2log (problem 3.3.6)

Average Error: 29.3 → 0.1

Time: 29.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 7250.408229147694:\\ \;\;\;\;\log \left(\frac{\frac{1 + N}{\sqrt{N}}}{\sqrt{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{N}}{N}, \frac{-1}{2}, \mathsf{fma}\left(\frac{1}{N} \cdot \left(\frac{1}{N} \cdot \frac{1}{N}\right), \frac{1}{3}, \frac{1}{N}\right)\right)\\ \end{array}\]

C

double f(double N) {
        double r1619102 = N;
        double r1619103 = 1.0;
        double r1619104 = r1619102 + r1619103;
        double r1619105 = log(r1619104);
        double r1619106 = log(r1619102);
        double r1619107 = r1619105 - r1619106;
        return r1619107;
}

↓

double f(double N) {
        double r1619108 = N;
        double r1619109 = 7250.408229147694;
        bool r1619110 = r1619108 <= r1619109;
        double r1619111 = 1.0;
        double r1619112 = r1619111 + r1619108;
        double r1619113 = sqrt(r1619108);
        double r1619114 = r1619112 / r1619113;
        double r1619115 = r1619114 / r1619113;
        double r1619116 = log(r1619115);
        double r1619117 = r1619111 / r1619108;
        double r1619118 = r1619117 / r1619108;
        double r1619119 = -0.5;
        double r1619120 = r1619117 * r1619117;
        double r1619121 = r1619117 * r1619120;
        double r1619122 = 0.3333333333333333;
        double r1619123 = fma(r1619121, r1619122, r1619117);
        double r1619124 = fma(r1619118, r1619119, r1619123);
        double r1619125 = r1619110 ? r1619116 : r1619124;
        return r1619125;
}

Error

Bits error versus N

Derivation

1. Split input into 2 regimes

2. if N < 7250.408229147694

   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 add-sqr-sqrt 0.1

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

   8. Applied associate-/r* 0.1

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

   if 7250.408229147694 < N

   1. Initial program 59.7

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

   2. Simplified 59.7

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

   4. Applied log1p-udef 59.7

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

   5. Applied diff-log 59.5

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

   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

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7250.408229147694:\\ \;\;\;\;\log \left(\frac{\frac{1 + N}{\sqrt{N}}}{\sqrt{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{N}}{N}, \frac{-1}{2}, \mathsf{fma}\left(\frac{1}{N} \cdot \left(\frac{1}{N} \cdot \frac{1}{N}\right), \frac{1}{3}, \frac{1}{N}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))