Average Error: 58.7 → 0.2
Time: 13.9s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \frac{-2}{3} - \mathsf{fma}\left(\frac{2}{5}, \left({\varepsilon}^{5}\right), \left(\varepsilon \cdot 2\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \frac{-2}{3} - \mathsf{fma}\left(\frac{2}{5}, \left({\varepsilon}^{5}\right), \left(\varepsilon \cdot 2\right)\right)
double f(double eps) {
        double r1228598 = 1.0;
        double r1228599 = eps;
        double r1228600 = r1228598 - r1228599;
        double r1228601 = r1228598 + r1228599;
        double r1228602 = r1228600 / r1228601;
        double r1228603 = log(r1228602);
        return r1228603;
}
↓
double f(double eps) {
        double r1228604 = eps;
        double r1228605 = r1228604 * r1228604;
        double r1228606 = r1228604 * r1228605;
        double r1228607 = -0.6666666666666666;
        double r1228608 = r1228606 * r1228607;
        double r1228609 = 0.4;
        double r1228610 = 5.0;
        double r1228611 = pow(r1228604, r1228610);
        double r1228612 = 2.0;
        double r1228613 = r1228604 * r1228612;
        double r1228614 = fma(r1228609, r1228611, r1228613);
        double r1228615 = r1228608 - r1228614;
        return r1228615;
}

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

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019128 +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))))