Average Error: 58.7 → 0.6
Time: 19.9s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left(2, \varepsilon \cdot \varepsilon - \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right), \log 1\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left(2, \varepsilon \cdot \varepsilon - \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right), \log 1\right)
double f(double eps) {
        double r3896185 = 1.0;
        double r3896186 = eps;
        double r3896187 = r3896185 - r3896186;
        double r3896188 = r3896185 + r3896186;
        double r3896189 = r3896187 / r3896188;
        double r3896190 = log(r3896189);
        return r3896190;
}
↓
double f(double eps) {
        double r3896191 = 2.0;
        double r3896192 = eps;
        double r3896193 = r3896192 * r3896192;
        double r3896194 = 1.0;
        double r3896195 = r3896192 / r3896194;
        double r3896196 = fma(r3896195, r3896195, r3896192);
        double r3896197 = r3896193 - r3896196;
        double r3896198 = log(r3896194);
        double r3896199 = fma(r3896191, r3896197, r3896198);
        return r3896199;
}

Error

Bits error versus eps

Target

Original58.7
Target0.2
Herbie0.6
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.7

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019169 +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))))