Average Error: 58.6 → 0.3
Time: 15.4s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left({x}^{5}, \frac{2}{5}, \mathsf{fma}\left(\frac{2}{3} \cdot x, x, 2\right) \cdot x\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \mathsf{fma}\left({x}^{5}, \frac{2}{5}, \mathsf{fma}\left(\frac{2}{3} \cdot x, x, 2\right) \cdot x\right)
double f(double x) {
        double r3225887 = 1.0;
        double r3225888 = 2.0;
        double r3225889 = r3225887 / r3225888;
        double r3225890 = x;
        double r3225891 = r3225887 + r3225890;
        double r3225892 = r3225887 - r3225890;
        double r3225893 = r3225891 / r3225892;
        double r3225894 = log(r3225893);
        double r3225895 = r3225889 * r3225894;
        return r3225895;
}
↓
double f(double x) {
        double r3225896 = 0.5;
        double r3225897 = x;
        double r3225898 = 5.0;
        double r3225899 = pow(r3225897, r3225898);
        double r3225900 = 0.4;
        double r3225901 = 0.6666666666666666;
        double r3225902 = r3225901 * r3225897;
        double r3225903 = 2.0;
        double r3225904 = fma(r3225902, r3225897, r3225903);
        double r3225905 = r3225904 * r3225897;
        double r3225906 = fma(r3225899, r3225900, r3225905);
        double r3225907 = r3225896 * r3225906;
        return r3225907;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.6

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

  2. Simplified58.6

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

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))