2log (problem 3.3.6)

Average Error: 29.4 → 0.1

Time: 2.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 244787.715706025687:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} \cdot \left(1 - \frac{0.5}{N}\right)\\ \end{array}\]

C

double code(double N) {
	return ((double) (((double) log(((double) (N + 1.0)))) - ((double) log(N))));
}

↓

double code(double N) {
	double VAR;
	if ((N <= 244787.7157060257)) {
		VAR = ((double) log(((double) (((double) (N + 1.0)) / N))));
	} else {
		VAR = ((double) (((double) (1.0 / N)) * ((double) (1.0 - ((double) (0.5 / N))))));
	}
	return VAR;
}

Error

Bits error versus N

Derivation

1. Split input into 2 regimes

2. if N < 244787.715706025687

   1. Initial program 0.2

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

   3. Applied diff-log 0.2

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

   if 244787.715706025687 < N

   1. Initial program 59.7

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

   2. Taylor expanded around -inf 64.0

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

   3. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 244787.715706025687:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} \cdot \left(1 - \frac{0.5}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020174 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))