2log (problem 3.3.6)

Average Error: 29.7 → 0.1

Time: 17.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 7951.823313362932822201400995254516601562:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{N}, \frac{\frac{1}{N}}{N} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right)\right)\\ \end{array}\]

C

double f(double N) {
        double r2232385 = N;
        double r2232386 = 1.0;
        double r2232387 = r2232385 + r2232386;
        double r2232388 = log(r2232387);
        double r2232389 = log(r2232385);
        double r2232390 = r2232388 - r2232389;
        return r2232390;
}

↓

double f(double N) {
        double r2232391 = N;
        double r2232392 = 7951.823313362933;
        bool r2232393 = r2232391 <= r2232392;
        double r2232394 = 1.0;
        double r2232395 = r2232394 + r2232391;
        double r2232396 = r2232395 / r2232391;
        double r2232397 = log(r2232396);
        double r2232398 = 1.0;
        double r2232399 = r2232398 / r2232391;
        double r2232400 = r2232399 / r2232391;
        double r2232401 = 0.3333333333333333;
        double r2232402 = r2232401 / r2232391;
        double r2232403 = 0.5;
        double r2232404 = r2232402 - r2232403;
        double r2232405 = r2232400 * r2232404;
        double r2232406 = fma(r2232394, r2232399, r2232405);
        double r2232407 = r2232393 ? r2232397 : r2232406;
        return r2232407;
}

Error

Bits error versus N

Derivation

1. Split input into 2 regimes

2. if N < 7951.823313362933

   1. Initial program 0.1

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

   3. Applied add-log-exp 0.1

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

   4. Simplified 0.1

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

   if 7951.823313362933 < N

   1. Initial program 59.5

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

   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 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7951.823313362932822201400995254516601562:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{N}, \frac{\frac{1}{N}}{N} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1.0)) (log N)))