Average Error: 58.5 → 0.2
Time: 5.2s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left(\frac{-2}{3}, \frac{{\varepsilon}^{3}}{{1}^{3}}, -\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}}\right) - 2 \cdot \varepsilon\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left(\frac{-2}{3}, \frac{{\varepsilon}^{3}}{{1}^{3}}, -\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}}\right) - 2 \cdot \varepsilon
double f(double eps) {
        double r85047 = 1.0;
        double r85048 = eps;
        double r85049 = r85047 - r85048;
        double r85050 = r85047 + r85048;
        double r85051 = r85049 / r85050;
        double r85052 = log(r85051);
        return r85052;
}
↓
double f(double eps) {
        double r85053 = -0.6666666666666666;
        double r85054 = eps;
        double r85055 = 3.0;
        double r85056 = pow(r85054, r85055);
        double r85057 = 1.0;
        double r85058 = pow(r85057, r85055);
        double r85059 = r85056 / r85058;
        double r85060 = 0.4;
        double r85061 = 5.0;
        double r85062 = pow(r85054, r85061);
        double r85063 = pow(r85057, r85061);
        double r85064 = r85062 / r85063;
        double r85065 = r85060 * r85064;
        double r85066 = -r85065;
        double r85067 = fma(r85053, r85059, r85066);
        double r85068 = 2.0;
        double r85069 = r85068 * r85054;
        double r85070 = r85067 - r85069;
        return r85070;
}

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. Using strategy rm

  7. Applied fma-udef0.2

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

  8. Applied associate--r+0.2

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

  9. Simplified0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019362 +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))))