Average Error: 58.7 → 0.2
Time: 14.0s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[-\mathsf{fma}\left(2, \varepsilon, \mathsf{fma}\left(0.6666666666666666296592325124947819858789, {\varepsilon}^{3}, 0.4000000000000000222044604925031308084726 \cdot {\varepsilon}^{5}\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
-\mathsf{fma}\left(2, \varepsilon, \mathsf{fma}\left(0.6666666666666666296592325124947819858789, {\varepsilon}^{3}, 0.4000000000000000222044604925031308084726 \cdot {\varepsilon}^{5}\right)\right)
double f(double eps) {
        double r66541 = 1.0;
        double r66542 = eps;
        double r66543 = r66541 - r66542;
        double r66544 = r66541 + r66542;
        double r66545 = r66543 / r66544;
        double r66546 = log(r66545);
        return r66546;
}
↓
double f(double eps) {
        double r66547 = 2.0;
        double r66548 = eps;
        double r66549 = 0.6666666666666666;
        double r66550 = 3.0;
        double r66551 = pow(r66548, r66550);
        double r66552 = 0.4;
        double r66553 = 5.0;
        double r66554 = pow(r66548, r66553);
        double r66555 = r66552 * r66554;
        double r66556 = fma(r66549, r66551, r66555);
        double r66557 = fma(r66547, r66548, r66556);
        double r66558 = -r66557;
        return r66558;
}

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. Using strategy rm

  3. Applied add-exp-log58.7

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

  4. Applied add-exp-log58.7

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

  5. Applied div-exp58.7

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

  6. Applied rem-log-exp58.6

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

  7. Taylor expanded around 0 0.2

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

  8. Simplified0.2

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

  9. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{-\left(2 \cdot \varepsilon + \left(0.6666666666666666296592325124947819858789 \cdot {\varepsilon}^{3} + 0.4000000000000000222044604925031308084726 \cdot {\varepsilon}^{5}\right)\right)}\]

  10. Simplified0.2

    \[\leadsto \color{blue}{-\mathsf{fma}\left(2, \varepsilon, \mathsf{fma}\left(0.6666666666666666296592325124947819858789, {\varepsilon}^{3}, 0.4000000000000000222044604925031308084726 \cdot {\varepsilon}^{5}\right)\right)}\]

  11. Final simplification0.2

    \[\leadsto -\mathsf{fma}\left(2, \varepsilon, \mathsf{fma}\left(0.6666666666666666296592325124947819858789, {\varepsilon}^{3}, 0.4000000000000000222044604925031308084726 \cdot {\varepsilon}^{5}\right)\right)\]

Reproduce

herbie shell --seed 2019323 +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))))