Average Error: 58.7 → 0.2
Time: 13.9s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[-\mathsf{fma}\left(2, \varepsilon, \mathsf{fma}\left(0.6666666666666666296592325124947819858789, {\varepsilon}^{3}, 0.4000000000000000222044604925031308084726 \cdot {\varepsilon}^{5}\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
-\mathsf{fma}\left(2, \varepsilon, \mathsf{fma}\left(0.6666666666666666296592325124947819858789, {\varepsilon}^{3}, 0.4000000000000000222044604925031308084726 \cdot {\varepsilon}^{5}\right)\right)
double f(double eps) {
        double r76694 = 1.0;
        double r76695 = eps;
        double r76696 = r76694 - r76695;
        double r76697 = r76694 + r76695;
        double r76698 = r76696 / r76697;
        double r76699 = log(r76698);
        return r76699;
}
↓
double f(double eps) {
        double r76700 = 2.0;
        double r76701 = eps;
        double r76702 = 0.6666666666666666;
        double r76703 = 3.0;
        double r76704 = pow(r76701, r76703);
        double r76705 = 0.4;
        double r76706 = 5.0;
        double r76707 = pow(r76701, r76706);
        double r76708 = r76705 * r76707;
        double r76709 = fma(r76702, r76704, r76708);
        double r76710 = fma(r76700, r76701, r76709);
        double r76711 = -r76710;
        return r76711;
}

Error

Bits error versus eps

Target

Original58.7
Target0.2
Herbie0.2
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.7

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

  3. Applied log-div58.6

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

  4. Simplified58.6

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

  5. Taylor expanded around 0 0.2

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

  6. Simplified0.2

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

  7. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{-\left(2 \cdot \varepsilon + \left(0.6666666666666666296592325124947819858789 \cdot {\varepsilon}^{3} + 0.4000000000000000222044604925031308084726 \cdot {\varepsilon}^{5}\right)\right)}\]

  8. Simplified0.2

    \[\leadsto \color{blue}{-\mathsf{fma}\left(2, \varepsilon, \mathsf{fma}\left(0.6666666666666666296592325124947819858789, {\varepsilon}^{3}, 0.4000000000000000222044604925031308084726 \cdot {\varepsilon}^{5}\right)\right)}\]

  9. Final simplification0.2

    \[\leadsto -\mathsf{fma}\left(2, \varepsilon, \mathsf{fma}\left(0.6666666666666666296592325124947819858789, {\varepsilon}^{3}, 0.4000000000000000222044604925031308084726 \cdot {\varepsilon}^{5}\right)\right)\]

Reproduce

herbie shell --seed 2019323 +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))))