Average Error: 58.7 → 0.2
Time: 16.7s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right), \frac{-2}{3}, \mathsf{fma}\left(\frac{-2}{5}, {\varepsilon}^{5}, -2 \cdot \varepsilon\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right), \frac{-2}{3}, \mathsf{fma}\left(\frac{-2}{5}, {\varepsilon}^{5}, -2 \cdot \varepsilon\right)\right)
double f(double eps) {
        double r2340660 = 1.0;
        double r2340661 = eps;
        double r2340662 = r2340660 - r2340661;
        double r2340663 = r2340660 + r2340661;
        double r2340664 = r2340662 / r2340663;
        double r2340665 = log(r2340664);
        return r2340665;
}
↓
double f(double eps) {
        double r2340666 = eps;
        double r2340667 = r2340666 * r2340666;
        double r2340668 = r2340666 * r2340667;
        double r2340669 = -0.6666666666666666;
        double r2340670 = -0.4;
        double r2340671 = 5.0;
        double r2340672 = pow(r2340666, r2340671);
        double r2340673 = -2.0;
        double r2340674 = r2340673 * r2340666;
        double r2340675 = fma(r2340670, r2340672, r2340674);
        double r2340676 = fma(r2340668, r2340669, r2340675);
        return r2340676;
}

Error

Bits error versus eps

Target

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

Derivation

  1. Initial program 58.7

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

  2. Taylor expanded around 0 0.2

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

  3. Simplified0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019164 +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))))