Average Error: 58.6 → 0.6
Time: 11.7s
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 r89652 = 1.0;
        double r89653 = eps;
        double r89654 = r89652 - r89653;
        double r89655 = r89652 + r89653;
        double r89656 = r89654 / r89655;
        double r89657 = log(r89656);
        return r89657;
}
↓
double f(double eps) {
        double r89658 = 2.0;
        double r89659 = eps;
        double r89660 = 1.0;
        double r89661 = r89660 * r89660;
        double r89662 = r89659 / r89661;
        double r89663 = r89659 - r89662;
        double r89664 = r89659 * r89663;
        double r89665 = r89664 - r89659;
        double r89666 = r89658 * r89665;
        double r89667 = log(r89660);
        double r89668 = r89666 + r89667;
        return r89668;
}

Error

Bits error versus eps

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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

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019326 
(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))))