logq (problem 3.4.3)

Average Error: 58.6 → 0.2

Time: 12.7s

Precision: 64

Math

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

↓

\[-\mathsf{fma}\left({\varepsilon}^{5}, \frac{2}{5}, \frac{\varepsilon \cdot \left(4 - \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{2}{3}\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{2}{3}\right)\right)}{2 - \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{2}{3}}\right)\]

C

double f(double eps) {
        double r1782182 = 1.0;
        double r1782183 = eps;
        double r1782184 = r1782182 - r1782183;
        double r1782185 = r1782182 + r1782183;
        double r1782186 = r1782184 / r1782185;
        double r1782187 = log(r1782186);
        return r1782187;
}

↓

double f(double eps) {
        double r1782188 = eps;
        double r1782189 = 5.0;
        double r1782190 = pow(r1782188, r1782189);
        double r1782191 = 0.4;
        double r1782192 = 4.0;
        double r1782193 = r1782188 * r1782188;
        double r1782194 = 0.6666666666666666;
        double r1782195 = r1782193 * r1782194;
        double r1782196 = r1782195 * r1782195;
        double r1782197 = r1782192 - r1782196;
        double r1782198 = r1782188 * r1782197;
        double r1782199 = 2.0;
        double r1782200 = r1782199 - r1782195;
        double r1782201 = r1782198 / r1782200;
        double r1782202 = fma(r1782190, r1782191, r1782201);
        double r1782203 = -r1782202;
        return r1782203;
}

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. Taylor expanded around 0 0.2

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

3. Simplified 0.2

   \[\leadsto \color{blue}{-\mathsf{fma}\left({\varepsilon}^{5}, \frac{2}{5}, \varepsilon \cdot \left(2 + \frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)}\]
4. Using strategy rm

5. Applied flip-+ 0.2

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

6. Applied associate-*r/ 0.2

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

7. Final simplification 0.2

   \[\leadsto -\mathsf{fma}\left({\varepsilon}^{5}, \frac{2}{5}, \frac{\varepsilon \cdot \left(4 - \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{2}{3}\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{2}{3}\right)\right)}{2 - \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{2}{3}}\right)\]

Reproduce

herbie shell --seed 2019153 +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))))