Average Error: 58.7 → 0.2
Time: 43.1s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\mathsf{fma}\left(\frac{2}{5}, \left({x}^{5}\right), \left(x \cdot \mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right), 2\right)\right)\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\mathsf{fma}\left(\frac{2}{5}, \left({x}^{5}\right), \left(x \cdot \mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right), 2\right)\right)\right) \cdot \frac{1}{2}
double f(double x) {
        double r5676425 = 1.0;
        double r5676426 = 2.0;
        double r5676427 = r5676425 / r5676426;
        double r5676428 = x;
        double r5676429 = r5676425 + r5676428;
        double r5676430 = r5676425 - r5676428;
        double r5676431 = r5676429 / r5676430;
        double r5676432 = log(r5676431);
        double r5676433 = r5676427 * r5676432;
        return r5676433;
}
↓
double f(double x) {
        double r5676434 = 0.4;
        double r5676435 = x;
        double r5676436 = 5.0;
        double r5676437 = pow(r5676435, r5676436);
        double r5676438 = 0.6666666666666666;
        double r5676439 = r5676435 * r5676435;
        double r5676440 = 2.0;
        double r5676441 = fma(r5676438, r5676439, r5676440);
        double r5676442 = r5676435 * r5676441;
        double r5676443 = fma(r5676434, r5676437, r5676442);
        double r5676444 = 0.5;
        double r5676445 = r5676443 * r5676444;
        return r5676445;
}

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

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

  3. Taylor expanded around 0 0.2

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))