Average Error: 58.7 → 0.5
Time: 6.0s
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 r43750 = 1.0;
        double r43751 = 2.0;
        double r43752 = r43750 / r43751;
        double r43753 = x;
        double r43754 = r43750 + r43753;
        double r43755 = r43750 - r43753;
        double r43756 = r43754 / r43755;
        double r43757 = log(r43756);
        double r43758 = r43752 * r43757;
        return r43758;
}
↓
double f(double x) {
        double r43759 = 1.0;
        double r43760 = 2.0;
        double r43761 = r43759 / r43760;
        double r43762 = x;
        double r43763 = 2.0;
        double r43764 = pow(r43762, r43763);
        double r43765 = r43764 + r43762;
        double r43766 = r43760 * r43765;
        double r43767 = log(r43759);
        double r43768 = pow(r43759, r43763);
        double r43769 = r43764 / r43768;
        double r43770 = r43760 * r43769;
        double r43771 = r43767 - r43770;
        double r43772 = r43766 + r43771;
        double r43773 = r43761 * r43772;
        return r43773;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.7

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

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2020033 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))