Average Error: 29.6 → 0.1
Time: 3.6s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 3438.6849377257358:\\ \;\;\;\;e^{\log \left(\log \left(N + 1\right)\right)} - \log N\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{1}{{N}^{2}}, \mathsf{fma}\left(0.333333333333333315, \frac{1}{{N}^{3}}, \frac{1}{N}\right)\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 3438.6849377257358:\\
\;\;\;\;e^{\log \left(\log \left(N + 1\right)\right)} - \log N\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(-0.5, \frac{1}{{N}^{2}}, \mathsf{fma}\left(0.333333333333333315, \frac{1}{{N}^{3}}, \frac{1}{N}\right)\right)\\

\end{array}
double f(double N) {
        double r34600 = N;
        double r34601 = 1.0;
        double r34602 = r34600 + r34601;
        double r34603 = log(r34602);
        double r34604 = log(r34600);
        double r34605 = r34603 - r34604;
        return r34605;
}
↓
double f(double N) {
        double r34606 = N;
        double r34607 = 3438.684937725736;
        bool r34608 = r34606 <= r34607;
        double r34609 = 1.0;
        double r34610 = r34606 + r34609;
        double r34611 = log(r34610);
        double r34612 = log(r34611);
        double r34613 = exp(r34612);
        double r34614 = log(r34606);
        double r34615 = r34613 - r34614;
        double r34616 = 0.5;
        double r34617 = -r34616;
        double r34618 = 1.0;
        double r34619 = 2.0;
        double r34620 = pow(r34606, r34619);
        double r34621 = r34618 / r34620;
        double r34622 = 0.3333333333333333;
        double r34623 = 3.0;
        double r34624 = pow(r34606, r34623);
        double r34625 = r34618 / r34624;
        double r34626 = r34609 / r34606;
        double r34627 = fma(r34622, r34625, r34626);
        double r34628 = fma(r34617, r34621, r34627);
        double r34629 = r34608 ? r34615 : r34628;
        return r34629;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 3438.684937725736

    1. Initial program 0.1

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

    3. Applied add-exp-log0.1

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

    if 3438.684937725736 < N

    1. Initial program 59.4

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

    3. Applied add-exp-log60.2

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

    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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 3438.6849377257358:\\ \;\;\;\;e^{\log \left(\log \left(N + 1\right)\right)} - \log N\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{1}{{N}^{2}}, \mathsf{fma}\left(0.333333333333333315, \frac{1}{{N}^{3}}, \frac{1}{N}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))