Average Error: 30.0 → 0.1
Time: 16.2s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8885.992756934142:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, \left(\frac{\frac{1}{N}}{N}\right), \left(\mathsf{fma}\left(\frac{1}{3}, \left(\frac{\frac{\frac{1}{N}}{N}}{N}\right), \left(\frac{1}{N}\right)\right)\right)\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8885.992756934142:\\
\;\;\;\;\log \left(\frac{1 + N}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, \left(\frac{\frac{1}{N}}{N}\right), \left(\mathsf{fma}\left(\frac{1}{3}, \left(\frac{\frac{\frac{1}{N}}{N}}{N}\right), \left(\frac{1}{N}\right)\right)\right)\right)\\

\end{array}
double f(double N) {
        double r1003905 = N;
        double r1003906 = 1.0;
        double r1003907 = r1003905 + r1003906;
        double r1003908 = log(r1003907);
        double r1003909 = log(r1003905);
        double r1003910 = r1003908 - r1003909;
        return r1003910;
}
↓
double f(double N) {
        double r1003911 = N;
        double r1003912 = 8885.992756934142;
        bool r1003913 = r1003911 <= r1003912;
        double r1003914 = 1.0;
        double r1003915 = r1003914 + r1003911;
        double r1003916 = r1003915 / r1003911;
        double r1003917 = log(r1003916);
        double r1003918 = -0.5;
        double r1003919 = r1003914 / r1003911;
        double r1003920 = r1003919 / r1003911;
        double r1003921 = 0.3333333333333333;
        double r1003922 = r1003920 / r1003911;
        double r1003923 = fma(r1003921, r1003922, r1003919);
        double r1003924 = fma(r1003918, r1003920, r1003923);
        double r1003925 = r1003913 ? r1003917 : r1003924;
        return r1003925;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 8885.992756934142

    1. Initial program 0.1

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

    2. Simplified0.1

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N}\]
    3. Using strategy rm

    4. Applied log1p-udef0.1

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

    5. Applied diff-log0.1

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

    if 8885.992756934142 < N

    1. Initial program 59.4

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

    2. Simplified59.4

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N}\]
    3. Using strategy rm

    4. Applied add-cbrt-cube59.4

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\mathsf{log1p}\left(N\right) - \log N\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)}}\]

    5. Taylor expanded around inf 0.0

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

    6. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8885.992756934142:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, \left(\frac{\frac{1}{N}}{N}\right), \left(\mathsf{fma}\left(\frac{1}{3}, \left(\frac{\frac{\frac{1}{N}}{N}}{N}\right), \left(\frac{1}{N}\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))