Average Error: 58.6 → 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 r47144 = 1.0;
        double r47145 = 2.0;
        double r47146 = r47144 / r47145;
        double r47147 = x;
        double r47148 = r47144 + r47147;
        double r47149 = r47144 - r47147;
        double r47150 = r47148 / r47149;
        double r47151 = log(r47150);
        double r47152 = r47146 * r47151;
        return r47152;
}
↓
double f(double x) {
        double r47153 = 1.0;
        double r47154 = 2.0;
        double r47155 = r47153 / r47154;
        double r47156 = 0.6666666666666666;
        double r47157 = x;
        double r47158 = 3.0;
        double r47159 = pow(r47157, r47158);
        double r47160 = pow(r47153, r47158);
        double r47161 = r47159 / r47160;
        double r47162 = 0.4;
        double r47163 = 5.0;
        double r47164 = pow(r47157, r47163);
        double r47165 = pow(r47153, r47163);
        double r47166 = r47164 / r47165;
        double r47167 = r47162 * r47166;
        double r47168 = fma(r47154, r47157, r47167);
        double r47169 = fma(r47156, r47161, r47168);
        double r47170 = r47155 * r47169;
        return r47170;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.6

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

  3. Applied log-div58.6

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

  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 2020039 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))