Average Error: 58.5 → 0.2
Time: 5.3s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[-\left(\frac{2}{3} \cdot \frac{{\varepsilon}^{3}}{{1}^{3}} + \left(\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
-\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)
double f(double eps) {
        double r80702 = 1.0;
        double r80703 = eps;
        double r80704 = r80702 - r80703;
        double r80705 = r80702 + r80703;
        double r80706 = r80704 / r80705;
        double r80707 = log(r80706);
        return r80707;
}
↓
double f(double eps) {
        double r80708 = 0.6666666666666666;
        double r80709 = eps;
        double r80710 = 3.0;
        double r80711 = pow(r80709, r80710);
        double r80712 = 1.0;
        double r80713 = pow(r80712, r80710);
        double r80714 = r80711 / r80713;
        double r80715 = r80708 * r80714;
        double r80716 = 0.4;
        double r80717 = 5.0;
        double r80718 = pow(r80709, r80717);
        double r80719 = pow(r80712, r80717);
        double r80720 = r80718 / r80719;
        double r80721 = r80716 * r80720;
        double r80722 = 2.0;
        double r80723 = r80722 * r80709;
        double r80724 = r80721 + r80723;
        double r80725 = r80715 + r80724;
        double r80726 = -r80725;
        return r80726;
}

Error

Bits error versus eps

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.5

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

  3. Applied log-div58.5

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

  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. Final simplification0.2

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

Reproduce

herbie shell --seed 2019347 
(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))))