Average Error: 58.6 → 0.2
Time: 24.3s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[{\varepsilon}^{5} \cdot \frac{-2}{5} - \frac{\left(\left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) - 4\right) \cdot \varepsilon}{\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon - 2}\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
{\varepsilon}^{5} \cdot \frac{-2}{5} - \frac{\left(\left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) - 4\right) \cdot \varepsilon}{\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon - 2}
double f(double eps) {
        double r6791796 = 1.0;
        double r6791797 = eps;
        double r6791798 = r6791796 - r6791797;
        double r6791799 = r6791796 + r6791797;
        double r6791800 = r6791798 / r6791799;
        double r6791801 = log(r6791800);
        return r6791801;
}
↓
double f(double eps) {
        double r6791802 = eps;
        double r6791803 = 5.0;
        double r6791804 = pow(r6791802, r6791803);
        double r6791805 = -0.4;
        double r6791806 = r6791804 * r6791805;
        double r6791807 = 0.6666666666666666;
        double r6791808 = r6791807 * r6791802;
        double r6791809 = r6791808 * r6791802;
        double r6791810 = r6791809 * r6791809;
        double r6791811 = 4.0;
        double r6791812 = r6791810 - r6791811;
        double r6791813 = r6791812 * r6791802;
        double r6791814 = 2.0;
        double r6791815 = r6791809 - r6791814;
        double r6791816 = r6791813 / r6791815;
        double r6791817 = r6791806 - r6791816;
        return r6791817;
}

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

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

  5. Applied flip-+0.2

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

  6. Applied associate-*l/0.2

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

  7. Final simplification0.2

    \[\leadsto {\varepsilon}^{5} \cdot \frac{-2}{5} - \frac{\left(\left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) - 4\right) \cdot \varepsilon}{\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon - 2}\]

Reproduce

herbie shell --seed 2019119 
(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))))