Average Error: 58.7 → 0.2
Time: 11.7s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(\frac{2}{3}, \frac{{x}^{3}}{{1}^{3}}, \mathsf{fma}\left(2, x, \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \mathsf{fma}\left(\frac{2}{3}, \frac{{x}^{3}}{{1}^{3}}, \mathsf{fma}\left(2, x, \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)
double f(double x) {
        double r54911 = 1.0;
        double r54912 = 2.0;
        double r54913 = r54911 / r54912;
        double r54914 = x;
        double r54915 = r54911 + r54914;
        double r54916 = r54911 - r54914;
        double r54917 = r54915 / r54916;
        double r54918 = log(r54917);
        double r54919 = r54913 * r54918;
        return r54919;
}
↓
double f(double x) {
        double r54920 = 1.0;
        double r54921 = 2.0;
        double r54922 = r54920 / r54921;
        double r54923 = 0.6666666666666666;
        double r54924 = x;
        double r54925 = 3.0;
        double r54926 = pow(r54924, r54925);
        double r54927 = pow(r54920, r54925);
        double r54928 = r54926 / r54927;
        double r54929 = 0.4;
        double r54930 = 5.0;
        double r54931 = pow(r54924, r54930);
        double r54932 = pow(r54920, r54930);
        double r54933 = r54931 / r54932;
        double r54934 = r54929 * r54933;
        double r54935 = fma(r54921, r54924, r54934);
        double r54936 = fma(r54923, r54928, r54935);
        double r54937 = r54922 * r54936;
        return r54937;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.7

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

  3. Applied div-inv58.7

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

  4. Applied log-prod58.7

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

  5. Simplified58.7

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

  6. Taylor expanded around 0 0.2

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{2}{3} \cdot \frac{{x}^{3}}{{1}^{3}} + \left(2 \cdot x + \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)}\]

  7. Simplified0.2

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{2}{3}, \frac{{x}^{3}}{{1}^{3}}, \mathsf{fma}\left(2, x, \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)}\]

  8. Final simplification0.2

    \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(\frac{2}{3}, \frac{{x}^{3}}{{1}^{3}}, \mathsf{fma}\left(2, x, \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)\]

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))