Average Error: 58.4 → 0.3
Time: 18.7s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), {x}^{5} \cdot \frac{2}{5}\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), {x}^{5} \cdot \frac{2}{5}\right)
double f(double x) {
        double r2549675 = 1.0;
        double r2549676 = 2.0;
        double r2549677 = r2549675 / r2549676;
        double r2549678 = x;
        double r2549679 = r2549675 + r2549678;
        double r2549680 = r2549675 - r2549678;
        double r2549681 = r2549679 / r2549680;
        double r2549682 = log(r2549681);
        double r2549683 = r2549677 * r2549682;
        return r2549683;
}
↓
double f(double x) {
        double r2549684 = 0.5;
        double r2549685 = x;
        double r2549686 = r2549685 * r2549685;
        double r2549687 = 0.6666666666666666;
        double r2549688 = 2.0;
        double r2549689 = fma(r2549686, r2549687, r2549688);
        double r2549690 = 5.0;
        double r2549691 = pow(r2549685, r2549690);
        double r2549692 = 0.4;
        double r2549693 = r2549691 * r2549692;
        double r2549694 = fma(r2549685, r2549689, r2549693);
        double r2549695 = r2549684 * r2549694;
        return r2549695;
}

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

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

  5. Final simplification0.3

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

Reproduce

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