Average Error: 58.6 → 0.6
Time: 8.9s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[2 \cdot \left({\varepsilon}^{2} - \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right)\right) + \log 1\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
2 \cdot \left({\varepsilon}^{2} - \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right)\right) + \log 1
double f(double eps) {
        double r97175 = 1.0;
        double r97176 = eps;
        double r97177 = r97175 - r97176;
        double r97178 = r97175 + r97176;
        double r97179 = r97177 / r97178;
        double r97180 = log(r97179);
        return r97180;
}
↓
double f(double eps) {
        double r97181 = 2.0;
        double r97182 = eps;
        double r97183 = 2.0;
        double r97184 = pow(r97182, r97183);
        double r97185 = 1.0;
        double r97186 = r97182 / r97185;
        double r97187 = fma(r97186, r97186, r97182);
        double r97188 = r97184 - r97187;
        double r97189 = r97181 * r97188;
        double r97190 = log(r97185);
        double r97191 = r97189 + r97190;
        return r97191;
}

Error

Bits error versus eps

Target

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

Derivation

  1. Initial program 58.6

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

  2. 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

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2020024 +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))))