Average Error: 58.7 → 0.2
Time: 13.8s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\mathsf{fma}\left(2, x, \mathsf{fma}\left(\frac{2}{5}, {x}^{5}, x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{3}\right)\right)\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\mathsf{fma}\left(2, x, \mathsf{fma}\left(\frac{2}{5}, {x}^{5}, x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{3}\right)\right)\right) \cdot \frac{1}{2}
double f(double x) {
        double r1569887 = 1.0;
        double r1569888 = 2.0;
        double r1569889 = r1569887 / r1569888;
        double r1569890 = x;
        double r1569891 = r1569887 + r1569890;
        double r1569892 = r1569887 - r1569890;
        double r1569893 = r1569891 / r1569892;
        double r1569894 = log(r1569893);
        double r1569895 = r1569889 * r1569894;
        return r1569895;
}
↓
double f(double x) {
        double r1569896 = 2.0;
        double r1569897 = x;
        double r1569898 = 0.4;
        double r1569899 = 5.0;
        double r1569900 = pow(r1569897, r1569899);
        double r1569901 = r1569897 * r1569897;
        double r1569902 = 0.6666666666666666;
        double r1569903 = r1569901 * r1569902;
        double r1569904 = r1569897 * r1569903;
        double r1569905 = fma(r1569898, r1569900, r1569904);
        double r1569906 = fma(r1569896, r1569897, r1569905);
        double r1569907 = 0.5;
        double r1569908 = r1569906 * r1569907;
        return r1569908;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.7

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

  2. Simplified58.7

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

  3. Taylor expanded around 0 0.2

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))