2log (problem 3.3.6)

Average Error: 29.1 → 0.1

Time: 25.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 9556.535660077324791927821934223175048828:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N \cdot N}, \frac{0.3333333333333333148296162562473909929395}{N} + \left(-0.5\right), \frac{1}{N}\right)\\ \end{array}\]

C

double f(double N) {
        double r2126250 = N;
        double r2126251 = 1.0;
        double r2126252 = r2126250 + r2126251;
        double r2126253 = log(r2126252);
        double r2126254 = log(r2126250);
        double r2126255 = r2126253 - r2126254;
        return r2126255;
}

↓

double f(double N) {
        double r2126256 = N;
        double r2126257 = 9556.535660077325;
        bool r2126258 = r2126256 <= r2126257;
        double r2126259 = 1.0;
        double r2126260 = r2126259 + r2126256;
        double r2126261 = r2126260 / r2126256;
        double r2126262 = log(r2126261);
        double r2126263 = 1.0;
        double r2126264 = r2126256 * r2126256;
        double r2126265 = r2126263 / r2126264;
        double r2126266 = 0.3333333333333333;
        double r2126267 = r2126266 / r2126256;
        double r2126268 = 0.5;
        double r2126269 = -r2126268;
        double r2126270 = r2126267 + r2126269;
        double r2126271 = r2126259 / r2126256;
        double r2126272 = fma(r2126265, r2126270, r2126271);
        double r2126273 = r2126258 ? r2126262 : r2126272;
        return r2126273;
}

Error

Bits error versus N

Derivation

1. Split input into 2 regimes

2. if N < 9556.535660077325

   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 9556.535660077325 < 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

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9556.535660077324791927821934223175048828:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N \cdot N}, \frac{0.3333333333333333148296162562473909929395}{N} + \left(-0.5\right), \frac{1}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1.0)) (log N)))