Average Error: 58.5 → 0.7
Time: 17.5s
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 r7936045 = 1.0;
        double r7936046 = eps;
        double r7936047 = r7936045 - r7936046;
        double r7936048 = r7936045 + r7936046;
        double r7936049 = r7936047 / r7936048;
        double r7936050 = log(r7936049);
        return r7936050;
}
↓
double f(double eps) {
        double r7936051 = 2.0;
        double r7936052 = eps;
        double r7936053 = r7936052 * r7936052;
        double r7936054 = 1.0;
        double r7936055 = r7936052 / r7936054;
        double r7936056 = fma(r7936055, r7936055, r7936052);
        double r7936057 = r7936053 - r7936056;
        double r7936058 = log(r7936054);
        double r7936059 = fma(r7936051, r7936057, r7936058);
        return r7936059;
}

Error

Bits error versus eps

Target

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

Derivation

  1. Initial program 58.5

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019173 +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))))