Average Error: 28.9 → 0.1
Time: 16.8s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8041.189519456958:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{N}}{N}, \frac{\frac{1}{3}}{N} + \frac{-1}{2}, \frac{1}{N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8041.189519456958:\\
\;\;\;\;\log \left(\frac{1 + N}{N}\right)\\

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

\end{array}
double f(double N) {
        double r2040831 = N;
        double r2040832 = 1.0;
        double r2040833 = r2040831 + r2040832;
        double r2040834 = log(r2040833);
        double r2040835 = log(r2040831);
        double r2040836 = r2040834 - r2040835;
        return r2040836;
}
↓
double f(double N) {
        double r2040837 = N;
        double r2040838 = 8041.189519456958;
        bool r2040839 = r2040837 <= r2040838;
        double r2040840 = 1.0;
        double r2040841 = r2040840 + r2040837;
        double r2040842 = r2040841 / r2040837;
        double r2040843 = log(r2040842);
        double r2040844 = r2040840 / r2040837;
        double r2040845 = r2040844 / r2040837;
        double r2040846 = 0.3333333333333333;
        double r2040847 = r2040846 / r2040837;
        double r2040848 = -0.5;
        double r2040849 = r2040847 + r2040848;
        double r2040850 = fma(r2040845, r2040849, r2040844);
        double r2040851 = r2040839 ? r2040843 : r2040850;
        return r2040851;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 8041.189519456958

    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 8041.189519456958 < N

    1. Initial program 59.6

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

    2. Simplified59.6

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

    3. Taylor expanded around inf 0.0

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

    4. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8041.189519456958:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{N}}{N}, \frac{\frac{1}{3}}{N} + \frac{-1}{2}, \frac{1}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))