Average Error: 58.7 → 0.2
Time: 5.7s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(-2 \cdot \varepsilon\right) - \mathsf{fma}\left(0.66666666666666663, {\varepsilon}^{3}, 0.40000000000000002 \cdot {\varepsilon}^{5}\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\left(-2 \cdot \varepsilon\right) - \mathsf{fma}\left(0.66666666666666663, {\varepsilon}^{3}, 0.40000000000000002 \cdot {\varepsilon}^{5}\right)
double f(double eps) {
        double r89552 = 1.0;
        double r89553 = eps;
        double r89554 = r89552 - r89553;
        double r89555 = r89552 + r89553;
        double r89556 = r89554 / r89555;
        double r89557 = log(r89556);
        return r89557;
}
↓
double f(double eps) {
        double r89558 = 2.0;
        double r89559 = eps;
        double r89560 = r89558 * r89559;
        double r89561 = -r89560;
        double r89562 = 0.6666666666666666;
        double r89563 = 3.0;
        double r89564 = pow(r89559, r89563);
        double r89565 = 0.4;
        double r89566 = 5.0;
        double r89567 = pow(r89559, r89566);
        double r89568 = r89565 * r89567;
        double r89569 = fma(r89562, r89564, r89568);
        double r89570 = r89561 - r89569;
        return r89570;
}

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 log-div58.7

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

  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

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

  6. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{-\left(2 \cdot \varepsilon + \left(0.66666666666666663 \cdot {\varepsilon}^{3} + 0.40000000000000002 \cdot {\varepsilon}^{5}\right)\right)}\]

  7. Simplified0.2

    \[\leadsto \color{blue}{\left(-2 \cdot \varepsilon\right) - \mathsf{fma}\left(0.66666666666666663, {\varepsilon}^{3}, 0.40000000000000002 \cdot {\varepsilon}^{5}\right)}\]

  8. Final simplification0.2

    \[\leadsto \left(-2 \cdot \varepsilon\right) - \mathsf{fma}\left(0.66666666666666663, {\varepsilon}^{3}, 0.40000000000000002 \cdot {\varepsilon}^{5}\right)\]

Reproduce

herbie shell --seed 2020081 +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))))