2log (problem 3.3.6)

Average Error: 29.7 → 0.1

Time: 4.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 9843.297794559382964507676661014556884766:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N}, 1 - \frac{0.5}{N}, \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\right)\\ \end{array}\]

C

double f(double N) {
        double r54660 = N;
        double r54661 = 1.0;
        double r54662 = r54660 + r54661;
        double r54663 = log(r54662);
        double r54664 = log(r54660);
        double r54665 = r54663 - r54664;
        return r54665;
}

↓

double f(double N) {
        double r54666 = N;
        double r54667 = 9843.297794559383;
        bool r54668 = r54666 <= r54667;
        double r54669 = 1.0;
        double r54670 = r54666 + r54669;
        double r54671 = r54670 / r54666;
        double r54672 = log(r54671);
        double r54673 = 1.0;
        double r54674 = r54673 / r54666;
        double r54675 = 0.5;
        double r54676 = r54675 / r54666;
        double r54677 = r54669 - r54676;
        double r54678 = 0.3333333333333333;
        double r54679 = 3.0;
        double r54680 = pow(r54666, r54679);
        double r54681 = r54678 / r54680;
        double r54682 = fma(r54674, r54677, r54681);
        double r54683 = r54668 ? r54672 : r54682;
        return r54683;
}

Error

Bits error versus N

Derivation

1. Split input into 2 regimes

2. if N < 9843.297794559383

   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 9843.297794559383 < N

   1. Initial program 59.5

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

   3. Applied diff-log 59.3

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

   4. Taylor expanded around inf 0.0

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

   5. Simplified 0.0

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9843.297794559382964507676661014556884766:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N}, 1 - \frac{0.5}{N}, \frac{0.3333333333333333148296162562473909929395}{{N}^{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))