Average Error: 58.4 → 0.2
Time: 37.7s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\frac{-2}{3} \cdot \left(\frac{\varepsilon}{1} \cdot \left(\frac{\varepsilon}{1} \cdot \frac{\varepsilon}{1}\right)\right) - \mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, \varepsilon \cdot 2\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\frac{-2}{3} \cdot \left(\frac{\varepsilon}{1} \cdot \left(\frac{\varepsilon}{1} \cdot \frac{\varepsilon}{1}\right)\right) - \mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, \varepsilon \cdot 2\right)
double f(double eps) {
        double r2200268 = 1.0;
        double r2200269 = eps;
        double r2200270 = r2200268 - r2200269;
        double r2200271 = r2200268 + r2200269;
        double r2200272 = r2200270 / r2200271;
        double r2200273 = log(r2200272);
        return r2200273;
}
↓
double f(double eps) {
        double r2200274 = -0.6666666666666666;
        double r2200275 = eps;
        double r2200276 = 1.0;
        double r2200277 = r2200275 / r2200276;
        double r2200278 = r2200277 * r2200277;
        double r2200279 = r2200277 * r2200278;
        double r2200280 = r2200274 * r2200279;
        double r2200281 = 0.4;
        double r2200282 = 5.0;
        double r2200283 = pow(r2200275, r2200282);
        double r2200284 = pow(r2200276, r2200282);
        double r2200285 = r2200283 / r2200284;
        double r2200286 = 2.0;
        double r2200287 = r2200275 * r2200286;
        double r2200288 = fma(r2200281, r2200285, r2200287);
        double r2200289 = r2200280 - r2200288;
        return r2200289;
}

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. Using strategy rm

  3. Applied log-div58.4

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

  4. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{-\left(\frac{2}{3} \cdot \frac{{\varepsilon}^{3}}{{1}^{3}} + \left(\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)\right)}\]

  5. Simplified0.2

    \[\leadsto \color{blue}{\left(\frac{\varepsilon}{1} \cdot \left(\frac{\varepsilon}{1} \cdot \frac{\varepsilon}{1}\right)\right) \cdot \frac{-2}{3} - \mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, \varepsilon \cdot 2\right)}\]

  6. Final simplification0.2

    \[\leadsto \frac{-2}{3} \cdot \left(\frac{\varepsilon}{1} \cdot \left(\frac{\varepsilon}{1} \cdot \frac{\varepsilon}{1}\right)\right) - \mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, \varepsilon \cdot 2\right)\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (eps)
  :name "logq (problem 3.4.3)"

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

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