Average Error: 28.9 → 0.1
Time: 23.3s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8114.657537075721:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{3}, \frac{\frac{1}{N \cdot N}}{N}, \mathsf{fma}\left(\frac{1}{N \cdot N}, \frac{-1}{2}, \frac{1}{N}\right)\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8114.657537075721:\\
\;\;\;\;\log \left(\frac{1 + N}{N}\right)\\

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

\end{array}
double f(double N) {
        double r935613 = N;
        double r935614 = 1.0;
        double r935615 = r935613 + r935614;
        double r935616 = log(r935615);
        double r935617 = log(r935613);
        double r935618 = r935616 - r935617;
        return r935618;
}
↓
double f(double N) {
        double r935619 = N;
        double r935620 = 8114.657537075721;
        bool r935621 = r935619 <= r935620;
        double r935622 = 1.0;
        double r935623 = r935622 + r935619;
        double r935624 = r935623 / r935619;
        double r935625 = log(r935624);
        double r935626 = 0.3333333333333333;
        double r935627 = r935619 * r935619;
        double r935628 = r935622 / r935627;
        double r935629 = r935628 / r935619;
        double r935630 = -0.5;
        double r935631 = r935622 / r935619;
        double r935632 = fma(r935628, r935630, r935631);
        double r935633 = fma(r935626, r935629, r935632);
        double r935634 = r935621 ? r935625 : r935633;
        return r935634;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 8114.657537075721

    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 8114.657537075721 < N

    1. Initial program 59.5

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

    2. Simplified59.5

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

    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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8114.657537075721:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{3}, \frac{\frac{1}{N \cdot N}}{N}, \mathsf{fma}\left(\frac{1}{N \cdot N}, \frac{-1}{2}, \frac{1}{N}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))