Average Error: 29.6 → 0.1
Time: 23.4s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8386.707428118045:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{N}}{N}, \frac{-1}{2}, \mathsf{fma}\left(\frac{1}{N} \cdot \left(\frac{1}{N} \cdot \frac{1}{N}\right), \frac{1}{3}, \frac{1}{N}\right)\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8386.707428118045:\\
\;\;\;\;\log \left(\frac{1 + N}{N}\right)\\

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

\end{array}
double f(double N) {
        double r1164848 = N;
        double r1164849 = 1.0;
        double r1164850 = r1164848 + r1164849;
        double r1164851 = log(r1164850);
        double r1164852 = log(r1164848);
        double r1164853 = r1164851 - r1164852;
        return r1164853;
}
↓
double f(double N) {
        double r1164854 = N;
        double r1164855 = 8386.707428118045;
        bool r1164856 = r1164854 <= r1164855;
        double r1164857 = 1.0;
        double r1164858 = r1164857 + r1164854;
        double r1164859 = r1164858 / r1164854;
        double r1164860 = log(r1164859);
        double r1164861 = r1164857 / r1164854;
        double r1164862 = r1164861 / r1164854;
        double r1164863 = -0.5;
        double r1164864 = r1164861 * r1164861;
        double r1164865 = r1164861 * r1164864;
        double r1164866 = 0.3333333333333333;
        double r1164867 = fma(r1164865, r1164866, r1164861);
        double r1164868 = fma(r1164862, r1164863, r1164867);
        double r1164869 = r1164856 ? r1164860 : r1164868;
        return r1164869;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 8386.707428118045

    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 8386.707428118045 < 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 log1p-udef59.4

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

    5. Applied diff-log59.2

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

    6. Taylor expanded around inf 0.0

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

    7. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8386.707428118045:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{N}}{N}, \frac{-1}{2}, \mathsf{fma}\left(\frac{1}{N} \cdot \left(\frac{1}{N} \cdot \frac{1}{N}\right), \frac{1}{3}, \frac{1}{N}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))