Average Error: 29.8 → 0.1
Time: 13.3s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 9562.6451805155593:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \left(\frac{0.333333333333333315}{{N}^{3}} - \frac{\frac{0.5}{N}}{N}\right)\right) + \mathsf{fma}\left(-\frac{0.5}{N \cdot N}, 1, \frac{0.5}{N \cdot N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 9562.6451805155593:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{N} + \left(\frac{0.333333333333333315}{{N}^{3}} - \frac{\frac{0.5}{N}}{N}\right)\right) + \mathsf{fma}\left(-\frac{0.5}{N \cdot N}, 1, \frac{0.5}{N \cdot N}\right)\\

\end{array}
double f(double N) {
        double r69687 = N;
        double r69688 = 1.0;
        double r69689 = r69687 + r69688;
        double r69690 = log(r69689);
        double r69691 = log(r69687);
        double r69692 = r69690 - r69691;
        return r69692;
}
↓
double f(double N) {
        double r69693 = N;
        double r69694 = 9562.64518051556;
        bool r69695 = r69693 <= r69694;
        double r69696 = 1.0;
        double r69697 = r69693 + r69696;
        double r69698 = r69697 / r69693;
        double r69699 = log(r69698);
        double r69700 = r69696 / r69693;
        double r69701 = 0.3333333333333333;
        double r69702 = 3.0;
        double r69703 = pow(r69693, r69702);
        double r69704 = r69701 / r69703;
        double r69705 = 0.5;
        double r69706 = r69705 / r69693;
        double r69707 = r69706 / r69693;
        double r69708 = r69704 - r69707;
        double r69709 = r69700 + r69708;
        double r69710 = r69693 * r69693;
        double r69711 = r69705 / r69710;
        double r69712 = -r69711;
        double r69713 = 1.0;
        double r69714 = fma(r69712, r69713, r69711);
        double r69715 = r69709 + r69714;
        double r69716 = r69695 ? r69699 : r69715;
        return r69716;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 9562.64518051556

    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 9562.64518051556 < 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

      \[\leadsto \color{blue}{\frac{0.333333333333333315}{{N}^{3}} + \left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right)}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity0.0

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

    6. Applied add-sqr-sqrt0.5

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

    7. Applied prod-diff0.5

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

    8. Applied associate-+r+0.5

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

    9. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9562.6451805155593:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \left(\frac{0.333333333333333315}{{N}^{3}} - \frac{\frac{0.5}{N}}{N}\right)\right) + \mathsf{fma}\left(-\frac{0.5}{N \cdot N}, 1, \frac{0.5}{N \cdot N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))