Average Error: 58.6 → 0.3
Time: 6.4s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left(\frac{2}{3} \cdot \frac{{x}^{3}}{{1}^{3}} + \left(2 \cdot x + \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \left(\frac{2}{3} \cdot \frac{{x}^{3}}{{1}^{3}} + \left(2 \cdot x + \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)
double f(double x) {
        double r80701 = 1.0;
        double r80702 = 2.0;
        double r80703 = r80701 / r80702;
        double r80704 = x;
        double r80705 = r80701 + r80704;
        double r80706 = r80701 - r80704;
        double r80707 = r80705 / r80706;
        double r80708 = log(r80707);
        double r80709 = r80703 * r80708;
        return r80709;
}
↓
double f(double x) {
        double r80710 = 1.0;
        double r80711 = 2.0;
        double r80712 = r80710 / r80711;
        double r80713 = 0.6666666666666666;
        double r80714 = x;
        double r80715 = 3.0;
        double r80716 = pow(r80714, r80715);
        double r80717 = pow(r80710, r80715);
        double r80718 = r80716 / r80717;
        double r80719 = r80713 * r80718;
        double r80720 = r80711 * r80714;
        double r80721 = 0.4;
        double r80722 = 5.0;
        double r80723 = pow(r80714, r80722);
        double r80724 = pow(r80710, r80722);
        double r80725 = r80723 / r80724;
        double r80726 = r80721 * r80725;
        double r80727 = r80720 + r80726;
        double r80728 = r80719 + r80727;
        double r80729 = r80712 * r80728;
        return r80729;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.6

    \[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
  2. Using strategy rm

  3. Applied log-div58.6

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\log \left(1 + x\right) - \log \left(1 - x\right)\right)}\]

  4. Taylor expanded around 0 0.3

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{2}{3} \cdot \frac{{x}^{3}}{{1}^{3}} + \left(2 \cdot x + \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)}\]

  5. Final simplification0.3

    \[\leadsto \frac{1}{2} \cdot \left(\frac{2}{3} \cdot \frac{{x}^{3}}{{1}^{3}} + \left(2 \cdot x + \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)\]

Reproduce

herbie shell --seed 2019356 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))