2log (problem 3.3.6)

Average Error: 29.0 → 0.1

Time: 18.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 5040.279255067729:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{log1p}\left(N\right)}, \sqrt{\mathsf{log1p}\left(N\right)}, -\log N\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{N}}{N \cdot N}, \frac{1}{3}, \frac{1}{N}\right) + \frac{\frac{-1}{2}}{N \cdot N}\\ \end{array}\]

C

double f(double N) {
        double r1119994 = N;
        double r1119995 = 1.0;
        double r1119996 = r1119994 + r1119995;
        double r1119997 = log(r1119996);
        double r1119998 = log(r1119994);
        double r1119999 = r1119997 - r1119998;
        return r1119999;
}

↓

double f(double N) {
        double r1120000 = N;
        double r1120001 = 5040.279255067729;
        bool r1120002 = r1120000 <= r1120001;
        double r1120003 = log1p(r1120000);
        double r1120004 = sqrt(r1120003);
        double r1120005 = log(r1120000);
        double r1120006 = -r1120005;
        double r1120007 = fma(r1120004, r1120004, r1120006);
        double r1120008 = 1.0;
        double r1120009 = r1120008 / r1120000;
        double r1120010 = r1120000 * r1120000;
        double r1120011 = r1120009 / r1120010;
        double r1120012 = 0.3333333333333333;
        double r1120013 = fma(r1120011, r1120012, r1120009);
        double r1120014 = -0.5;
        double r1120015 = r1120014 / r1120010;
        double r1120016 = r1120013 + r1120015;
        double r1120017 = r1120002 ? r1120007 : r1120016;
        return r1120017;
}

Error

Bits error versus N

Derivation

1. Split input into 2 regimes

2. if N < 5040.279255067729

   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 add-sqr-sqrt 0.1

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

   5. Applied fma-neg 0.1

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

   if 5040.279255067729 < N

   1. Initial program 59.5

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

   2. Simplified 59.5

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

   4. Applied add-sqr-sqrt 59.9

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

   5. Applied fma-neg 60.5

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

   6. Taylor expanded around -inf 0.0

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

   7. Simplified 0.0

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 5040.279255067729:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{log1p}\left(N\right)}, \sqrt{\mathsf{log1p}\left(N\right)}, -\log N\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{N}}{N \cdot N}, \frac{1}{3}, \frac{1}{N}\right) + \frac{\frac{-1}{2}}{N \cdot N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))