Average Error: 58.7 → 0.6
Time: 14.8s
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 r83522 = 1.0;
        double r83523 = eps;
        double r83524 = r83522 - r83523;
        double r83525 = r83522 + r83523;
        double r83526 = r83524 / r83525;
        double r83527 = log(r83526);
        return r83527;
}
↓
double f(double eps) {
        double r83528 = 2.0;
        double r83529 = eps;
        double r83530 = 2.0;
        double r83531 = pow(r83529, r83530);
        double r83532 = r83528 * r83531;
        double r83533 = 1.0;
        double r83534 = log(r83533);
        double r83535 = pow(r83533, r83530);
        double r83536 = r83531 / r83535;
        double r83537 = r83536 + r83529;
        double r83538 = r83528 * r83537;
        double r83539 = r83534 - r83538;
        double r83540 = r83532 + r83539;
        return r83540;
}

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 2019305 
(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))))