Average Error: 58.5 → 0.7
Time: 6.3s
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 r83333 = 1.0;
        double r83334 = 2.0;
        double r83335 = r83333 / r83334;
        double r83336 = x;
        double r83337 = r83333 + r83336;
        double r83338 = r83333 - r83336;
        double r83339 = r83337 / r83338;
        double r83340 = log(r83339);
        double r83341 = r83335 * r83340;
        return r83341;
}
↓
double f(double x) {
        double r83342 = 1.0;
        double r83343 = 2.0;
        double r83344 = r83342 / r83343;
        double r83345 = x;
        double r83346 = fma(r83345, r83345, r83345);
        double r83347 = log(r83342);
        double r83348 = 2.0;
        double r83349 = pow(r83345, r83348);
        double r83350 = pow(r83342, r83348);
        double r83351 = r83349 / r83350;
        double r83352 = r83343 * r83351;
        double r83353 = r83347 - r83352;
        double r83354 = fma(r83346, r83343, r83353);
        double r83355 = r83344 * r83354;
        return r83355;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.5

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))