Average Error: 58.6 → 0.6
Time: 7.2s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, x\right), 2, \log 1 - 2 \cdot \frac{{x}^{2}}{{1}^{2}}\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, x\right), 2, \log 1 - 2 \cdot \frac{{x}^{2}}{{1}^{2}}\right)
double f(double x) {
        double r86606 = 1.0;
        double r86607 = 2.0;
        double r86608 = r86606 / r86607;
        double r86609 = x;
        double r86610 = r86606 + r86609;
        double r86611 = r86606 - r86609;
        double r86612 = r86610 / r86611;
        double r86613 = log(r86612);
        double r86614 = r86608 * r86613;
        return r86614;
}
↓
double f(double x) {
        double r86615 = 1.0;
        double r86616 = 2.0;
        double r86617 = r86615 / r86616;
        double r86618 = x;
        double r86619 = fma(r86618, r86618, r86618);
        double r86620 = log(r86615);
        double r86621 = 2.0;
        double r86622 = pow(r86618, r86621);
        double r86623 = pow(r86615, r86621);
        double r86624 = r86622 / r86623;
        double r86625 = r86616 * r86624;
        double r86626 = r86620 - r86625;
        double r86627 = fma(r86619, r86616, r86626);
        double r86628 = r86617 * r86627;
        return r86628;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.6

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))