Average Error: 58.4 → 0.7
Time: 14.9s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(\frac{{x}^{2}}{{1}^{2}}, -2, \mathsf{fma}\left(2, \mathsf{fma}\left(x, x, x\right), \log 1\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \mathsf{fma}\left(\frac{{x}^{2}}{{1}^{2}}, -2, \mathsf{fma}\left(2, \mathsf{fma}\left(x, x, x\right), \log 1\right)\right)
double f(double x) {
        double r51784 = 1.0;
        double r51785 = 2.0;
        double r51786 = r51784 / r51785;
        double r51787 = x;
        double r51788 = r51784 + r51787;
        double r51789 = r51784 - r51787;
        double r51790 = r51788 / r51789;
        double r51791 = log(r51790);
        double r51792 = r51786 * r51791;
        return r51792;
}
↓
double f(double x) {
        double r51793 = 1.0;
        double r51794 = 2.0;
        double r51795 = r51793 / r51794;
        double r51796 = x;
        double r51797 = 2.0;
        double r51798 = pow(r51796, r51797);
        double r51799 = pow(r51793, r51797);
        double r51800 = r51798 / r51799;
        double r51801 = -r51794;
        double r51802 = fma(r51796, r51796, r51796);
        double r51803 = log(r51793);
        double r51804 = fma(r51794, r51802, r51803);
        double r51805 = fma(r51800, r51801, r51804);
        double r51806 = r51795 * r51805;
        return r51806;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.4

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))