2log (problem 3.3.6)

Average Error: 29.7 → 0.1

Time: 15.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 5705.402280155900371028110384941101074219:\\ \;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \left(\log \left(\sqrt{1 + N}\right) - \log \left(\sqrt{N}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N}, 1, \frac{\frac{1}{N}}{N} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right)\right)\\ \end{array}\]

C

double f(double N) {
        double r3015077 = N;
        double r3015078 = 1.0;
        double r3015079 = r3015077 + r3015078;
        double r3015080 = log(r3015079);
        double r3015081 = log(r3015077);
        double r3015082 = r3015080 - r3015081;
        return r3015082;
}

↓

double f(double N) {
        double r3015083 = N;
        double r3015084 = 5705.4022801559;
        bool r3015085 = r3015083 <= r3015084;
        double r3015086 = 1.0;
        double r3015087 = r3015086 + r3015083;
        double r3015088 = r3015087 / r3015083;
        double r3015089 = sqrt(r3015088);
        double r3015090 = log(r3015089);
        double r3015091 = sqrt(r3015087);
        double r3015092 = log(r3015091);
        double r3015093 = sqrt(r3015083);
        double r3015094 = log(r3015093);
        double r3015095 = r3015092 - r3015094;
        double r3015096 = r3015090 + r3015095;
        double r3015097 = 1.0;
        double r3015098 = r3015097 / r3015083;
        double r3015099 = r3015098 / r3015083;
        double r3015100 = 0.3333333333333333;
        double r3015101 = r3015100 / r3015083;
        double r3015102 = 0.5;
        double r3015103 = r3015101 - r3015102;
        double r3015104 = r3015099 * r3015103;
        double r3015105 = fma(r3015098, r3015086, r3015104);
        double r3015106 = r3015085 ? r3015096 : r3015105;
        return r3015106;
}

Error

Bits error versus N

Derivation

1. Split input into 2 regimes

2. if N < 5705.4022801559

   1. Initial program 0.1

      \[\log \left(N + 1\right) - \log N\]
   2. Using strategy rm

   3. Applied diff-log 0.1

      \[\leadsto \color{blue}{\log \left(\frac{N + 1}{N}\right)}\]
   4. Using strategy rm

   5. Applied add-sqr-sqrt 0.1

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

   6. Applied log-prod 0.1

      \[\leadsto \color{blue}{\log \left(\sqrt{\frac{N + 1}{N}}\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)}\]
   7. Using strategy rm

   8. Applied sqrt-div 0.1

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

   9. Applied log-div 0.1

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

   if 5705.4022801559 < N

   1. Initial program 59.3

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

   2. Taylor expanded around inf 0.1

      \[\leadsto \color{blue}{\left(0.3333333333333333148296162562473909929395 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

   3. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 5705.402280155900371028110384941101074219:\\ \;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \left(\log \left(\sqrt{1 + N}\right) - \log \left(\sqrt{N}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N}, 1, \frac{\frac{1}{N}}{N} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1.0)) (log N)))