Average Error: 58.9 → 0.5
Time: 21.3s
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 r51882 = 1.0;
        double r51883 = 2.0;
        double r51884 = r51882 / r51883;
        double r51885 = x;
        double r51886 = r51882 + r51885;
        double r51887 = r51882 - r51885;
        double r51888 = r51886 / r51887;
        double r51889 = log(r51888);
        double r51890 = r51884 * r51889;
        return r51890;
}
↓
double f(double x) {
        double r51891 = 1.0;
        double r51892 = 2.0;
        double r51893 = r51891 / r51892;
        double r51894 = x;
        double r51895 = 2.0;
        double r51896 = pow(r51894, r51895);
        double r51897 = pow(r51891, r51895);
        double r51898 = r51896 / r51897;
        double r51899 = -r51892;
        double r51900 = fma(r51894, r51894, r51894);
        double r51901 = log(r51891);
        double r51902 = fma(r51892, r51900, r51901);
        double r51903 = fma(r51898, r51899, r51902);
        double r51904 = r51893 * r51903;
        return r51904;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.9

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

  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

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))