2log (problem 3.3.6)

Average Error: 29.5 → 0.1

Time: 15.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 5400.000739830978:\\ \;\;\;\;\log \left(\frac{\sqrt{1 + N}}{\sqrt{N}}\right) + \left(\log \left(\sqrt{1 + N}\right) - \log \left(\sqrt{N}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \frac{\frac{\frac{1}{3}}{N} - \frac{1}{2}}{N \cdot N}\\ \end{array}\]

C

double f(double N) {
        double r2501355 = N;
        double r2501356 = 1.0;
        double r2501357 = r2501355 + r2501356;
        double r2501358 = log(r2501357);
        double r2501359 = log(r2501355);
        double r2501360 = r2501358 - r2501359;
        return r2501360;
}

↓

double f(double N) {
        double r2501361 = N;
        double r2501362 = 5400.000739830978;
        bool r2501363 = r2501361 <= r2501362;
        double r2501364 = 1.0;
        double r2501365 = r2501364 + r2501361;
        double r2501366 = sqrt(r2501365);
        double r2501367 = sqrt(r2501361);
        double r2501368 = r2501366 / r2501367;
        double r2501369 = log(r2501368);
        double r2501370 = log(r2501366);
        double r2501371 = log(r2501367);
        double r2501372 = r2501370 - r2501371;
        double r2501373 = r2501369 + r2501372;
        double r2501374 = r2501364 / r2501361;
        double r2501375 = 0.3333333333333333;
        double r2501376 = r2501375 / r2501361;
        double r2501377 = 0.5;
        double r2501378 = r2501376 - r2501377;
        double r2501379 = r2501361 * r2501361;
        double r2501380 = r2501378 / r2501379;
        double r2501381 = r2501374 + r2501380;
        double r2501382 = r2501363 ? r2501373 : r2501381;
        return r2501382;
}

Error

Bits error versus N

Derivation

1. Split input into 2 regimes

2. if N < 5400.000739830978

   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 *-un-lft-identity 0.1

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

   6. Applied add-sqr-sqrt 0.1

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

   7. Applied times-frac 0.1

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

   8. Applied log-prod 0.1

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

   10. Applied add-sqr-sqrt 0.1

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

   11. Applied *-un-lft-identity 0.1

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

   12. Applied sqrt-prod 0.1

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

   13. Applied times-frac 0.1

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

   14. Applied log-prod 0.1

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

   15. Applied associate-+r+ 0.1

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

   16. Simplified 0.1

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

   if 5400.000739830978 < N

   1. Initial program 59.4

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

   2. Taylor expanded around inf 0.0

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

   3. Simplified 0.0

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{3}}{N} - \frac{1}{2}}{N \cdot N} + \frac{1}{N}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 5400.000739830978:\\ \;\;\;\;\log \left(\frac{\sqrt{1 + N}}{\sqrt{N}}\right) + \left(\log \left(\sqrt{1 + N}\right) - \log \left(\sqrt{N}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \frac{\frac{\frac{1}{3}}{N} - \frac{1}{2}}{N \cdot N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))