Average Error: 29.7 → 0.1
Time: 2.6s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7877.19975446712033:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N}, 1 - \frac{0.5}{N}, \frac{0.333333333333333315}{{N}^{3}}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7877.19975446712033:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

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

\end{array}
double f(double N) {
        double r39618 = N;
        double r39619 = 1.0;
        double r39620 = r39618 + r39619;
        double r39621 = log(r39620);
        double r39622 = log(r39618);
        double r39623 = r39621 - r39622;
        return r39623;
}
↓
double f(double N) {
        double r39624 = N;
        double r39625 = 7877.19975446712;
        bool r39626 = r39624 <= r39625;
        double r39627 = 1.0;
        double r39628 = r39624 + r39627;
        double r39629 = r39628 / r39624;
        double r39630 = log(r39629);
        double r39631 = 1.0;
        double r39632 = r39631 / r39624;
        double r39633 = 0.5;
        double r39634 = r39633 / r39624;
        double r39635 = r39627 - r39634;
        double r39636 = 0.3333333333333333;
        double r39637 = 3.0;
        double r39638 = pow(r39624, r39637);
        double r39639 = r39636 / r39638;
        double r39640 = fma(r39632, r39635, r39639);
        double r39641 = r39626 ? r39630 : r39640;
        return r39641;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 7877.19975446712

    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 7877.19975446712 < N

    1. Initial program 59.5

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

    3. Applied diff-log59.3

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

    4. Taylor expanded around inf 0.0

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

    5. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7877.19975446712033:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N}, 1 - \frac{0.5}{N}, \frac{0.333333333333333315}{{N}^{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))