2log (problem 3.3.6)

Average Error: 29.1 → 0.1

Time: 18.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 4585.464779854794:\\ \;\;\;\;\frac{\sqrt[3]{\left(\mathsf{log1p}\left(N\right) \cdot \mathsf{log1p}\left(N\right)\right) \cdot \left(\left(\mathsf{log1p}\left(N\right) \cdot \mathsf{log1p}\left(N\right)\right) \cdot \left(\mathsf{log1p}\left(N\right) \cdot \mathsf{log1p}\left(N\right)\right)\right)} - \log N \cdot \log N}{\mathsf{log1p}\left(N\right) + \log N}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N} \cdot \frac{1}{N}, \frac{\frac{1}{3}}{N} - \frac{1}{2}, \frac{1}{N}\right)\\ \end{array}\]

C

double f(double N) {
        double r1914669 = N;
        double r1914670 = 1.0;
        double r1914671 = r1914669 + r1914670;
        double r1914672 = log(r1914671);
        double r1914673 = log(r1914669);
        double r1914674 = r1914672 - r1914673;
        return r1914674;
}

↓

double f(double N) {
        double r1914675 = N;
        double r1914676 = 4585.464779854794;
        bool r1914677 = r1914675 <= r1914676;
        double r1914678 = log1p(r1914675);
        double r1914679 = r1914678 * r1914678;
        double r1914680 = r1914679 * r1914679;
        double r1914681 = r1914679 * r1914680;
        double r1914682 = cbrt(r1914681);
        double r1914683 = log(r1914675);
        double r1914684 = r1914683 * r1914683;
        double r1914685 = r1914682 - r1914684;
        double r1914686 = r1914678 + r1914683;
        double r1914687 = r1914685 / r1914686;
        double r1914688 = 1.0;
        double r1914689 = r1914688 / r1914675;
        double r1914690 = r1914689 * r1914689;
        double r1914691 = 0.3333333333333333;
        double r1914692 = r1914691 / r1914675;
        double r1914693 = 0.5;
        double r1914694 = r1914692 - r1914693;
        double r1914695 = fma(r1914690, r1914694, r1914689);
        double r1914696 = r1914677 ? r1914687 : r1914695;
        return r1914696;
}

Error

Bits error versus N

Derivation

1. Split input into 2 regimes

2. if N < 4585.464779854794

   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 flip-- 0.2

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

   6. Applied add-cbrt-cube 0.2

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\mathsf{log1p}\left(N\right) \cdot \mathsf{log1p}\left(N\right)\right) \cdot \left(\mathsf{log1p}\left(N\right) \cdot \mathsf{log1p}\left(N\right)\right)\right) \cdot \left(\mathsf{log1p}\left(N\right) \cdot \mathsf{log1p}\left(N\right)\right)}} - \log N \cdot \log N}{\mathsf{log1p}\left(N\right) + \log N}\]

   if 4585.464779854794 < 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. Using strategy rm

   4. Applied log1p-udef 59.4

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

   5. Applied diff-log 59.2

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

   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

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 4585.464779854794:\\ \;\;\;\;\frac{\sqrt[3]{\left(\mathsf{log1p}\left(N\right) \cdot \mathsf{log1p}\left(N\right)\right) \cdot \left(\left(\mathsf{log1p}\left(N\right) \cdot \mathsf{log1p}\left(N\right)\right) \cdot \left(\mathsf{log1p}\left(N\right) \cdot \mathsf{log1p}\left(N\right)\right)\right)} - \log N \cdot \log N}{\mathsf{log1p}\left(N\right) + \log N}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N} \cdot \frac{1}{N}, \frac{\frac{1}{3}}{N} - \frac{1}{2}, \frac{1}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))