2log (problem 3.3.6)

Average Error: 29.3 → 0.1

Time: 11.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 11278.66194821723365748766809701919555664:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.3333333333333333148296162562473909929395}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]

C

double f(double N) {
        double r50469 = N;
        double r50470 = 1.0;
        double r50471 = r50469 + r50470;
        double r50472 = log(r50471);
        double r50473 = log(r50469);
        double r50474 = r50472 - r50473;
        return r50474;
}

↓

double f(double N) {
        double r50475 = N;
        double r50476 = 11278.661948217234;
        bool r50477 = r50475 <= r50476;
        double r50478 = 1.0;
        double r50479 = r50475 + r50478;
        double r50480 = r50479 / r50475;
        double r50481 = log(r50480);
        double r50482 = 1.0;
        double r50483 = 2.0;
        double r50484 = pow(r50475, r50483);
        double r50485 = r50482 / r50484;
        double r50486 = 0.3333333333333333;
        double r50487 = r50486 / r50475;
        double r50488 = 0.5;
        double r50489 = r50487 - r50488;
        double r50490 = r50478 / r50475;
        double r50491 = fma(r50485, r50489, r50490);
        double r50492 = r50477 ? r50481 : r50491;
        return r50492;
}

Error

Bits error versus N

Derivation

1. Split input into 2 regimes

2. if N < 11278.661948217234

   1. Initial program 0.1

      \[\log \left(N + 1\right) - \log N\]
   2. Using strategy rm

   3. Applied diff-log 0.1

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

   if 11278.661948217234 < N

   1. Initial program 59.6

      \[\log \left(N + 1\right) - \log N\]
   2. Using strategy rm

   3. Applied diff-log 59.3

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

   4. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.3333333333333333148296162562473909929395 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

   5. Simplified 0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.3333333333333333148296162562473909929395}{N} - 0.5, \frac{1}{N}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 11278.66194821723365748766809701919555664:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{N}^{2}}, \frac{0.3333333333333333148296162562473909929395}{N} - 0.5, \frac{1}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))