Average Error: 58.6 → 0.2
Time: 17.0s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[-\mathsf{fma}\left(\frac{2}{3}, \frac{{\varepsilon}^{3}}{{1}^{3}}, \mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, 2 \cdot \varepsilon\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
-\mathsf{fma}\left(\frac{2}{3}, \frac{{\varepsilon}^{3}}{{1}^{3}}, \mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, 2 \cdot \varepsilon\right)\right)
double f(double eps) {
        double r72669 = 1.0;
        double r72670 = eps;
        double r72671 = r72669 - r72670;
        double r72672 = r72669 + r72670;
        double r72673 = r72671 / r72672;
        double r72674 = log(r72673);
        return r72674;
}
↓
double f(double eps) {
        double r72675 = 0.6666666666666666;
        double r72676 = eps;
        double r72677 = 3.0;
        double r72678 = pow(r72676, r72677);
        double r72679 = 1.0;
        double r72680 = pow(r72679, r72677);
        double r72681 = r72678 / r72680;
        double r72682 = 0.4;
        double r72683 = 5.0;
        double r72684 = pow(r72676, r72683);
        double r72685 = pow(r72679, r72683);
        double r72686 = r72684 / r72685;
        double r72687 = 2.0;
        double r72688 = r72687 * r72676;
        double r72689 = fma(r72682, r72686, r72688);
        double r72690 = fma(r72675, r72681, r72689);
        double r72691 = -r72690;
        return r72691;
}

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. Using strategy rm

  3. Applied log-div58.6

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

  4. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{-\left(\frac{2}{3} \cdot \frac{{\varepsilon}^{3}}{{1}^{3}} + \left(\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)\right)}\]

  5. Simplified0.2

    \[\leadsto \color{blue}{-\mathsf{fma}\left(\frac{2}{3}, \frac{{\varepsilon}^{3}}{{1}^{3}}, \mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, 2 \cdot \varepsilon\right)\right)}\]

  6. Final simplification0.2

    \[\leadsto -\mathsf{fma}\left(\frac{2}{3}, \frac{{\varepsilon}^{3}}{{1}^{3}}, \mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, 2 \cdot \varepsilon\right)\right)\]

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (eps)
  :name "logq (problem 3.4.3)"
  :precision binary64

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

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