Average Error: 58.5 → 0.7
Time: 15.0s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[2 \cdot \left(\varepsilon \cdot \left(\varepsilon - \frac{\varepsilon}{1 \cdot 1}\right) - \varepsilon\right) + \log 1\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
2 \cdot \left(\varepsilon \cdot \left(\varepsilon - \frac{\varepsilon}{1 \cdot 1}\right) - \varepsilon\right) + \log 1
double f(double eps) {
        double r78825 = 1.0;
        double r78826 = eps;
        double r78827 = r78825 - r78826;
        double r78828 = r78825 + r78826;
        double r78829 = r78827 / r78828;
        double r78830 = log(r78829);
        return r78830;
}
↓
double f(double eps) {
        double r78831 = 2.0;
        double r78832 = eps;
        double r78833 = 1.0;
        double r78834 = r78833 * r78833;
        double r78835 = r78832 / r78834;
        double r78836 = r78832 - r78835;
        double r78837 = r78832 * r78836;
        double r78838 = r78837 - r78832;
        double r78839 = r78831 * r78838;
        double r78840 = log(r78833);
        double r78841 = r78839 + r78840;
        return r78841;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.5

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019322 
(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))))