2log (problem 3.3.6)

Average Error: 29.9 → 0.0

Time: 17.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 9148.879937239351420430466532707214355469:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{N}, \frac{\frac{1}{N}}{N} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right)\right)\\ \end{array}\]

C

double f(double N) {
        double r3503368 = N;
        double r3503369 = 1.0;
        double r3503370 = r3503368 + r3503369;
        double r3503371 = log(r3503370);
        double r3503372 = log(r3503368);
        double r3503373 = r3503371 - r3503372;
        return r3503373;
}

↓

double f(double N) {
        double r3503374 = N;
        double r3503375 = 9148.879937239351;
        bool r3503376 = r3503374 <= r3503375;
        double r3503377 = 1.0;
        double r3503378 = r3503377 + r3503374;
        double r3503379 = r3503378 / r3503374;
        double r3503380 = log(r3503379);
        double r3503381 = 1.0;
        double r3503382 = r3503381 / r3503374;
        double r3503383 = r3503382 / r3503374;
        double r3503384 = 0.3333333333333333;
        double r3503385 = r3503384 / r3503374;
        double r3503386 = 0.5;
        double r3503387 = r3503385 - r3503386;
        double r3503388 = r3503383 * r3503387;
        double r3503389 = fma(r3503377, r3503382, r3503388);
        double r3503390 = r3503376 ? r3503380 : r3503389;
        return r3503390;
}

Error

Bits error versus N

Derivation

1. Split input into 2 regimes

2. if N < 9148.879937239351

   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 9148.879937239351 < N

   1. Initial program 59.7

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

   2. Taylor expanded around inf 0.0

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

   3. Simplified 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9148.879937239351420430466532707214355469:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{N}, \frac{\frac{1}{N}}{N} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1.0)) (log N)))