2log (problem 3.3.6)

Average Error: 29.5 → 0.1

Time: 17.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 4842.291926013775:\\ \;\;\;\;\log \left(\frac{\sqrt{1 + N}}{N}\right) + \log \left(\sqrt{1 + N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{3}, \frac{\frac{\frac{1}{N}}{N}}{N}, \mathsf{fma}\left(\frac{-1}{2}, \frac{\frac{1}{N}}{N}, \frac{1}{N}\right)\right)\\ \end{array}\]

C

double f(double N) {
        double r2036976 = N;
        double r2036977 = 1.0;
        double r2036978 = r2036976 + r2036977;
        double r2036979 = log(r2036978);
        double r2036980 = log(r2036976);
        double r2036981 = r2036979 - r2036980;
        return r2036981;
}

↓

double f(double N) {
        double r2036982 = N;
        double r2036983 = 4842.291926013775;
        bool r2036984 = r2036982 <= r2036983;
        double r2036985 = 1.0;
        double r2036986 = r2036985 + r2036982;
        double r2036987 = sqrt(r2036986);
        double r2036988 = r2036987 / r2036982;
        double r2036989 = log(r2036988);
        double r2036990 = log(r2036987);
        double r2036991 = r2036989 + r2036990;
        double r2036992 = 0.3333333333333333;
        double r2036993 = r2036985 / r2036982;
        double r2036994 = r2036993 / r2036982;
        double r2036995 = r2036994 / r2036982;
        double r2036996 = -0.5;
        double r2036997 = fma(r2036996, r2036994, r2036993);
        double r2036998 = fma(r2036992, r2036995, r2036997);
        double r2036999 = r2036984 ? r2036991 : r2036998;
        return r2036999;
}

Error

Bits error versus N

Derivation

1. Split input into 2 regimes

2. if N < 4842.291926013775

   1. Initial program 0.1

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N}\]
   3. Using strategy rm

   4. Applied log1p-udef 0.1

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

   5. Applied diff-log 0.1

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

   7. Applied *-un-lft-identity 0.1

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

   8. Applied add-sqr-sqrt 0.1

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

   9. Applied times-frac 0.1

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

   10. Applied log-prod 0.1

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

   11. Simplified 0.1

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

   if 4842.291926013775 < N

   1. Initial program 59.4

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

   2. Simplified 59.4

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N}\]

   3. Taylor expanded around inf 0.0

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

   4. Simplified 0.0

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 4842.291926013775:\\ \;\;\;\;\log \left(\frac{\sqrt{1 + N}}{N}\right) + \log \left(\sqrt{1 + N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{3}, \frac{\frac{\frac{1}{N}}{N}}{N}, \mathsf{fma}\left(\frac{-1}{2}, \frac{\frac{1}{N}}{N}, \frac{1}{N}\right)\right)\\ \end{array}\]

Reproduce

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