Average Error: 58.6 → 0.2
Time: 19.1s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right), \frac{-2}{3}, \mathsf{fma}\left(\varepsilon, -2, {\varepsilon}^{5} \cdot \frac{-2}{5}\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right), \frac{-2}{3}, \mathsf{fma}\left(\varepsilon, -2, {\varepsilon}^{5} \cdot \frac{-2}{5}\right)\right)
double f(double eps) {
        double r1451393 = 1.0;
        double r1451394 = eps;
        double r1451395 = r1451393 - r1451394;
        double r1451396 = r1451393 + r1451394;
        double r1451397 = r1451395 / r1451396;
        double r1451398 = log(r1451397);
        return r1451398;
}
↓
double f(double eps) {
        double r1451399 = eps;
        double r1451400 = r1451399 * r1451399;
        double r1451401 = r1451399 * r1451400;
        double r1451402 = -0.6666666666666666;
        double r1451403 = -2.0;
        double r1451404 = 5.0;
        double r1451405 = pow(r1451399, r1451404);
        double r1451406 = -0.4;
        double r1451407 = r1451405 * r1451406;
        double r1451408 = fma(r1451399, r1451403, r1451407);
        double r1451409 = fma(r1451401, r1451402, r1451408);
        return r1451409;
}

Error

Bits error versus eps

Target

Original58.6
Target0.2
Herbie0.2
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.6

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

  2. Taylor expanded around 0 0.2

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

  3. Simplified0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019139 +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))))