2log (problem 3.3.6)

Average Error: 29.6 → 0.1

Time: 4.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 13931.9619516847724:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.333333333333333315}{N} \cdot \log \left(e^{\frac{\frac{1}{N}}{N}}\right) + \left(\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\right)\\ \end{array}\]

C

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

↓

double code(double N) {
	double temp;
	if ((N <= 13931.961951684772)) {
		temp = log(((N + 1.0) / N));
	} else {
		temp = (((0.3333333333333333 / N) * log(exp(((1.0 / N) / 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 < 13931.961951684772

   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 13931.961951684772 < N

   1. Initial program 59.5

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

   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-rgt-in 0.0

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

   7. Applied associate-+l+ 0.0

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

   8. Simplified 0.0

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

   10. Applied add-log-exp 0.0

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

   11. Simplified 0.0

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 13931.9619516847724:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.333333333333333315}{N} \cdot \log \left(e^{\frac{\frac{1}{N}}{N}}\right) + \left(\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))