2log (problem 3.3.6)

Average Error: 29.2 → 0.1

Time: 16.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 8915.293301236255501862615346908569335938:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N}, 1, \frac{0.3333333333333333148296162562473909929395}{N} \cdot \frac{\frac{1}{N}}{N} - \frac{\frac{1}{N}}{N} \cdot 0.5\right)\\ \end{array}\]

C

double f(double N) {
        double r3422627 = N;
        double r3422628 = 1.0;
        double r3422629 = r3422627 + r3422628;
        double r3422630 = log(r3422629);
        double r3422631 = log(r3422627);
        double r3422632 = r3422630 - r3422631;
        return r3422632;
}

↓

double f(double N) {
        double r3422633 = N;
        double r3422634 = 8915.293301236256;
        bool r3422635 = r3422633 <= r3422634;
        double r3422636 = 1.0;
        double r3422637 = r3422636 + r3422633;
        double r3422638 = r3422637 / r3422633;
        double r3422639 = log(r3422638);
        double r3422640 = 1.0;
        double r3422641 = r3422640 / r3422633;
        double r3422642 = 0.3333333333333333;
        double r3422643 = r3422642 / r3422633;
        double r3422644 = r3422641 / r3422633;
        double r3422645 = r3422643 * r3422644;
        double r3422646 = 0.5;
        double r3422647 = r3422644 * r3422646;
        double r3422648 = r3422645 - r3422647;
        double r3422649 = fma(r3422641, r3422636, r3422648);
        double r3422650 = r3422635 ? r3422639 : r3422649;
        return r3422650;
}

Error

Bits error versus N

Derivation

1. Split input into 2 regimes

2. if N < 8915.293301236256

   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 8915.293301236256 < 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

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8915.293301236255501862615346908569335938:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N}, 1, \frac{0.3333333333333333148296162562473909929395}{N} \cdot \frac{\frac{1}{N}}{N} - \frac{\frac{1}{N}}{N} \cdot 0.5\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1.0)) (log N)))