Average Error: 58.7 → 0.6
Time: 18.3s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[2 \cdot {\varepsilon}^{2} + \left(\log 1 - 2 \cdot \left(\frac{{\varepsilon}^{2}}{{1}^{2}} + \varepsilon\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
2 \cdot {\varepsilon}^{2} + \left(\log 1 - 2 \cdot \left(\frac{{\varepsilon}^{2}}{{1}^{2}} + \varepsilon\right)\right)
double f(double eps) {
        double r70908 = 1.0;
        double r70909 = eps;
        double r70910 = r70908 - r70909;
        double r70911 = r70908 + r70909;
        double r70912 = r70910 / r70911;
        double r70913 = log(r70912);
        return r70913;
}
↓
double f(double eps) {
        double r70914 = 2.0;
        double r70915 = eps;
        double r70916 = 2.0;
        double r70917 = pow(r70915, r70916);
        double r70918 = r70914 * r70917;
        double r70919 = 1.0;
        double r70920 = log(r70919);
        double r70921 = pow(r70919, r70916);
        double r70922 = r70917 / r70921;
        double r70923 = r70922 + r70915;
        double r70924 = r70914 * r70923;
        double r70925 = r70920 - r70924;
        double r70926 = r70918 + r70925;
        return r70926;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.7
Target0.2
Herbie0.6
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.7

    \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

    \[\leadsto 2 \cdot {\varepsilon}^{2} + \left(\log 1 - 2 \cdot \left(\frac{{\varepsilon}^{2}}{{1}^{2}} + \varepsilon\right)\right)\]

Reproduce

herbie shell --seed 2019323 
(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))))