Average Error: 58.6 → 0.3
Time: 19.2s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(\frac{-2}{3} \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \left(-2 \cdot \varepsilon - {\varepsilon}^{5} \cdot \frac{2}{5}\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\left(\frac{-2}{3} \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \left(-2 \cdot \varepsilon - {\varepsilon}^{5} \cdot \frac{2}{5}\right)
double f(double eps) {
        double r2803231 = 1.0;
        double r2803232 = eps;
        double r2803233 = r2803231 - r2803232;
        double r2803234 = r2803231 + r2803232;
        double r2803235 = r2803233 / r2803234;
        double r2803236 = log(r2803235);
        return r2803236;
}
↓
double f(double eps) {
        double r2803237 = -0.6666666666666666;
        double r2803238 = eps;
        double r2803239 = r2803237 * r2803238;
        double r2803240 = r2803238 * r2803238;
        double r2803241 = r2803239 * r2803240;
        double r2803242 = -2.0;
        double r2803243 = r2803242 * r2803238;
        double r2803244 = 5.0;
        double r2803245 = pow(r2803238, r2803244);
        double r2803246 = 0.4;
        double r2803247 = r2803245 * r2803246;
        double r2803248 = r2803243 - r2803247;
        double r2803249 = r2803241 + r2803248;
        return r2803249;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.6

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

  2. Taylor expanded around 0 0.3

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019168 
(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))))