Average Error: 58.7 → 0.6
Time: 10.5s
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 r70932 = 1.0;
        double r70933 = eps;
        double r70934 = r70932 - r70933;
        double r70935 = r70932 + r70933;
        double r70936 = r70934 / r70935;
        double r70937 = log(r70936);
        return r70937;
}
↓
double f(double eps) {
        double r70938 = 2.0;
        double r70939 = eps;
        double r70940 = 1.0;
        double r70941 = r70940 * r70940;
        double r70942 = r70939 / r70941;
        double r70943 = r70939 - r70942;
        double r70944 = r70939 * r70943;
        double r70945 = r70944 - r70939;
        double r70946 = r70938 * r70945;
        double r70947 = log(r70940);
        double r70948 = r70946 + r70947;
        return r70948;
}

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

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

  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 2019350 
(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))))