Average Error: 30.0 → 0.1
Time: 3.0s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7004.8538811289454:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.333333333333333315}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7004.8538811289454:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.333333333333333315}{N} - 0.5, \frac{1}{N}\right)\\

\end{array}
double f(double N) {
        double r66780 = N;
        double r66781 = 1.0;
        double r66782 = r66780 + r66781;
        double r66783 = log(r66782);
        double r66784 = log(r66780);
        double r66785 = r66783 - r66784;
        return r66785;
}
↓
double f(double N) {
        double r66786 = N;
        double r66787 = 7004.853881128945;
        bool r66788 = r66786 <= r66787;
        double r66789 = 1.0;
        double r66790 = r66786 + r66789;
        double r66791 = r66790 / r66786;
        double r66792 = log(r66791);
        double r66793 = 1.0;
        double r66794 = 2.0;
        double r66795 = pow(r66786, r66794);
        double r66796 = r66793 / r66795;
        double r66797 = 0.3333333333333333;
        double r66798 = r66797 / r66786;
        double r66799 = 0.5;
        double r66800 = r66798 - r66799;
        double r66801 = r66789 / r66786;
        double r66802 = fma(r66796, r66800, r66801);
        double r66803 = r66788 ? r66792 : r66802;
        return r66803;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 7004.853881128945

    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 7004.853881128945 < N

    1. Initial program 59.5

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

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.333333333333333315 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.333333333333333315}{N} - 0.5, \frac{1}{N}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7004.8538811289454:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.333333333333333315}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))