Average Error: 58.3 → 0.8
Time: 48.1s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{\left(\left(2 \cdot x + \log 1\right) - {x}^{2} \cdot \left(\frac{2}{{1}^{2}} - 2\right)\right) \cdot 1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{\left(\left(2 \cdot x + \log 1\right) - {x}^{2} \cdot \left(\frac{2}{{1}^{2}} - 2\right)\right) \cdot 1}{2}
double f(double x) {
        double r57674 = 1.0;
        double r57675 = 2.0;
        double r57676 = r57674 / r57675;
        double r57677 = x;
        double r57678 = r57674 + r57677;
        double r57679 = r57674 - r57677;
        double r57680 = r57678 / r57679;
        double r57681 = log(r57680);
        double r57682 = r57676 * r57681;
        return r57682;
}
↓
double f(double x) {
        double r57683 = 2.0;
        double r57684 = x;
        double r57685 = r57683 * r57684;
        double r57686 = 1.0;
        double r57687 = log(r57686);
        double r57688 = r57685 + r57687;
        double r57689 = 2.0;
        double r57690 = pow(r57684, r57689);
        double r57691 = pow(r57686, r57689);
        double r57692 = r57683 / r57691;
        double r57693 = r57692 - r57683;
        double r57694 = r57690 * r57693;
        double r57695 = r57688 - r57694;
        double r57696 = r57695 * r57686;
        double r57697 = r57696 / r57683;
        return r57697;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.3

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019294 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))