Average Error: 29.5 → 0.1
Time: 18.8s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 4683.545058486523:\\ \;\;\;\;\frac{\mathsf{log1p}\left(N\right) \cdot \mathsf{log1p}\left(N\right) - \log N \cdot \log N}{\mathsf{log1p}\left(N\right) + \log N}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{N}}{N \cdot N}, \frac{1}{3}, \frac{1}{N} - \frac{\frac{1}{2}}{N \cdot N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 4683.545058486523:\\
\;\;\;\;\frac{\mathsf{log1p}\left(N\right) \cdot \mathsf{log1p}\left(N\right) - \log N \cdot \log N}{\mathsf{log1p}\left(N\right) + \log N}\\

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

\end{array}
double f(double N) {
        double r1033732 = N;
        double r1033733 = 1.0;
        double r1033734 = r1033732 + r1033733;
        double r1033735 = log(r1033734);
        double r1033736 = log(r1033732);
        double r1033737 = r1033735 - r1033736;
        return r1033737;
}
↓
double f(double N) {
        double r1033738 = N;
        double r1033739 = 4683.545058486523;
        bool r1033740 = r1033738 <= r1033739;
        double r1033741 = log1p(r1033738);
        double r1033742 = r1033741 * r1033741;
        double r1033743 = log(r1033738);
        double r1033744 = r1033743 * r1033743;
        double r1033745 = r1033742 - r1033744;
        double r1033746 = r1033741 + r1033743;
        double r1033747 = r1033745 / r1033746;
        double r1033748 = 1.0;
        double r1033749 = r1033748 / r1033738;
        double r1033750 = r1033738 * r1033738;
        double r1033751 = r1033749 / r1033750;
        double r1033752 = 0.3333333333333333;
        double r1033753 = 0.5;
        double r1033754 = r1033753 / r1033750;
        double r1033755 = r1033749 - r1033754;
        double r1033756 = fma(r1033751, r1033752, r1033755);
        double r1033757 = r1033740 ? r1033747 : r1033756;
        return r1033757;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 4683.545058486523

    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 flip--0.1

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

    if 4683.545058486523 < 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. Using strategy rm

    4. Applied flip--59.5

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

    5. Taylor expanded around inf 0.1

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

    6. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 4683.545058486523:\\ \;\;\;\;\frac{\mathsf{log1p}\left(N\right) \cdot \mathsf{log1p}\left(N\right) - \log N \cdot \log N}{\mathsf{log1p}\left(N\right) + \log N}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{N}}{N \cdot N}, \frac{1}{3}, \frac{1}{N} - \frac{\frac{1}{2}}{N \cdot N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))