Average Error: 58.6 → 0.6
Time: 5.9s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left(2 \cdot \left({x}^{2} + x\right) + \left(\log 1 - 2 \cdot \frac{{x}^{2}}{{1}^{2}}\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \left(2 \cdot \left({x}^{2} + x\right) + \left(\log 1 - 2 \cdot \frac{{x}^{2}}{{1}^{2}}\right)\right)
double f(double x) {
        double r51737 = 1.0;
        double r51738 = 2.0;
        double r51739 = r51737 / r51738;
        double r51740 = x;
        double r51741 = r51737 + r51740;
        double r51742 = r51737 - r51740;
        double r51743 = r51741 / r51742;
        double r51744 = log(r51743);
        double r51745 = r51739 * r51744;
        return r51745;
}
↓
double f(double x) {
        double r51746 = 1.0;
        double r51747 = 2.0;
        double r51748 = r51746 / r51747;
        double r51749 = x;
        double r51750 = 2.0;
        double r51751 = pow(r51749, r51750);
        double r51752 = r51751 + r51749;
        double r51753 = r51747 * r51752;
        double r51754 = log(r51746);
        double r51755 = pow(r51746, r51750);
        double r51756 = r51751 / r51755;
        double r51757 = r51747 * r51756;
        double r51758 = r51754 - r51757;
        double r51759 = r51753 + r51758;
        double r51760 = r51748 * r51759;
        return r51760;
}

Error

Bits error versus x

Try it out

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. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2020047 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))