Average Error: 29.3 → 0.0
Time: 19.2s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7336.587493395303:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{N}}{N}, \frac{-1}{2}, \mathsf{fma}\left(\frac{\frac{1}{N}}{N \cdot N}, \frac{1}{3}, \frac{1}{N}\right)\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7336.587493395303:\\
\;\;\;\;\log \left(\frac{1 + N}{N}\right)\\

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

\end{array}
double f(double N) {
        double r1229306 = N;
        double r1229307 = 1.0;
        double r1229308 = r1229306 + r1229307;
        double r1229309 = log(r1229308);
        double r1229310 = log(r1229306);
        double r1229311 = r1229309 - r1229310;
        return r1229311;
}
↓
double f(double N) {
        double r1229312 = N;
        double r1229313 = 7336.587493395303;
        bool r1229314 = r1229312 <= r1229313;
        double r1229315 = 1.0;
        double r1229316 = r1229315 + r1229312;
        double r1229317 = r1229316 / r1229312;
        double r1229318 = log(r1229317);
        double r1229319 = r1229315 / r1229312;
        double r1229320 = r1229319 / r1229312;
        double r1229321 = -0.5;
        double r1229322 = r1229312 * r1229312;
        double r1229323 = r1229319 / r1229322;
        double r1229324 = 0.3333333333333333;
        double r1229325 = fma(r1229323, r1229324, r1229319);
        double r1229326 = fma(r1229320, r1229321, r1229325);
        double r1229327 = r1229314 ? r1229318 : r1229326;
        return r1229327;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 7336.587493395303

    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 7336.587493395303 < 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

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7336.587493395303:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{N}}{N}, \frac{-1}{2}, \mathsf{fma}\left(\frac{\frac{1}{N}}{N \cdot N}, \frac{1}{3}, \frac{1}{N}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))