Average Error: 58.5 → 0.3
Time: 13.4s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left(\frac{2}{3} \cdot {\left(\frac{x}{1}\right)}^{3} + \left(2 \cdot x + \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \left(\frac{2}{3} \cdot {\left(\frac{x}{1}\right)}^{3} + \left(2 \cdot x + \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)
double f(double x) {
        double r58811 = 1.0;
        double r58812 = 2.0;
        double r58813 = r58811 / r58812;
        double r58814 = x;
        double r58815 = r58811 + r58814;
        double r58816 = r58811 - r58814;
        double r58817 = r58815 / r58816;
        double r58818 = log(r58817);
        double r58819 = r58813 * r58818;
        return r58819;
}
↓
double f(double x) {
        double r58820 = 1.0;
        double r58821 = 2.0;
        double r58822 = r58820 / r58821;
        double r58823 = 0.6666666666666666;
        double r58824 = x;
        double r58825 = r58824 / r58820;
        double r58826 = 3.0;
        double r58827 = pow(r58825, r58826);
        double r58828 = r58823 * r58827;
        double r58829 = r58821 * r58824;
        double r58830 = 0.4;
        double r58831 = 5.0;
        double r58832 = pow(r58824, r58831);
        double r58833 = pow(r58820, r58831);
        double r58834 = r58832 / r58833;
        double r58835 = r58830 * r58834;
        double r58836 = r58829 + r58835;
        double r58837 = r58828 + r58836;
        double r58838 = r58822 * r58837;
        return r58838;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.5

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

  3. Applied log-div58.5

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

  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. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019303 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))