Average Error: 58.6 → 0.2
Time: 6.7s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left(\frac{-2}{3}, {\left(\frac{\varepsilon}{1}\right)}^{3}, -\mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, 2 \cdot \varepsilon\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left(\frac{-2}{3}, {\left(\frac{\varepsilon}{1}\right)}^{3}, -\mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, 2 \cdot \varepsilon\right)\right)
double f(double eps) {
        double r117994 = 1.0;
        double r117995 = eps;
        double r117996 = r117994 - r117995;
        double r117997 = r117994 + r117995;
        double r117998 = r117996 / r117997;
        double r117999 = log(r117998);
        return r117999;
}
↓
double f(double eps) {
        double r118000 = -0.6666666666666666;
        double r118001 = eps;
        double r118002 = 1.0;
        double r118003 = r118001 / r118002;
        double r118004 = 3.0;
        double r118005 = pow(r118003, r118004);
        double r118006 = 0.4;
        double r118007 = 5.0;
        double r118008 = pow(r118001, r118007);
        double r118009 = pow(r118002, r118007);
        double r118010 = r118008 / r118009;
        double r118011 = 2.0;
        double r118012 = r118011 * r118001;
        double r118013 = fma(r118006, r118010, r118012);
        double r118014 = -r118013;
        double r118015 = fma(r118000, r118005, r118014);
        return r118015;
}

Error

Bits error versus eps

Target

Original58.6
Target0.2
Herbie0.2
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.6

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

  3. Applied log-div58.6

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

  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

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (eps)
  :name "logq (problem 3.4.3)"
  :precision binary64

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

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