2log (problem 3.3.6)

Average Error: 29.8 → 0.1

Time: 3.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 172828.268406431453:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array}\]

C

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

↓

double code(double N) {
	double VAR;
	if ((N <= 172828.26840643145)) {
		VAR = log(((N + 1.0) / N));
	} else {
		VAR = ((1.0 - (0.5 / N)) / N);
	}
	return VAR;
}

Error

Bits error versus N

Derivation

1. Split input into 2 regimes

2. if N < 172828.26840643145

   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 172828.26840643145 < N

   1. Initial program 59.8

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

   3. Applied add-sqr-sqrt 60.2

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

   4. Applied fma-neg 60.8

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

   5. Simplified 60.8

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

   6. Taylor expanded around inf 27.9

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

   7. Simplified 0.1

      \[\leadsto \color{blue}{\frac{1}{N} \cdot \left(1 - \frac{0.5}{N}\right)}\]
   8. Using strategy rm

   9. Applied associate-*l/ 0.1

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

   10. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 172828.268406431453:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020102 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))