Average Error: 58.6 → 0.6
Time: 7.1s
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 r111888 = 1.0;
        double r111889 = 2.0;
        double r111890 = r111888 / r111889;
        double r111891 = x;
        double r111892 = r111888 + r111891;
        double r111893 = r111888 - r111891;
        double r111894 = r111892 / r111893;
        double r111895 = log(r111894);
        double r111896 = r111890 * r111895;
        return r111896;
}
↓
double f(double x) {
        double r111897 = 1.0;
        double r111898 = 2.0;
        double r111899 = r111897 / r111898;
        double r111900 = x;
        double r111901 = fma(r111900, r111900, r111900);
        double r111902 = log(r111897);
        double r111903 = 2.0;
        double r111904 = pow(r111900, r111903);
        double r111905 = pow(r111897, r111903);
        double r111906 = r111904 / r111905;
        double r111907 = r111898 * r111906;
        double r111908 = r111902 - r111907;
        double r111909 = fma(r111901, r111898, r111908);
        double r111910 = r111899 * r111909;
        return r111910;
}

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 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))