2log (problem 3.3.6)

Average Error: 29.8 → 0.1

Time: 5.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 8116.362270388288379763253033161163330078:\\ \;\;\;\;\log \left(\frac{\sqrt{N + 1}}{\sqrt{N}}\right) + \log \left(\frac{\sqrt{N + 1}}{\sqrt{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N}, 1 - \frac{0.5}{N}, \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\right)\\ \end{array}\]

C

double f(double N) {
        double r55702 = N;
        double r55703 = 1.0;
        double r55704 = r55702 + r55703;
        double r55705 = log(r55704);
        double r55706 = log(r55702);
        double r55707 = r55705 - r55706;
        return r55707;
}

↓

double f(double N) {
        double r55708 = N;
        double r55709 = 8116.362270388288;
        bool r55710 = r55708 <= r55709;
        double r55711 = 1.0;
        double r55712 = r55708 + r55711;
        double r55713 = sqrt(r55712);
        double r55714 = sqrt(r55708);
        double r55715 = r55713 / r55714;
        double r55716 = log(r55715);
        double r55717 = r55716 + r55716;
        double r55718 = 1.0;
        double r55719 = r55718 / r55708;
        double r55720 = 0.5;
        double r55721 = r55720 / r55708;
        double r55722 = r55711 - r55721;
        double r55723 = 0.3333333333333333;
        double r55724 = 3.0;
        double r55725 = pow(r55708, r55724);
        double r55726 = r55723 / r55725;
        double r55727 = fma(r55719, r55722, r55726);
        double r55728 = r55710 ? r55717 : r55727;
        return r55728;
}

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-log 0.1

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

   5. Applied add-sqr-sqrt 0.1

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

   6. Applied add-sqr-sqrt 0.1

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

   7. Applied times-frac 0.1

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

   8. Applied log-prod 0.1

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

   if 8116.362270388288 < N

   1. Initial program 59.4

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

   3. Applied diff-log 59.2

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

   4. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.3333333333333333148296162562473909929395 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

   5. Simplified 0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{N}, 1 - \frac{0.5}{N}, \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8116.362270388288379763253033161163330078:\\ \;\;\;\;\log \left(\frac{\sqrt{N + 1}}{\sqrt{N}}\right) + \log \left(\frac{\sqrt{N + 1}}{\sqrt{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N}, 1 - \frac{0.5}{N}, \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\right)\\ \end{array}\]

Reproduce

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