Average Error: 58.5 → 0.2
Time: 13.1s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left(\frac{-2}{3}, {\left(\frac{\varepsilon}{1}\right)}^{3}, -\mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, 2 \cdot \varepsilon\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left(\frac{-2}{3}, {\left(\frac{\varepsilon}{1}\right)}^{3}, -\mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, 2 \cdot \varepsilon\right)\right)
double f(double eps) {
        double r45560 = 1.0;
        double r45561 = eps;
        double r45562 = r45560 - r45561;
        double r45563 = r45560 + r45561;
        double r45564 = r45562 / r45563;
        double r45565 = log(r45564);
        return r45565;
}
↓
double f(double eps) {
        double r45566 = -0.6666666666666666;
        double r45567 = eps;
        double r45568 = 1.0;
        double r45569 = r45567 / r45568;
        double r45570 = 3.0;
        double r45571 = pow(r45569, r45570);
        double r45572 = 0.4;
        double r45573 = 5.0;
        double r45574 = pow(r45567, r45573);
        double r45575 = pow(r45568, r45573);
        double r45576 = r45574 / r45575;
        double r45577 = 2.0;
        double r45578 = r45577 * r45567;
        double r45579 = fma(r45572, r45576, r45578);
        double r45580 = -r45579;
        double r45581 = fma(r45566, r45571, r45580);
        return r45581;
}

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. Simplified58.5

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

  5. Taylor expanded around 0 0.2

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

  6. Simplified0.2

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

  7. Final simplification0.2

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

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))))