Average Error: 29.1 → 0.1
Time: 9.8s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 3829.6646277712093:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{log1p}\left(N\right)}, \sqrt{\mathsf{log1p}\left(N\right)}, -\log N\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{3}, \frac{\frac{1}{N}}{N \cdot N}, \frac{1}{N}\right) + \frac{\frac{-1}{2}}{N \cdot N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 3829.6646277712093:\\
\;\;\;\;\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}{3}, \frac{\frac{1}{N}}{N \cdot N}, \frac{1}{N}\right) + \frac{\frac{-1}{2}}{N \cdot N}\\

\end{array}
double f(double N) {
        double r1082377 = N;
        double r1082378 = 1.0;
        double r1082379 = r1082377 + r1082378;
        double r1082380 = log(r1082379);
        double r1082381 = log(r1082377);
        double r1082382 = r1082380 - r1082381;
        return r1082382;
}
↓
double f(double N) {
        double r1082383 = N;
        double r1082384 = 3829.6646277712093;
        bool r1082385 = r1082383 <= r1082384;
        double r1082386 = log1p(r1082383);
        double r1082387 = sqrt(r1082386);
        double r1082388 = log(r1082383);
        double r1082389 = -r1082388;
        double r1082390 = fma(r1082387, r1082387, r1082389);
        double r1082391 = 0.3333333333333333;
        double r1082392 = 1.0;
        double r1082393 = r1082392 / r1082383;
        double r1082394 = r1082383 * r1082383;
        double r1082395 = r1082393 / r1082394;
        double r1082396 = fma(r1082391, r1082395, r1082393);
        double r1082397 = -0.5;
        double r1082398 = r1082397 / r1082394;
        double r1082399 = r1082396 + r1082398;
        double r1082400 = r1082385 ? r1082390 : r1082399;
        return r1082400;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 3829.6646277712093

    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 3829.6646277712093 < 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 add-sqr-sqrt59.9

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

    5. Applied fma-neg60.5

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

    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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 3829.6646277712093:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{log1p}\left(N\right)}, \sqrt{\mathsf{log1p}\left(N\right)}, -\log N\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{3}, \frac{\frac{1}{N}}{N \cdot N}, \frac{1}{N}\right) + \frac{\frac{-1}{2}}{N \cdot N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019155 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))