Average Error: 29.8 → 0.1
Time: 5.4s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8116.362270388288379763253033161163330078:\\ \;\;\;\;\log \left(\frac{\sqrt{N + 1}}{\sqrt{N}}\right) + \log \left(\frac{\sqrt{N + 1}}{\sqrt{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.3333333333333333148296162562473909929395}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8116.362270388288379763253033161163330078:\\
\;\;\;\;\log \left(\frac{\sqrt{N + 1}}{\sqrt{N}}\right) + \log \left(\frac{\sqrt{N + 1}}{\sqrt{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 r55606 = N;
        double r55607 = 1.0;
        double r55608 = r55606 + r55607;
        double r55609 = log(r55608);
        double r55610 = log(r55606);
        double r55611 = r55609 - r55610;
        return r55611;
}
↓
double f(double N) {
        double r55612 = N;
        double r55613 = 8116.362270388288;
        bool r55614 = r55612 <= r55613;
        double r55615 = 1.0;
        double r55616 = r55612 + r55615;
        double r55617 = sqrt(r55616);
        double r55618 = sqrt(r55612);
        double r55619 = r55617 / r55618;
        double r55620 = log(r55619);
        double r55621 = r55620 + r55620;
        double r55622 = 1.0;
        double r55623 = 2.0;
        double r55624 = pow(r55612, r55623);
        double r55625 = r55622 / r55624;
        double r55626 = 0.3333333333333333;
        double r55627 = r55626 / r55612;
        double r55628 = 0.5;
        double r55629 = r55627 - r55628;
        double r55630 = r55615 / r55612;
        double r55631 = fma(r55625, r55629, r55630);
        double r55632 = r55614 ? r55621 : r55631;
        return r55632;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 8116.362270388288

    1. Initial program 0.1

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

    3. Applied diff-log0.1

      \[\leadsto \color{blue}{\log \left(\frac{N + 1}{N}\right)}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt0.1

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

    6. Applied add-sqr-sqrt0.1

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

    7. Applied times-frac0.1

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

    8. Applied log-prod0.1

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

    if 8116.362270388288 < 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

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8116.362270388288379763253033161163330078:\\ \;\;\;\;\log \left(\frac{\sqrt{N + 1}}{\sqrt{N}}\right) + \log \left(\frac{\sqrt{N + 1}}{\sqrt{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.3333333333333333148296162562473909929395}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))