Average Error: 29.1 → 0.1
Time: 15.4s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 4148.853859252381880651228129863739013672:\\ \;\;\;\;\sqrt[3]{{\left(\log \left(N + 1\right)\right)}^{3}} - \log N\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N}, 1 - \frac{0.5}{N}, \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 4148.853859252381880651228129863739013672:\\
\;\;\;\;\sqrt[3]{{\left(\log \left(N + 1\right)\right)}^{3}} - \log N\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{N}, 1 - \frac{0.5}{N}, \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\right)\\

\end{array}
double f(double N) {
        double r56894 = N;
        double r56895 = 1.0;
        double r56896 = r56894 + r56895;
        double r56897 = log(r56896);
        double r56898 = log(r56894);
        double r56899 = r56897 - r56898;
        return r56899;
}
↓
double f(double N) {
        double r56900 = N;
        double r56901 = 4148.853859252382;
        bool r56902 = r56900 <= r56901;
        double r56903 = 1.0;
        double r56904 = r56900 + r56903;
        double r56905 = log(r56904);
        double r56906 = 3.0;
        double r56907 = pow(r56905, r56906);
        double r56908 = cbrt(r56907);
        double r56909 = log(r56900);
        double r56910 = r56908 - r56909;
        double r56911 = 1.0;
        double r56912 = r56911 / r56900;
        double r56913 = 0.5;
        double r56914 = r56913 / r56900;
        double r56915 = r56903 - r56914;
        double r56916 = 0.3333333333333333;
        double r56917 = pow(r56900, r56906);
        double r56918 = r56916 / r56917;
        double r56919 = fma(r56912, r56915, r56918);
        double r56920 = r56902 ? r56910 : r56919;
        return r56920;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 4148.853859252382

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube0.1

      \[\leadsto \color{blue}{\sqrt[3]{\left(\log \left(N + 1\right) \cdot \log \left(N + 1\right)\right) \cdot \log \left(N + 1\right)}} - \log N\]

    4. Simplified0.1

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\log \left(N + 1\right)\right)}^{3}}} - \log N\]

    if 4148.853859252382 < 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

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{N}, 1 - \frac{0.5}{N}, \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 4148.853859252381880651228129863739013672:\\ \;\;\;\;\sqrt[3]{{\left(\log \left(N + 1\right)\right)}^{3}} - \log N\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N}, 1 - \frac{0.5}{N}, \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))