Average Error: 58.6 → 0.2
Time: 12.0s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\varepsilon \cdot -2 + \left(\frac{-2}{3} \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) + \frac{-2}{5} \cdot {\varepsilon}^{5}\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\varepsilon \cdot -2 + \left(\frac{-2}{3} \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) + \frac{-2}{5} \cdot {\varepsilon}^{5}\right)
double f(double eps) {
        double r1321081 = 1.0;
        double r1321082 = eps;
        double r1321083 = r1321081 - r1321082;
        double r1321084 = r1321081 + r1321082;
        double r1321085 = r1321083 / r1321084;
        double r1321086 = log(r1321085);
        return r1321086;
}
↓
double f(double eps) {
        double r1321087 = eps;
        double r1321088 = -2.0;
        double r1321089 = r1321087 * r1321088;
        double r1321090 = -0.6666666666666666;
        double r1321091 = r1321087 * r1321087;
        double r1321092 = r1321091 * r1321087;
        double r1321093 = r1321090 * r1321092;
        double r1321094 = -0.4;
        double r1321095 = 5.0;
        double r1321096 = pow(r1321087, r1321095);
        double r1321097 = r1321094 * r1321096;
        double r1321098 = r1321093 + r1321097;
        double r1321099 = r1321089 + r1321098;
        return r1321099;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.6

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

  2. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{-\left(\frac{2}{3} \cdot {\varepsilon}^{3} + \left(\frac{2}{5} \cdot {\varepsilon}^{5} + 2 \cdot \varepsilon\right)\right)}\]

  3. Simplified0.2

    \[\leadsto \color{blue}{\varepsilon \cdot -2 + \left(\frac{-2}{5} \cdot {\varepsilon}^{5} + \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \frac{-2}{3}\right)}\]

  4. Final simplification0.2

    \[\leadsto \varepsilon \cdot -2 + \left(\frac{-2}{3} \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) + \frac{-2}{5} \cdot {\varepsilon}^{5}\right)\]

Reproduce

herbie shell --seed 2019135 
(FPCore (eps)
  :name "logq (problem 3.4.3)"

  :herbie-target
  (* -2 (+ (+ eps (/ (pow eps 3) 3)) (/ (pow eps 5) 5)))

  (log (/ (- 1 eps) (+ 1 eps))))