Average Error: 58.5 → 0.2
Time: 5.6s
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 r51910 = 1.0;
        double r51911 = 2.0;
        double r51912 = r51910 / r51911;
        double r51913 = x;
        double r51914 = r51910 + r51913;
        double r51915 = r51910 - r51913;
        double r51916 = r51914 / r51915;
        double r51917 = log(r51916);
        double r51918 = r51912 * r51917;
        return r51918;
}
↓
double f(double x) {
        double r51919 = 1.0;
        double r51920 = 2.0;
        double r51921 = r51919 / r51920;
        double r51922 = 0.6666666666666666;
        double r51923 = x;
        double r51924 = 3.0;
        double r51925 = pow(r51923, r51924);
        double r51926 = pow(r51919, r51924);
        double r51927 = r51925 / r51926;
        double r51928 = 0.4;
        double r51929 = 5.0;
        double r51930 = pow(r51923, r51929);
        double r51931 = pow(r51919, r51929);
        double r51932 = r51930 / r51931;
        double r51933 = r51928 * r51932;
        double r51934 = fma(r51920, r51923, r51933);
        double r51935 = fma(r51922, r51927, r51934);
        double r51936 = r51921 * r51935;
        return r51936;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.5

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

  3. Applied log-div58.5

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

  4. Taylor expanded around 0 0.2

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

  5. Simplified0.2

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

  6. 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 2020025 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))