logq (problem 3.4.3)

Average Error: 58.6 → 0.3

Time: 5.2s

Precision: binary64

Math

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

↓

\[\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.6666666666666666, -2\right) \]

FPCore

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

↓

(FPCore (eps)
 :precision binary64
 (* eps (fma eps (* eps -0.6666666666666666) -2.0)))

C

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

↓

double code(double eps) {
	return eps * fma(eps, (eps * -0.6666666666666666), -2.0);
}

Error

Bits error versus eps

Target

Original	58.6
Target	0.2
Herbie	0.3

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

Derivation

1. Initial program 58.6

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

2. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{log1p}\left(-\varepsilon\right) - \mathsf{log1p}\left(\varepsilon\right)} \]

3. Taylor expanded in eps around 0 0.3

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

4. Simplified 0.3

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.6666666666666666, -2\right)} \]

5. Final simplification 0.3

   \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.6666666666666666, -2\right) \]

Reproduce

herbie shell --seed 2021275 
(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))))