Average Error: 29.3 → 0.1
Time: 21.4s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 6916.42260176444:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{log1p}\left(N\right)}, \sqrt{\mathsf{log1p}\left(N\right)}, -\log N\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N \cdot N}, \frac{-1}{2}, \frac{\frac{\frac{1}{3}}{N}}{N \cdot N}\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 6916.42260176444:\\
\;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{log1p}\left(N\right)}, \sqrt{\mathsf{log1p}\left(N\right)}, -\log N\right)\\

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

\end{array}
double f(double N) {
        double r1520923 = N;
        double r1520924 = 1.0;
        double r1520925 = r1520923 + r1520924;
        double r1520926 = log(r1520925);
        double r1520927 = log(r1520923);
        double r1520928 = r1520926 - r1520927;
        return r1520928;
}
↓
double f(double N) {
        double r1520929 = N;
        double r1520930 = 6916.42260176444;
        bool r1520931 = r1520929 <= r1520930;
        double r1520932 = log1p(r1520929);
        double r1520933 = sqrt(r1520932);
        double r1520934 = log(r1520929);
        double r1520935 = -r1520934;
        double r1520936 = fma(r1520933, r1520933, r1520935);
        double r1520937 = 1.0;
        double r1520938 = r1520929 * r1520929;
        double r1520939 = r1520937 / r1520938;
        double r1520940 = -0.5;
        double r1520941 = 0.3333333333333333;
        double r1520942 = r1520941 / r1520929;
        double r1520943 = r1520942 / r1520938;
        double r1520944 = fma(r1520939, r1520940, r1520943);
        double r1520945 = r1520937 / r1520929;
        double r1520946 = r1520944 + r1520945;
        double r1520947 = r1520931 ? r1520936 : r1520946;
        return r1520947;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 6916.42260176444

    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 add-sqr-sqrt0.1

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

    5. Applied fma-neg0.1

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

    if 6916.42260176444 < N

    1. Initial program 59.2

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

    2. Simplified59.2

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

    3. Taylor expanded around inf 0.1

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

    4. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 6916.42260176444:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{log1p}\left(N\right)}, \sqrt{\mathsf{log1p}\left(N\right)}, -\log N\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N \cdot N}, \frac{-1}{2}, \frac{\frac{\frac{1}{3}}{N}}{N \cdot N}\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))