logq (problem 3.4.3)

Average Error: 58.6 → 0.2

Time: 5.0s

Precision: 64

Math

\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]

↓

\[-\left(\frac{2}{3} \cdot \frac{{\varepsilon}^{3}}{{1}^{3}} + \left(\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)\right)\]

C

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

↓

double code(double eps) {
	return ((double) -(((double) (((double) (0.6666666666666666 * ((double) (((double) pow(eps, 3.0)) / ((double) pow(1.0, 3.0)))))) + ((double) (((double) (0.4 * ((double) (((double) pow(eps, 5.0)) / ((double) pow(1.0, 5.0)))))) + ((double) (2.0 * eps))))))));
}

Error

Bits error versus eps

Target

Original	58.6
Target	0.2
Herbie	0.2

\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

1. Initial program 58.6

   \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
2. Using strategy rm

3. Applied log-div 58.6

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

4. Taylor expanded around 0 0.2

   \[\leadsto \color{blue}{-\left(\frac{2}{3} \cdot \frac{{\varepsilon}^{3}}{{1}^{3}} + \left(\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)\right)}\]

5. Final simplification 0.2

   \[\leadsto -\left(\frac{2}{3} \cdot \frac{{\varepsilon}^{3}}{{1}^{3}} + \left(\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)\right)\]

Reproduce

herbie shell --seed 2020120 
(FPCore (eps)
  :name "logq (problem 3.4.3)"
  :precision binary64

  :herbie-target
  (* -2 (+ (+ eps (/ (pow eps 3) 3)) (/ (pow eps 5) 5)))

  (log (/ (- 1 eps) (+ 1 eps))))