Average Error: 58.6 → 0.7
Time: 5.8s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, x\right), 2, \log 1 - 2 \cdot \frac{{x}^{2}}{{1}^{2}}\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, x\right), 2, \log 1 - 2 \cdot \frac{{x}^{2}}{{1}^{2}}\right)
double f(double x) {
        double r75890 = 1.0;
        double r75891 = 2.0;
        double r75892 = r75890 / r75891;
        double r75893 = x;
        double r75894 = r75890 + r75893;
        double r75895 = r75890 - r75893;
        double r75896 = r75894 / r75895;
        double r75897 = log(r75896);
        double r75898 = r75892 * r75897;
        return r75898;
}
↓
double f(double x) {
        double r75899 = 1.0;
        double r75900 = 2.0;
        double r75901 = r75899 / r75900;
        double r75902 = x;
        double r75903 = fma(r75902, r75902, r75902);
        double r75904 = log(r75899);
        double r75905 = 2.0;
        double r75906 = pow(r75902, r75905);
        double r75907 = pow(r75899, r75905);
        double r75908 = r75906 / r75907;
        double r75909 = r75900 * r75908;
        double r75910 = r75904 - r75909;
        double r75911 = fma(r75903, r75900, r75910);
        double r75912 = r75901 * r75911;
        return r75912;
}

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 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))