Average Error: 58.4 → 0.3
Time: 17.1s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[-2 \cdot \varepsilon + \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\right) \cdot \frac{-2}{5} - \varepsilon \cdot \left(\frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
-2 \cdot \varepsilon + \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\right) \cdot \frac{-2}{5} - \varepsilon \cdot \left(\frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)
double f(double eps) {
        double r2566126 = 1.0;
        double r2566127 = eps;
        double r2566128 = r2566126 - r2566127;
        double r2566129 = r2566126 + r2566127;
        double r2566130 = r2566128 / r2566129;
        double r2566131 = log(r2566130);
        return r2566131;
}
↓
double f(double eps) {
        double r2566132 = -2.0;
        double r2566133 = eps;
        double r2566134 = r2566132 * r2566133;
        double r2566135 = r2566133 * r2566133;
        double r2566136 = r2566135 * r2566133;
        double r2566137 = r2566135 * r2566136;
        double r2566138 = -0.4;
        double r2566139 = r2566137 * r2566138;
        double r2566140 = 0.6666666666666666;
        double r2566141 = r2566140 * r2566135;
        double r2566142 = r2566133 * r2566141;
        double r2566143 = r2566139 - r2566142;
        double r2566144 = r2566134 + r2566143;
        return r2566144;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.4
Target0.3
Herbie0.3
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.4

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

  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. Taylor expanded around 0 0.3

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019163 
(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))))