Average Error: 58.7 → 0.2
Time: 5.3s
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 r84835 = 1.0;
        double r84836 = eps;
        double r84837 = r84835 - r84836;
        double r84838 = r84835 + r84836;
        double r84839 = r84837 / r84838;
        double r84840 = log(r84839);
        return r84840;
}
↓
double f(double eps) {
        double r84841 = -0.6666666666666666;
        double r84842 = eps;
        double r84843 = 3.0;
        double r84844 = pow(r84842, r84843);
        double r84845 = 1.0;
        double r84846 = pow(r84845, r84843);
        double r84847 = r84844 / r84846;
        double r84848 = 0.4;
        double r84849 = 5.0;
        double r84850 = pow(r84842, r84849);
        double r84851 = pow(r84845, r84849);
        double r84852 = r84850 / r84851;
        double r84853 = r84848 * r84852;
        double r84854 = -r84853;
        double r84855 = fma(r84841, r84847, r84854);
        double r84856 = 2.0;
        double r84857 = r84856 * r84842;
        double r84858 = r84855 - r84857;
        return r84858;
}

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-div58.6

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

  4. Taylor expanded around 0 0.2

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

  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 2020002 +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))))