Average Error: 58.7 → 0.6
Time: 7.9s
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 r89358 = 1.0;
        double r89359 = 2.0;
        double r89360 = r89358 / r89359;
        double r89361 = x;
        double r89362 = r89358 + r89361;
        double r89363 = r89358 - r89361;
        double r89364 = r89362 / r89363;
        double r89365 = log(r89364);
        double r89366 = r89360 * r89365;
        return r89366;
}
↓
double f(double x) {
        double r89367 = 1.0;
        double r89368 = 2.0;
        double r89369 = r89367 / r89368;
        double r89370 = x;
        double r89371 = fma(r89370, r89370, r89370);
        double r89372 = log(r89367);
        double r89373 = 2.0;
        double r89374 = pow(r89370, r89373);
        double r89375 = pow(r89367, r89373);
        double r89376 = r89374 / r89375;
        double r89377 = r89368 * r89376;
        double r89378 = r89372 - r89377;
        double r89379 = fma(r89371, r89368, r89378);
        double r89380 = r89369 * r89379;
        return r89380;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.7

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

  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 2020083 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))