\left(1 \cdot \frac{1}{N} + \left(0.333333333333333315 \cdot \frac{1}{\left(1 - \log \left(\frac{1}{N}\right)\right) \cdot {N}^{3}} + 5.55112 \cdot 10^{-17} \cdot \frac{\log \left(\frac{1}{N}\right)}{{\left(1 - \log \left(\frac{1}{N}\right)\right)}^{3} \cdot {N}^{3}}\right)\right) - \left(5.55112 \cdot 10^{-17} \cdot \frac{1}{{\left(1 - \log \left(\frac{1}{N}\right)\right)}^{3} \cdot {N}^{3}} + \left(0.333333333333333315 \cdot \frac{\log \left(\frac{1}{N}\right)}{\left(1 - \log \left(\frac{1}{N}\right)\right) \cdot {N}^{3}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)\right)}\]

    5. Simplified0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, \frac{1}{N}, \mathsf{fma}\left(0.333333333333333315, \frac{1}{\left(1 - \log \left(\frac{1}{N}\right)\right) \cdot {N}^{3}}, 5.55112 \cdot 10^{-17} \cdot \frac{\log \left(\frac{1}{N}\right)}{{\left(1 - \log \left(\frac{1}{N}\right)\right)}^{3} \cdot {N}^{3}}\right) - \mathsf{fma}\left(5.55112 \cdot 10^{-17}, \frac{1}{{\left(1 - \log \left(\frac{1}{N}\right)\right)}^{3} \cdot {N}^{3}}, \mathsf{fma}\left(0.333333333333333315, \frac{\log \left(\frac{1}{N}\right)}{\left(1 - \log \left(\frac{1}{N}\right)\right) \cdot {N}^{3}}, 0.5 \cdot \frac{1}{{N}^{2}}\right)\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 3680.1889511455383:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(N + 1\right)\right)\right) - \log N\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{N}, \mathsf{fma}\left(0.333333333333333315, \frac{1}{\left(1 - \log \left(\frac{1}{N}\right)\right) \cdot {N}^{3}}, 5.55112 \cdot 10^{-17} \cdot \frac{\log \left(\frac{1}{N}\right)}{{\left(1 - \log \left(\frac{1}{N}\right)\right)}^{3} \cdot {N}^{3}}\right) - \mathsf{fma}\left(5.55112 \cdot 10^{-17}, \frac{1}{{\left(1 - \log \left(\frac{1}{N}\right)\right)}^{3} \cdot {N}^{3}}, \mathsf{fma}\left(0.333333333333333315, \frac{\log \left(\frac{1}{N}\right)}{\left(1 - \log \left(\frac{1}{N}\right)\right) \cdot {N}^{3}}, 0.5 \cdot \frac{1}{{N}^{2}}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))