Average Error: 30.1 → 0.1
Time: 20.1s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 9454.187458193411657703109085559844970703:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.3333333333333333148296162562473909929395}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 9454.187458193411657703109085559844970703:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.3333333333333333148296162562473909929395}{N} - 0.5, \frac{1}{N}\right)\\

\end{array}
double f(double N) {
        double r38514 = N;
        double r38515 = 1.0;
        double r38516 = r38514 + r38515;
        double r38517 = log(r38516);
        double r38518 = log(r38514);
        double r38519 = r38517 - r38518;
        return r38519;
}
↓
double f(double N) {
        double r38520 = N;
        double r38521 = 9454.187458193412;
        bool r38522 = r38520 <= r38521;
        double r38523 = 1.0;
        double r38524 = r38520 + r38523;
        double r38525 = r38524 / r38520;
        double r38526 = log(r38525);
        double r38527 = 1.0;
        double r38528 = 2.0;
        double r38529 = pow(r38520, r38528);
        double r38530 = r38527 / r38529;
        double r38531 = 0.3333333333333333;
        double r38532 = r38531 / r38520;
        double r38533 = 0.5;
        double r38534 = r38532 - r38533;
        double r38535 = r38523 / r38520;
        double r38536 = fma(r38530, r38534, r38535);
        double r38537 = r38522 ? r38526 : r38536;
        return r38537;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 9454.187458193412

    1. Initial program 0.1

      \[\log \left(N + 1\right) - \log N\]
    2. Using strategy rm

    3. Applied diff-log0.1

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

    if 9454.187458193412 < N

    1. Initial program 59.4

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

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.3333333333333333148296162562473909929395 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.3333333333333333148296162562473909929395}{N} - 0.5, \frac{1}{N}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9454.187458193411657703109085559844970703:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.3333333333333333148296162562473909929395}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))