Average Error: 58.6 → 0.6
Time: 14.5s
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 r4857203 = 1.0;
        double r4857204 = eps;
        double r4857205 = r4857203 - r4857204;
        double r4857206 = r4857203 + r4857204;
        double r4857207 = r4857205 / r4857206;
        double r4857208 = log(r4857207);
        return r4857208;
}
↓
double f(double eps) {
        double r4857209 = 2.0;
        double r4857210 = eps;
        double r4857211 = r4857210 * r4857210;
        double r4857212 = 1.0;
        double r4857213 = r4857210 / r4857212;
        double r4857214 = fma(r4857213, r4857213, r4857210);
        double r4857215 = r4857211 - r4857214;
        double r4857216 = log(r4857212);
        double r4857217 = fma(r4857209, r4857215, r4857216);
        return r4857217;
}

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))))