Average Error: 58.4 → 0.2
Time: 28.3s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left(\left({\varepsilon}^{5}\right), \frac{-2}{5}, \left(\varepsilon \cdot -2 + \varepsilon \cdot \left(\left(\frac{-2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right)\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left(\left({\varepsilon}^{5}\right), \frac{-2}{5}, \left(\varepsilon \cdot -2 + \varepsilon \cdot \left(\left(\frac{-2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right)\right)\right)
double f(double eps) {
        double r3143322 = 1.0;
        double r3143323 = eps;
        double r3143324 = r3143322 - r3143323;
        double r3143325 = r3143322 + r3143323;
        double r3143326 = r3143324 / r3143325;
        double r3143327 = log(r3143326);
        return r3143327;
}
↓
double f(double eps) {
        double r3143328 = eps;
        double r3143329 = 5.0;
        double r3143330 = pow(r3143328, r3143329);
        double r3143331 = -0.4;
        double r3143332 = -2.0;
        double r3143333 = r3143328 * r3143332;
        double r3143334 = -0.6666666666666666;
        double r3143335 = r3143334 * r3143328;
        double r3143336 = r3143335 * r3143328;
        double r3143337 = r3143328 * r3143336;
        double r3143338 = r3143333 + r3143337;
        double r3143339 = fma(r3143330, r3143331, r3143338);
        return r3143339;
}

Error

Bits error versus eps

Target

Original58.4
Target0.2
Herbie0.2
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.4

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

  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. Simplified0.2

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

  5. Applied sub-neg0.2

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

  6. Applied distribute-rgt-in0.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019125 +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))))