2log (problem 3.3.6)

Average Error: 29.5 → 0.1

Time: 12.7s

Precision: 64

Math

\[\log \left(N + 1\right) - \log N\]

↓

\[\begin{array}{l} \mathbf{if}\;N \le 4683.545058486523:\\ \;\;\;\;\frac{\mathsf{log1p}\left(N\right) \cdot \mathsf{log1p}\left(N\right) - \log N \cdot \log N}{\mathsf{log1p}\left(N\right) + \log N}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{N}}{N}, \frac{\frac{1}{3}}{N} - \frac{1}{2}, \frac{1}{N}\right)\\ \end{array}\]

C

double f(double N) {
        double r1198698 = N;
        double r1198699 = 1.0;
        double r1198700 = r1198698 + r1198699;
        double r1198701 = log(r1198700);
        double r1198702 = log(r1198698);
        double r1198703 = r1198701 - r1198702;
        return r1198703;
}

↓

double f(double N) {
        double r1198704 = N;
        double r1198705 = 4683.545058486523;
        bool r1198706 = r1198704 <= r1198705;
        double r1198707 = log1p(r1198704);
        double r1198708 = r1198707 * r1198707;
        double r1198709 = log(r1198704);
        double r1198710 = r1198709 * r1198709;
        double r1198711 = r1198708 - r1198710;
        double r1198712 = r1198707 + r1198709;
        double r1198713 = r1198711 / r1198712;
        double r1198714 = 1.0;
        double r1198715 = r1198714 / r1198704;
        double r1198716 = r1198715 / r1198704;
        double r1198717 = 0.3333333333333333;
        double r1198718 = r1198717 / r1198704;
        double r1198719 = 0.5;
        double r1198720 = r1198718 - r1198719;
        double r1198721 = fma(r1198716, r1198720, r1198715);
        double r1198722 = r1198706 ? r1198713 : r1198721;
        return r1198722;
}

Error

Bits error versus N

Derivation

1. Split input into 2 regimes

2. if N < 4683.545058486523

   1. Initial program 0.1

      \[\log \left(N + 1\right) - \log N\]

   2. Simplified 0.1

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N}\]
   3. Using strategy rm

   4. Applied flip-- 0.1

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(N\right) \cdot \mathsf{log1p}\left(N\right) - \log N \cdot \log N}{\mathsf{log1p}\left(N\right) + \log N}}\]

   if 4683.545058486523 < N

   1. Initial program 59.5

      \[\log \left(N + 1\right) - \log N\]

   2. Simplified 59.5

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N}\]

   3. Taylor expanded around inf 0.1

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

   4. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 4683.545058486523:\\ \;\;\;\;\frac{\mathsf{log1p}\left(N\right) \cdot \mathsf{log1p}\left(N\right) - \log N \cdot \log N}{\mathsf{log1p}\left(N\right) + \log N}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{N}}{N}, \frac{\frac{1}{3}}{N} - \frac{1}{2}, \frac{1}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))