Average Error: 58.7 → 0.2
Time: 17.5s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\varepsilon \cdot -2 + \left(\frac{-2}{3} \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) + \frac{-2}{5} \cdot {\varepsilon}^{5}\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\varepsilon \cdot -2 + \left(\frac{-2}{3} \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) + \frac{-2}{5} \cdot {\varepsilon}^{5}\right)
double f(double eps) {
        double r1580985 = 1.0;
        double r1580986 = eps;
        double r1580987 = r1580985 - r1580986;
        double r1580988 = r1580985 + r1580986;
        double r1580989 = r1580987 / r1580988;
        double r1580990 = log(r1580989);
        return r1580990;
}
↓
double f(double eps) {
        double r1580991 = eps;
        double r1580992 = -2.0;
        double r1580993 = r1580991 * r1580992;
        double r1580994 = -0.6666666666666666;
        double r1580995 = r1580991 * r1580991;
        double r1580996 = r1580995 * r1580991;
        double r1580997 = r1580994 * r1580996;
        double r1580998 = -0.4;
        double r1580999 = 5.0;
        double r1581000 = pow(r1580991, r1580999);
        double r1581001 = r1580998 * r1581000;
        double r1581002 = r1580997 + r1581001;
        double r1581003 = r1580993 + r1581002;
        return r1581003;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.7

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

  2. Taylor expanded around 0 0.2

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

  3. Simplified0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019137 
(FPCore (eps)
  :name "logq (problem 3.4.3)"

  :herbie-target
  (* -2 (+ (+ eps (/ (pow eps 3) 3)) (/ (pow eps 5) 5)))

  (log (/ (- 1 eps) (+ 1 eps))))