Average Error: 58.5 → 0.2
Time: 18.1s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left(\frac{-2}{3}, \left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, -2, \frac{-2}{5} \cdot {\varepsilon}^{5}\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left(\frac{-2}{3}, \left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, -2, \frac{-2}{5} \cdot {\varepsilon}^{5}\right)\right)
double f(double eps) {
        double r2293150 = 1.0;
        double r2293151 = eps;
        double r2293152 = r2293150 - r2293151;
        double r2293153 = r2293150 + r2293151;
        double r2293154 = r2293152 / r2293153;
        double r2293155 = log(r2293154);
        return r2293155;
}
↓
double f(double eps) {
        double r2293156 = -0.6666666666666666;
        double r2293157 = eps;
        double r2293158 = r2293157 * r2293157;
        double r2293159 = r2293158 * r2293157;
        double r2293160 = -2.0;
        double r2293161 = -0.4;
        double r2293162 = 5.0;
        double r2293163 = pow(r2293157, r2293162);
        double r2293164 = r2293161 * r2293163;
        double r2293165 = fma(r2293157, r2293160, r2293164);
        double r2293166 = fma(r2293156, r2293159, r2293165);
        return r2293166;
}

Error

Bits error versus eps

Target

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

Derivation

  1. Initial program 58.5

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

  2. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{-\left(\frac{2}{3} \cdot {\varepsilon}^{3} + \left(\frac{2}{5} \cdot {\varepsilon}^{5} + 2 \cdot \varepsilon\right)\right)}\]

  3. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-2}{3}, \left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, -2, {\varepsilon}^{5} \cdot \frac{-2}{5}\right)\right)}\]

  4. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(\frac{-2}{3}, \left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, -2, \frac{-2}{5} \cdot {\varepsilon}^{5}\right)\right)\]

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (eps)
  :name "logq (problem 3.4.3)"

  :herbie-target
  (* -2 (+ (+ eps (/ (pow eps 3) 3)) (/ (pow eps 5) 5)))

  (log (/ (- 1 eps) (+ 1 eps))))