Average Error: 58.4 → 0.3
Time: 18.9s
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 r2385919 = 1.0;
        double r2385920 = 2.0;
        double r2385921 = r2385919 / r2385920;
        double r2385922 = x;
        double r2385923 = r2385919 + r2385922;
        double r2385924 = r2385919 - r2385922;
        double r2385925 = r2385923 / r2385924;
        double r2385926 = log(r2385925);
        double r2385927 = r2385921 * r2385926;
        return r2385927;
}
↓
double f(double x) {
        double r2385928 = 0.5;
        double r2385929 = x;
        double r2385930 = 5.0;
        double r2385931 = pow(r2385929, r2385930);
        double r2385932 = 0.4;
        double r2385933 = 0.6666666666666666;
        double r2385934 = r2385933 * r2385929;
        double r2385935 = 2.0;
        double r2385936 = fma(r2385934, r2385929, r2385935);
        double r2385937 = r2385936 * r2385929;
        double r2385938 = fma(r2385931, r2385932, r2385937);
        double r2385939 = r2385928 * r2385938;
        return r2385939;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.4

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

  2. Simplified58.4

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

  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 2019163 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))