2log (problem 3.3.6)

Average Error: 29.2 → 0.1

Time: 12.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 7883.767016394317579397466033697128295898:\\ \;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N \cdot N}, \frac{0.3333333333333333148296162562473909929395}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]

C

double f(double N) {
        double r72885 = N;
        double r72886 = 1.0;
        double r72887 = r72885 + r72886;
        double r72888 = log(r72887);
        double r72889 = log(r72885);
        double r72890 = r72888 - r72889;
        return r72890;
}

↓

double f(double N) {
        double r72891 = N;
        double r72892 = 7883.767016394318;
        bool r72893 = r72891 <= r72892;
        double r72894 = 1.0;
        double r72895 = r72894 + r72891;
        double r72896 = r72895 / r72891;
        double r72897 = sqrt(r72896);
        double r72898 = log(r72897);
        double r72899 = r72898 + r72898;
        double r72900 = 1.0;
        double r72901 = r72891 * r72891;
        double r72902 = r72900 / r72901;
        double r72903 = 0.3333333333333333;
        double r72904 = r72903 / r72891;
        double r72905 = 0.5;
        double r72906 = r72904 - r72905;
        double r72907 = r72894 / r72891;
        double r72908 = fma(r72902, r72906, r72907);
        double r72909 = r72893 ? r72899 : r72908;
        return r72909;
}

Error

Bits error versus N

Derivation

1. Split input into 2 regimes

2. if N < 7883.767016394318

   1. Initial program 0.1

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

   2. Simplified 0.1

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

   4. Applied diff-log 0.1

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

   5. Simplified 0.1

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

   7. Applied add-sqr-sqrt 0.1

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

   8. Applied log-prod 0.1

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

   9. Simplified 0.1

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

   10. Simplified 0.1

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

   if 7883.767016394318 < N

   1. Initial program 59.5

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

   2. Simplified 59.5

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

   4. Applied diff-log 59.3

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

   5. Simplified 59.3

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

   6. Taylor expanded around inf 0.0

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

   7. Simplified 0.0

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7883.767016394317579397466033697128295898:\\ \;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N \cdot N}, \frac{0.3333333333333333148296162562473909929395}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1.0)) (log N)))