2log (problem 3.3.6)

Average Error: 30.1 → 0.1

Time: 3.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 3926.44294337209203:\\ \;\;\;\;e^{\log \left(\log \left(N + 1\right)\right)} - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \frac{0.333333333333333315}{N} + \left(\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\right)\\ \end{array}\]

C

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

↓

double code(double N) {
	double temp;
	if ((N <= 3926.442943372092)) {
		temp = (exp(log(log((N + 1.0)))) - log(N));
	} else {
		temp = (((1.0 / pow(N, 2.0)) * (0.3333333333333333 / N)) + ((1.0 / N) - ((0.5 / N) / N)));
	}
	return temp;
}

Error

Bits error versus N

Derivation

1. Split input into 2 regimes

2. if N < 3926.442943372092

   1. Initial program 0.1

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

   3. Applied add-exp-log 0.1

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

   if 3926.442943372092 < N

   1. Initial program 59.4

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

   2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.333333333333333315 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

   3. Simplified 0.0

      \[\leadsto \color{blue}{\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}}\]
   4. Using strategy rm

   5. Applied sub-neg 0.0

      \[\leadsto \frac{1}{{N}^{2}} \cdot \color{blue}{\left(\frac{0.333333333333333315}{N} + \left(-0.5\right)\right)} + \frac{1}{N}\]

   6. Applied distribute-lft-in 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{{N}^{2}} \cdot \frac{0.333333333333333315}{N} + \frac{1}{{N}^{2}} \cdot \left(-0.5\right)\right)} + \frac{1}{N}\]

   7. Applied associate-+l+ 0.0

      \[\leadsto \color{blue}{\frac{1}{{N}^{2}} \cdot \frac{0.333333333333333315}{N} + \left(\frac{1}{{N}^{2}} \cdot \left(-0.5\right) + \frac{1}{N}\right)}\]

   8. Simplified 0.0

      \[\leadsto \frac{1}{{N}^{2}} \cdot \frac{0.333333333333333315}{N} + \color{blue}{\left(\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 3926.44294337209203:\\ \;\;\;\;e^{\log \left(\log \left(N + 1\right)\right)} - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \frac{0.333333333333333315}{N} + \left(\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))