Average Error: 29.3 → 0.1
Time: 4.9s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8816.358696954577681026421487331390380859:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.3333333333333333148296162562473909929395}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8816.358696954577681026421487331390380859:\\
\;\;\;\;\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 r74801 = N;
        double r74802 = 1.0;
        double r74803 = r74801 + r74802;
        double r74804 = log(r74803);
        double r74805 = log(r74801);
        double r74806 = r74804 - r74805;
        return r74806;
}
↓
double f(double N) {
        double r74807 = N;
        double r74808 = 8816.358696954578;
        bool r74809 = r74807 <= r74808;
        double r74810 = 1.0;
        double r74811 = r74807 + r74810;
        double r74812 = r74811 / r74807;
        double r74813 = log(r74812);
        double r74814 = 1.0;
        double r74815 = 2.0;
        double r74816 = pow(r74807, r74815);
        double r74817 = r74814 / r74816;
        double r74818 = 0.3333333333333333;
        double r74819 = r74818 / r74807;
        double r74820 = 0.5;
        double r74821 = r74819 - r74820;
        double r74822 = r74810 / r74807;
        double r74823 = fma(r74817, r74821, r74822);
        double r74824 = r74809 ? r74813 : r74823;
        return r74824;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 8816.358696954578

    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 8816.358696954578 < N

    1. Initial program 59.7

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8816.358696954577681026421487331390380859:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.3333333333333333148296162562473909929395}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))