Average Error: 30.1 → 0.1
Time: 3.1s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 3926.44294337209203:\\ \;\;\;\;e^{\log \left(\log \left(N + 1\right)\right)} - \log N\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{1}{{N}^{2}}, \mathsf{fma}\left(0.333333333333333315, \frac{1}{{N}^{3}}, \frac{1}{N}\right)\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 3926.44294337209203:\\
\;\;\;\;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 r39647 = N;
        double r39648 = 1.0;
        double r39649 = r39647 + r39648;
        double r39650 = log(r39649);
        double r39651 = log(r39647);
        double r39652 = r39650 - r39651;
        return r39652;
}
↓
double f(double N) {
        double r39653 = N;
        double r39654 = 3926.442943372092;
        bool r39655 = r39653 <= r39654;
        double r39656 = 1.0;
        double r39657 = r39653 + r39656;
        double r39658 = log(r39657);
        double r39659 = log(r39658);
        double r39660 = exp(r39659);
        double r39661 = log(r39653);
        double r39662 = r39660 - r39661;
        double r39663 = 0.5;
        double r39664 = -r39663;
        double r39665 = 1.0;
        double r39666 = 2.0;
        double r39667 = pow(r39653, r39666);
        double r39668 = r39665 / r39667;
        double r39669 = 0.3333333333333333;
        double r39670 = 3.0;
        double r39671 = pow(r39653, r39670);
        double r39672 = r39665 / r39671;
        double r39673 = r39656 / r39653;
        double r39674 = fma(r39669, r39672, r39673);
        double r39675 = fma(r39664, r39668, r39674);
        double r39676 = r39655 ? r39662 : r39675;
        return r39676;
}

Error

Bits error versus N

Derivation

  1. Split input into 2 regimes

  2. if N < 3926.442943372092

    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 3926.442943372092 < N

    1. Initial program 59.4

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

    3. Applied add-exp-log60.4

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 3926.44294337209203:\\ \;\;\;\;e^{\log \left(\log \left(N + 1\right)\right)} - \log N\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{1}{{N}^{2}}, \mathsf{fma}\left(0.333333333333333315, \frac{1}{{N}^{3}}, \frac{1}{N}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))