Average Error: 29.1 → 0.1
Time: 17.3s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 9233.861277203294:\\ \;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{N}}{N}, \frac{\frac{1}{3}}{N} + \frac{-1}{2}, \frac{1}{N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 9233.861277203294:\\
\;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\\

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

\end{array}
double f(double N) {
        double r1665476 = N;
        double r1665477 = 1.0;
        double r1665478 = r1665476 + r1665477;
        double r1665479 = log(r1665478);
        double r1665480 = log(r1665476);
        double r1665481 = r1665479 - r1665480;
        return r1665481;
}
↓
double f(double N) {
        double r1665482 = N;
        double r1665483 = 9233.861277203294;
        bool r1665484 = r1665482 <= r1665483;
        double r1665485 = 1.0;
        double r1665486 = r1665485 + r1665482;
        double r1665487 = r1665486 / r1665482;
        double r1665488 = sqrt(r1665487);
        double r1665489 = log(r1665488);
        double r1665490 = r1665489 + r1665489;
        double r1665491 = r1665485 / r1665482;
        double r1665492 = r1665491 / r1665482;
        double r1665493 = 0.3333333333333333;
        double r1665494 = r1665493 / r1665482;
        double r1665495 = -0.5;
        double r1665496 = r1665494 + r1665495;
        double r1665497 = fma(r1665492, r1665496, r1665491);
        double r1665498 = r1665484 ? r1665490 : r1665497;
        return r1665498;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 9233.861277203294

    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

      \[\leadsto \color{blue}{\log \left(\frac{1 + N}{N}\right)}\]
    6. Using strategy rm

    7. Applied add-sqr-sqrt0.2

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

    8. Applied log-prod0.1

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

    if 9233.861277203294 < 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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9233.861277203294:\\ \;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{N}}{N}, \frac{\frac{1}{3}}{N} + \frac{-1}{2}, \frac{1}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))