2log (problem 3.3.6)

Average Error: 29.1 → 0.1

Time: 47.2s

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 r3605990 = N;
        double r3605991 = 1.0;
        double r3605992 = r3605990 + r3605991;
        double r3605993 = log(r3605992);
        double r3605994 = log(r3605990);
        double r3605995 = r3605993 - r3605994;
        return r3605995;
}

↓

double f(double N) {
        double r3605996 = N;
        double r3605997 = 9556.535660077325;
        bool r3605998 = r3605996 <= r3605997;
        double r3605999 = 1.0;
        double r3606000 = r3605999 + r3605996;
        double r3606001 = r3606000 / r3605996;
        double r3606002 = log(r3606001);
        double r3606003 = 1.0;
        double r3606004 = r3605996 * r3605996;
        double r3606005 = r3606003 / r3606004;
        double r3606006 = 0.3333333333333333;
        double r3606007 = r3606006 / r3605996;
        double r3606008 = 0.5;
        double r3606009 = -r3606008;
        double r3606010 = r3606007 + r3606009;
        double r3606011 = r3605999 / r3605996;
        double r3606012 = fma(r3606005, r3606010, r3606011);
        double r3606013 = r3605998 ? r3606002 : r3606012;
        return r3606013;
}

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)))