Average Error: 58.5 → 0.6
Time: 10.1s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[2 \cdot \left(\varepsilon \cdot \varepsilon - \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right)\right) + \log 1\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
2 \cdot \left(\varepsilon \cdot \varepsilon - \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right)\right) + \log 1
double f(double eps) {
        double r83171 = 1.0;
        double r83172 = eps;
        double r83173 = r83171 - r83172;
        double r83174 = r83171 + r83172;
        double r83175 = r83173 / r83174;
        double r83176 = log(r83175);
        return r83176;
}
↓
double f(double eps) {
        double r83177 = 2.0;
        double r83178 = eps;
        double r83179 = r83178 * r83178;
        double r83180 = 1.0;
        double r83181 = r83178 / r83180;
        double r83182 = fma(r83181, r83181, r83178);
        double r83183 = r83179 - r83182;
        double r83184 = r83177 * r83183;
        double r83185 = log(r83180);
        double r83186 = r83184 + r83185;
        return r83186;
}

Error

Bits error versus eps

Target

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

Derivation

  1. Initial program 58.5

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

    \[\leadsto 2 \cdot \left(\varepsilon \cdot \varepsilon - \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right)\right) + \log 1\]

Reproduce

herbie shell --seed 2020046 +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))))