logq (problem 3.4.3)

Average Error: 58.7 → 0.2

Time: 6.9s

Precision: binary64

Math

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

↓

\[-2 \cdot \varepsilon - \left(0.6666666666666666 \cdot {\varepsilon}^{3} + 0.4 \cdot {\varepsilon}^{5}\right)\]

FPCore

(FPCore (eps) :precision binary64 (log (/ (- 1.0 eps) (+ 1.0 eps))))

↓

(FPCore (eps)
 :precision binary64
 (-
  (* -2.0 eps)
  (+ (* 0.6666666666666666 (pow eps 3.0)) (* 0.4 (pow eps 5.0)))))

C

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

↓

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

Error

Bits error versus eps

Target

Original	58.7
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 58.7

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

2. Taylor expanded around 0 0.2

   \[\leadsto \color{blue}{-\left(0.4 \cdot {\varepsilon}^{5} + \left(2 \cdot \varepsilon + 0.6666666666666666 \cdot {\varepsilon}^{3}\right)\right)}\]

3. Simplified 0.2

   \[\leadsto \color{blue}{\left(-2 \cdot \varepsilon - 0.6666666666666666 \cdot {\varepsilon}^{3}\right) - 0.4 \cdot {\varepsilon}^{5}}\]
4. Using strategy rm

5. Applied associate--l-_binary64_1039 0.2

   \[\leadsto \color{blue}{-2 \cdot \varepsilon - \left(0.6666666666666666 \cdot {\varepsilon}^{3} + 0.4 \cdot {\varepsilon}^{5}\right)}\]

6. Final simplification 0.2

   \[\leadsto -2 \cdot \varepsilon - \left(0.6666666666666666 \cdot {\varepsilon}^{3} + 0.4 \cdot {\varepsilon}^{5}\right)\]

Reproduce

herbie shell --seed 2021050 
(FPCore (eps)
  :name "logq (problem 3.4.3)"
  :precision binary64

  :herbie-target
  (* -2.0 (+ (+ eps (/ (pow eps 3.0) 3.0)) (/ (pow eps 5.0) 5.0)))

  (log (/ (- 1.0 eps) (+ 1.0 eps))))