2log (problem 3.3.6)

Average Error: 29.9 → 0.1

Time: 16.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 10264.43105219115204818081110715866088867:\\ \;\;\;\;\log \left(\frac{\sqrt{1 + N}}{N}\right) + \log \left(\sqrt{1 + N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N}, 1, \frac{\frac{1}{N}}{N} \cdot \frac{0.3333333333333333148296162562473909929395}{N} - \frac{\frac{1}{N}}{N} \cdot 0.5\right)\\ \end{array}\]

C

double f(double N) {
        double r3565280 = N;
        double r3565281 = 1.0;
        double r3565282 = r3565280 + r3565281;
        double r3565283 = log(r3565282);
        double r3565284 = log(r3565280);
        double r3565285 = r3565283 - r3565284;
        return r3565285;
}

↓

double f(double N) {
        double r3565286 = N;
        double r3565287 = 10264.431052191152;
        bool r3565288 = r3565286 <= r3565287;
        double r3565289 = 1.0;
        double r3565290 = r3565289 + r3565286;
        double r3565291 = sqrt(r3565290);
        double r3565292 = r3565291 / r3565286;
        double r3565293 = log(r3565292);
        double r3565294 = log(r3565291);
        double r3565295 = r3565293 + r3565294;
        double r3565296 = 1.0;
        double r3565297 = r3565296 / r3565286;
        double r3565298 = r3565297 / r3565286;
        double r3565299 = 0.3333333333333333;
        double r3565300 = r3565299 / r3565286;
        double r3565301 = r3565298 * r3565300;
        double r3565302 = 0.5;
        double r3565303 = r3565298 * r3565302;
        double r3565304 = r3565301 - r3565303;
        double r3565305 = fma(r3565297, r3565289, r3565304);
        double r3565306 = r3565288 ? r3565295 : r3565305;
        return r3565306;
}

Error

Bits error versus N

Derivation

1. Split input into 2 regimes

2. if N < 10264.431052191152

   1. Initial program 0.1

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

   3. Applied add-log-exp 0.1

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

   4. Simplified 0.1

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

   6. Applied *-un-lft-identity 0.1

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

   7. Applied add-sqr-sqrt 0.1

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

   8. Applied times-frac 0.1

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

   9. Applied log-prod 0.1

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

   if 10264.431052191152 < N

   1. Initial program 59.6

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

   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. Simplified 0.0

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 10264.43105219115204818081110715866088867:\\ \;\;\;\;\log \left(\frac{\sqrt{1 + N}}{N}\right) + \log \left(\sqrt{1 + N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N}, 1, \frac{\frac{1}{N}}{N} \cdot \frac{0.3333333333333333148296162562473909929395}{N} - \frac{\frac{1}{N}}{N} \cdot 0.5\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1.0)) (log N)))