Average Error: 58.5 → 0.2
Time: 16.7s
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 r1847104 = 1.0;
        double r1847105 = eps;
        double r1847106 = r1847104 - r1847105;
        double r1847107 = r1847104 + r1847105;
        double r1847108 = r1847106 / r1847107;
        double r1847109 = log(r1847108);
        return r1847109;
}
↓
double f(double eps) {
        double r1847110 = eps;
        double r1847111 = 5.0;
        double r1847112 = pow(r1847110, r1847111);
        double r1847113 = -0.4;
        double r1847114 = r1847112 * r1847113;
        double r1847115 = r1847110 * r1847110;
        double r1847116 = r1847115 * r1847110;
        double r1847117 = -0.6666666666666666;
        double r1847118 = r1847116 * r1847117;
        double r1847119 = -2.0;
        double r1847120 = r1847119 * r1847110;
        double r1847121 = r1847118 + r1847120;
        double r1847122 = r1847114 + r1847121;
        return r1847122;
}

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 2019130 
(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))))