Average Error: 58.5 → 0.7
Time: 16.2s
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 r77950 = 1.0;
        double r77951 = eps;
        double r77952 = r77950 - r77951;
        double r77953 = r77950 + r77951;
        double r77954 = r77952 / r77953;
        double r77955 = log(r77954);
        return r77955;
}
↓
double f(double eps) {
        double r77956 = 2.0;
        double r77957 = eps;
        double r77958 = 1.0;
        double r77959 = r77958 * r77958;
        double r77960 = r77957 / r77959;
        double r77961 = r77957 - r77960;
        double r77962 = r77957 * r77961;
        double r77963 = r77962 - r77957;
        double r77964 = r77956 * r77963;
        double r77965 = log(r77958);
        double r77966 = r77964 + r77965;
        return r77966;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.5

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019306 
(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))))