Average Error: 58.6 → 0.2
Time: 14.4s
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 r891069 = 1.0;
        double r891070 = 2.0;
        double r891071 = r891069 / r891070;
        double r891072 = x;
        double r891073 = r891069 + r891072;
        double r891074 = r891069 - r891072;
        double r891075 = r891073 / r891074;
        double r891076 = log(r891075);
        double r891077 = r891071 * r891076;
        return r891077;
}
↓
double f(double x) {
        double r891078 = 2.0;
        double r891079 = x;
        double r891080 = 0.4;
        double r891081 = 5.0;
        double r891082 = pow(r891079, r891081);
        double r891083 = r891079 * r891079;
        double r891084 = 0.6666666666666666;
        double r891085 = r891083 * r891084;
        double r891086 = r891079 * r891085;
        double r891087 = fma(r891080, r891082, r891086);
        double r891088 = fma(r891078, r891079, r891087);
        double r891089 = 0.5;
        double r891090 = r891088 * r891089;
        return r891090;
}

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 0.2

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

  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 2019155 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))