Average Error: 58.6 → 0.6
Time: 13.7s
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 r3443280 = 1.0;
        double r3443281 = eps;
        double r3443282 = r3443280 - r3443281;
        double r3443283 = r3443280 + r3443281;
        double r3443284 = r3443282 / r3443283;
        double r3443285 = log(r3443284);
        return r3443285;
}
↓
double f(double eps) {
        double r3443286 = 2.0;
        double r3443287 = eps;
        double r3443288 = r3443287 * r3443287;
        double r3443289 = 1.0;
        double r3443290 = r3443287 / r3443289;
        double r3443291 = fma(r3443290, r3443290, r3443287);
        double r3443292 = r3443288 - r3443291;
        double r3443293 = log(r3443289);
        double r3443294 = fma(r3443286, r3443292, r3443293);
        return r3443294;
}

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

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

  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 2019172 +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))))