Average Error: 58.6 → 0.6
Time: 8.3s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(2, -\frac{{x}^{2}}{{1}^{2}}, \mathsf{fma}\left(2, \mathsf{fma}\left(x, x, x\right), \log 1\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \mathsf{fma}\left(2, -\frac{{x}^{2}}{{1}^{2}}, \mathsf{fma}\left(2, \mathsf{fma}\left(x, x, x\right), \log 1\right)\right)
double f(double x) {
        double r81825 = 1.0;
        double r81826 = 2.0;
        double r81827 = r81825 / r81826;
        double r81828 = x;
        double r81829 = r81825 + r81828;
        double r81830 = r81825 - r81828;
        double r81831 = r81829 / r81830;
        double r81832 = log(r81831);
        double r81833 = r81827 * r81832;
        return r81833;
}
↓
double f(double x) {
        double r81834 = 1.0;
        double r81835 = 2.0;
        double r81836 = r81834 / r81835;
        double r81837 = x;
        double r81838 = 2.0;
        double r81839 = pow(r81837, r81838);
        double r81840 = pow(r81834, r81838);
        double r81841 = r81839 / r81840;
        double r81842 = -r81841;
        double r81843 = fma(r81837, r81837, r81837);
        double r81844 = log(r81834);
        double r81845 = fma(r81835, r81843, r81844);
        double r81846 = fma(r81835, r81842, r81845);
        double r81847 = r81836 * r81846;
        return r81847;
}

Error

Bits error versus x

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

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

  4. Final simplification0.6

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

Reproduce

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