Average Error: 58.5 → 0.2
Time: 15.4s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[{\varepsilon}^{5} \cdot \frac{-2}{5} + \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \frac{-2}{3} + -2 \cdot \varepsilon\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
{\varepsilon}^{5} \cdot \frac{-2}{5} + \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \frac{-2}{3} + -2 \cdot \varepsilon\right)
double f(double eps) {
        double r2328375 = 1.0;
        double r2328376 = eps;
        double r2328377 = r2328375 - r2328376;
        double r2328378 = r2328375 + r2328376;
        double r2328379 = r2328377 / r2328378;
        double r2328380 = log(r2328379);
        return r2328380;
}
↓
double f(double eps) {
        double r2328381 = eps;
        double r2328382 = 5.0;
        double r2328383 = pow(r2328381, r2328382);
        double r2328384 = -0.4;
        double r2328385 = r2328383 * r2328384;
        double r2328386 = r2328381 * r2328381;
        double r2328387 = r2328386 * r2328381;
        double r2328388 = -0.6666666666666666;
        double r2328389 = r2328387 * r2328388;
        double r2328390 = -2.0;
        double r2328391 = r2328390 * r2328381;
        double r2328392 = r2328389 + r2328391;
        double r2328393 = r2328385 + r2328392;
        return r2328393;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.5

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

  2. Taylor expanded around 0 0.2

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

  3. Simplified0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019132 
(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))))