2log (problem 3.3.6)

Average Error: 29.5 → 0.1

Time: 16.6s

Precision: 64

Math

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

↓

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

C

double f(double N) {
        double r2428679 = N;
        double r2428680 = 1.0;
        double r2428681 = r2428679 + r2428680;
        double r2428682 = log(r2428681);
        double r2428683 = log(r2428679);
        double r2428684 = r2428682 - r2428683;
        return r2428684;
}

↓

double f(double N) {
        double r2428685 = N;
        double r2428686 = 5400.000739830978;
        bool r2428687 = r2428685 <= r2428686;
        double r2428688 = 1.0;
        double r2428689 = r2428688 + r2428685;
        double r2428690 = sqrt(r2428689);
        double r2428691 = sqrt(r2428685);
        double r2428692 = r2428690 / r2428691;
        double r2428693 = log(r2428692);
        double r2428694 = log(r2428690);
        double r2428695 = log(r2428691);
        double r2428696 = r2428694 - r2428695;
        double r2428697 = r2428693 + r2428696;
        double r2428698 = 0.3333333333333333;
        double r2428699 = r2428698 / r2428685;
        double r2428700 = -0.5;
        double r2428701 = r2428699 + r2428700;
        double r2428702 = r2428685 * r2428685;
        double r2428703 = r2428701 / r2428702;
        double r2428704 = r2428688 / r2428685;
        double r2428705 = r2428703 + r2428704;
        double r2428706 = r2428687 ? r2428697 : r2428705;
        return r2428706;
}

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

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

   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

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

4. Final simplification 0.1

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

Reproduce

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