2log (problem 3.3.6)

Average Error: 29.2 → 0.1

Time: 12.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 9550.567671573803:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{\frac{1}{N}}{N}\right), \frac{-1}{2}, \left(\mathsf{fma}\left(\left(\frac{\frac{\frac{1}{N}}{N}}{N}\right), \frac{1}{3}, \left(\frac{1}{N}\right)\right)\right)\right)\\ \end{array}\]

C

double f(double N) {
        double r553808 = N;
        double r553809 = 1.0;
        double r553810 = r553808 + r553809;
        double r553811 = log(r553810);
        double r553812 = log(r553808);
        double r553813 = r553811 - r553812;
        return r553813;
}

↓

double f(double N) {
        double r553814 = N;
        double r553815 = 9550.567671573803;
        bool r553816 = r553814 <= r553815;
        double r553817 = 1.0;
        double r553818 = r553817 + r553814;
        double r553819 = r553818 / r553814;
        double r553820 = log(r553819);
        double r553821 = r553817 / r553814;
        double r553822 = r553821 / r553814;
        double r553823 = -0.5;
        double r553824 = r553822 / r553814;
        double r553825 = 0.3333333333333333;
        double r553826 = fma(r553824, r553825, r553821);
        double r553827 = fma(r553822, r553823, r553826);
        double r553828 = r553816 ? r553820 : r553827;
        return r553828;
}

Error

Bits error versus N

Derivation

1. Split input into 2 regimes

2. if N < 9550.567671573803

   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

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

   if 9550.567671573803 < N

   1. Initial program 59.6

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

   2. Simplified 59.6

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

   4. Applied log1p-udef 59.6

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

   5. Applied diff-log 59.3

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

   7. Applied add-sqr-sqrt 59.3

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

   8. Taylor expanded around -inf 0.0

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

   9. Simplified 0.0

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9550.567671573803:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{\frac{1}{N}}{N}\right), \frac{-1}{2}, \left(\mathsf{fma}\left(\left(\frac{\frac{\frac{1}{N}}{N}}{N}\right), \frac{1}{3}, \left(\frac{1}{N}\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))