2log (problem 3.3.6)

Average Error: 29.2 → 0.1

Time: 13.0s

Precision: 64

Math

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

↓

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

C

double f(double N) {
        double r2258219 = N;
        double r2258220 = 1.0;
        double r2258221 = r2258219 + r2258220;
        double r2258222 = log(r2258221);
        double r2258223 = log(r2258219);
        double r2258224 = r2258222 - r2258223;
        return r2258224;
}

↓

double f(double N) {
        double r2258225 = N;
        double r2258226 = 8915.293301236256;
        bool r2258227 = r2258225 <= r2258226;
        double r2258228 = 1.0;
        double r2258229 = r2258228 + r2258225;
        double r2258230 = r2258229 / r2258225;
        double r2258231 = log(r2258230);
        double r2258232 = 1.0;
        double r2258233 = r2258232 / r2258225;
        double r2258234 = r2258233 / r2258225;
        double r2258235 = 0.3333333333333333;
        double r2258236 = r2258235 / r2258225;
        double r2258237 = 0.5;
        double r2258238 = r2258236 - r2258237;
        double r2258239 = r2258234 * r2258238;
        double r2258240 = fma(r2258233, r2258228, r2258239);
        double r2258241 = r2258227 ? r2258231 : r2258240;
        return r2258241;
}

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 diff-log 0.1

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

   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

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

4. Final simplification 0.1

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

Reproduce

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