Average Error: 58.4 → 0.3
Time: 15.3s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \frac{-2}{3} - 2 \cdot \varepsilon\right) - \frac{2}{5} \cdot {\varepsilon}^{5}\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \frac{-2}{3} - 2 \cdot \varepsilon\right) - \frac{2}{5} \cdot {\varepsilon}^{5}
double f(double eps) {
        double r2617124 = 1.0;
        double r2617125 = eps;
        double r2617126 = r2617124 - r2617125;
        double r2617127 = r2617124 + r2617125;
        double r2617128 = r2617126 / r2617127;
        double r2617129 = log(r2617128);
        return r2617129;
}
↓
double f(double eps) {
        double r2617130 = eps;
        double r2617131 = r2617130 * r2617130;
        double r2617132 = r2617131 * r2617130;
        double r2617133 = -0.6666666666666666;
        double r2617134 = r2617132 * r2617133;
        double r2617135 = 2.0;
        double r2617136 = r2617135 * r2617130;
        double r2617137 = r2617134 - r2617136;
        double r2617138 = 0.4;
        double r2617139 = 5.0;
        double r2617140 = pow(r2617130, r2617139);
        double r2617141 = r2617138 * r2617140;
        double r2617142 = r2617137 - r2617141;
        return r2617142;
}

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

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

  4. Final simplification0.3

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

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))))