Average Error: 58.6 → 0.2
Time: 12.1s
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 r1334762 = 1.0;
        double r1334763 = eps;
        double r1334764 = r1334762 - r1334763;
        double r1334765 = r1334762 + r1334763;
        double r1334766 = r1334764 / r1334765;
        double r1334767 = log(r1334766);
        return r1334767;
}
↓
double f(double eps) {
        double r1334768 = eps;
        double r1334769 = 5.0;
        double r1334770 = pow(r1334768, r1334769);
        double r1334771 = -0.4;
        double r1334772 = r1334770 * r1334771;
        double r1334773 = r1334768 * r1334768;
        double r1334774 = r1334773 * r1334768;
        double r1334775 = -0.6666666666666666;
        double r1334776 = r1334774 * r1334775;
        double r1334777 = -2.0;
        double r1334778 = r1334777 * r1334768;
        double r1334779 = r1334776 + r1334778;
        double r1334780 = r1334772 + r1334779;
        return r1334780;
}

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

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

  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 2019133 
(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))))