Average Error: 58.6 → 0.2
Time: 16.7s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left(\frac{-2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, \left(\frac{\varepsilon}{1} \cdot \left(\frac{\varepsilon}{1} \cdot \frac{\varepsilon}{1}\right)\right) \cdot \frac{-2}{3}\right) - \varepsilon \cdot 2\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left(\frac{-2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, \left(\frac{\varepsilon}{1} \cdot \left(\frac{\varepsilon}{1} \cdot \frac{\varepsilon}{1}\right)\right) \cdot \frac{-2}{3}\right) - \varepsilon \cdot 2
double f(double eps) {
        double r3809942 = 1.0;
        double r3809943 = eps;
        double r3809944 = r3809942 - r3809943;
        double r3809945 = r3809942 + r3809943;
        double r3809946 = r3809944 / r3809945;
        double r3809947 = log(r3809946);
        return r3809947;
}
↓
double f(double eps) {
        double r3809948 = -0.4;
        double r3809949 = eps;
        double r3809950 = 5.0;
        double r3809951 = pow(r3809949, r3809950);
        double r3809952 = 1.0;
        double r3809953 = pow(r3809952, r3809950);
        double r3809954 = r3809951 / r3809953;
        double r3809955 = r3809949 / r3809952;
        double r3809956 = r3809955 * r3809955;
        double r3809957 = r3809955 * r3809956;
        double r3809958 = -0.6666666666666666;
        double r3809959 = r3809957 * r3809958;
        double r3809960 = fma(r3809948, r3809954, r3809959);
        double r3809961 = 2.0;
        double r3809962 = r3809949 * r3809961;
        double r3809963 = r3809960 - r3809962;
        return r3809963;
}

Error

Bits error versus eps

Target

Original58.6
Target0.2
Herbie0.2
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.6

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

  3. Applied log-div58.6

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

  4. Taylor expanded around 0 0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (eps)
  :name "logq (problem 3.4.3)"

  :herbie-target
  (* -2.0 (+ (+ eps (/ (pow eps 3.0) 3.0)) (/ (pow eps 5.0) 5.0)))

  (log (/ (- 1.0 eps) (+ 1.0 eps))))