2log (problem 3.3.6)

Average Error: 29.2 → 0.1

Time: 16.7s

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 r3340752 = N;
        double r3340753 = 1.0;
        double r3340754 = r3340752 + r3340753;
        double r3340755 = log(r3340754);
        double r3340756 = log(r3340752);
        double r3340757 = r3340755 - r3340756;
        return r3340757;
}

↓

double f(double N) {
        double r3340758 = N;
        double r3340759 = 8915.293301236256;
        bool r3340760 = r3340758 <= r3340759;
        double r3340761 = 1.0;
        double r3340762 = r3340761 + r3340758;
        double r3340763 = r3340762 / r3340758;
        double r3340764 = log(r3340763);
        double r3340765 = 1.0;
        double r3340766 = r3340765 / r3340758;
        double r3340767 = 0.3333333333333333;
        double r3340768 = r3340767 / r3340758;
        double r3340769 = r3340766 / r3340758;
        double r3340770 = r3340768 * r3340769;
        double r3340771 = 0.5;
        double r3340772 = r3340769 * r3340771;
        double r3340773 = r3340770 - r3340772;
        double r3340774 = fma(r3340766, r3340761, r3340773);
        double r3340775 = r3340760 ? r3340764 : r3340774;
        return r3340775;
}

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)))