Average Error: 58.5 → 0.2
Time: 5.1s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(-\frac{2}{3} \cdot \frac{{\varepsilon}^{3}}{{1}^{3}}\right) - \mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, 2 \cdot \varepsilon\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\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)
double f(double eps) {
        double r79560 = 1.0;
        double r79561 = eps;
        double r79562 = r79560 - r79561;
        double r79563 = r79560 + r79561;
        double r79564 = r79562 / r79563;
        double r79565 = log(r79564);
        return r79565;
}
↓
double f(double eps) {
        double r79566 = 0.6666666666666666;
        double r79567 = eps;
        double r79568 = 3.0;
        double r79569 = pow(r79567, r79568);
        double r79570 = 1.0;
        double r79571 = pow(r79570, r79568);
        double r79572 = r79569 / r79571;
        double r79573 = r79566 * r79572;
        double r79574 = -r79573;
        double r79575 = 0.4;
        double r79576 = 5.0;
        double r79577 = pow(r79567, r79576);
        double r79578 = pow(r79570, r79576);
        double r79579 = r79577 / r79578;
        double r79580 = 2.0;
        double r79581 = r79580 * r79567;
        double r79582 = fma(r79575, r79579, r79581);
        double r79583 = r79574 - r79582;
        return r79583;
}

Error

Bits error versus eps

Target

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

Derivation

  1. Initial program 58.5

    \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
  2. Using strategy rm

  3. Applied log-div58.5

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

  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. Final simplification0.2

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

Reproduce

herbie shell --seed 2019347 +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))))