Average Error: 58.6 → 0.6
Time: 8.6s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\varepsilon \cdot \left(\left(\varepsilon - \frac{\varepsilon}{{1}^{2}}\right) \cdot 2 - 2\right) + \log 1\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\varepsilon \cdot \left(\left(\varepsilon - \frac{\varepsilon}{{1}^{2}}\right) \cdot 2 - 2\right) + \log 1
double f(double eps) {
        double r87650 = 1.0;
        double r87651 = eps;
        double r87652 = r87650 - r87651;
        double r87653 = r87650 + r87651;
        double r87654 = r87652 / r87653;
        double r87655 = log(r87654);
        return r87655;
}
↓
double f(double eps) {
        double r87656 = eps;
        double r87657 = 1.0;
        double r87658 = 2.0;
        double r87659 = pow(r87657, r87658);
        double r87660 = r87656 / r87659;
        double r87661 = r87656 - r87660;
        double r87662 = 2.0;
        double r87663 = r87661 * r87662;
        double r87664 = r87663 - r87662;
        double r87665 = r87656 * r87664;
        double r87666 = log(r87657);
        double r87667 = r87665 + r87666;
        return r87667;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.6

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

  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

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019308 
(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))))