logq (problem 3.4.3)

Average Error: 58.7 → 0.0

Time: 11.5s

Precision: 64

Math

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

↓

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

C

double f(double eps) {
        double r530840 = 1.0;
        double r530841 = eps;
        double r530842 = r530840 - r530841;
        double r530843 = r530840 + r530841;
        double r530844 = r530842 / r530843;
        double r530845 = log(r530844);
        return r530845;
}

↓

double f(double eps) {
        double r530846 = eps;
        double r530847 = -r530846;
        double r530848 = log1p(r530847);
        double r530849 = log1p(r530846);
        double r530850 = r530848 - r530849;
        return r530850;
}

Error

Bits error versus eps

Target

Original	58.7
Target	0.2
Herbie	0.0

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

Derivation

1. Initial program 58.7

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

3. Applied log-div 58.7

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

4. Simplified 50.6

   \[\leadsto \log \left(1 - \varepsilon\right) - \color{blue}{\mathsf{log1p}\left(\varepsilon\right)}\]
5. Using strategy rm

6. Applied sub-neg 50.6

   \[\leadsto \log \color{blue}{\left(1 + \left(-\varepsilon\right)\right)} - \mathsf{log1p}\left(\varepsilon\right)\]

7. Applied log1p-def 0.0

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

8. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (eps)
  :name "logq (problem 3.4.3)"

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

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