Average Error: 58.7 → 0.2
Time: 14.1s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \frac{-2}{3} - \left(\varepsilon \cdot 2 + {\varepsilon}^{5} \cdot \frac{2}{5}\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \frac{-2}{3} - \left(\varepsilon \cdot 2 + {\varepsilon}^{5} \cdot \frac{2}{5}\right)
double f(double eps) {
        double r1201212 = 1.0;
        double r1201213 = eps;
        double r1201214 = r1201212 - r1201213;
        double r1201215 = r1201212 + r1201213;
        double r1201216 = r1201214 / r1201215;
        double r1201217 = log(r1201216);
        return r1201217;
}
↓
double f(double eps) {
        double r1201218 = eps;
        double r1201219 = r1201218 * r1201218;
        double r1201220 = r1201219 * r1201218;
        double r1201221 = -0.6666666666666666;
        double r1201222 = r1201220 * r1201221;
        double r1201223 = 2.0;
        double r1201224 = r1201218 * r1201223;
        double r1201225 = 5.0;
        double r1201226 = pow(r1201218, r1201225);
        double r1201227 = 0.4;
        double r1201228 = r1201226 * r1201227;
        double r1201229 = r1201224 + r1201228;
        double r1201230 = r1201222 - r1201229;
        return r1201230;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.7

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

  2. Taylor expanded around 0 0.2

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

  3. Simplified0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019152 
(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))))