Average Error: 58.4 → 0.7
Time: 13.5s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[2 \cdot \left(\varepsilon \cdot \varepsilon - \left(\varepsilon + \frac{\varepsilon \cdot \varepsilon}{1 \cdot 1}\right)\right) + \log 1\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
2 \cdot \left(\varepsilon \cdot \varepsilon - \left(\varepsilon + \frac{\varepsilon \cdot \varepsilon}{1 \cdot 1}\right)\right) + \log 1
double f(double eps) {
        double r5058917 = 1.0;
        double r5058918 = eps;
        double r5058919 = r5058917 - r5058918;
        double r5058920 = r5058917 + r5058918;
        double r5058921 = r5058919 / r5058920;
        double r5058922 = log(r5058921);
        return r5058922;
}
↓
double f(double eps) {
        double r5058923 = 2.0;
        double r5058924 = eps;
        double r5058925 = r5058924 * r5058924;
        double r5058926 = 1.0;
        double r5058927 = r5058926 * r5058926;
        double r5058928 = r5058925 / r5058927;
        double r5058929 = r5058924 + r5058928;
        double r5058930 = r5058925 - r5058929;
        double r5058931 = r5058923 * r5058930;
        double r5058932 = log(r5058926);
        double r5058933 = r5058931 + r5058932;
        return r5058933;
}

Error

Bits error versus eps

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.4
Target0.2
Herbie0.7
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.4

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019174 
(FPCore (eps)
  :name "logq (problem 3.4.3)"

  :herbie-target
  (* -2.0 (+ (+ eps (/ (pow eps 3.0) 3.0)) (/ (pow eps 5.0) 5.0)))

  (log (/ (- 1.0 eps) (+ 1.0 eps))))