2log (problem 3.3.6)

Average Error: 29.9 → 0.0

Time: 16.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 7536.677708381748743704520165920257568359:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N}, 1, \frac{\frac{1}{N}}{N} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right)\right)\\ \end{array}\]

C

double f(double N) {
        double r3383502 = N;
        double r3383503 = 1.0;
        double r3383504 = r3383502 + r3383503;
        double r3383505 = log(r3383504);
        double r3383506 = log(r3383502);
        double r3383507 = r3383505 - r3383506;
        return r3383507;
}

↓

double f(double N) {
        double r3383508 = N;
        double r3383509 = 7536.677708381749;
        bool r3383510 = r3383508 <= r3383509;
        double r3383511 = 1.0;
        double r3383512 = r3383511 + r3383508;
        double r3383513 = r3383512 / r3383508;
        double r3383514 = log(r3383513);
        double r3383515 = 1.0;
        double r3383516 = r3383515 / r3383508;
        double r3383517 = r3383516 / r3383508;
        double r3383518 = 0.3333333333333333;
        double r3383519 = r3383518 / r3383508;
        double r3383520 = 0.5;
        double r3383521 = r3383519 - r3383520;
        double r3383522 = r3383517 * r3383521;
        double r3383523 = fma(r3383516, r3383511, r3383522);
        double r3383524 = r3383510 ? r3383514 : r3383523;
        return r3383524;
}

Error

Bits error versus N

Derivation

1. Split input into 2 regimes

2. if N < 7536.677708381749

   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 7536.677708381749 < N

   1. Initial program 59.6

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

   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

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7536.677708381748743704520165920257568359:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N}, 1, \frac{\frac{1}{N}}{N} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1.0)) (log N)))