Average Error: 58.4 → 0.7
Time: 6.8s
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 r71252 = 1.0;
        double r71253 = 2.0;
        double r71254 = r71252 / r71253;
        double r71255 = x;
        double r71256 = r71252 + r71255;
        double r71257 = r71252 - r71255;
        double r71258 = r71256 / r71257;
        double r71259 = log(r71258);
        double r71260 = r71254 * r71259;
        return r71260;
}
↓
double f(double x) {
        double r71261 = 1.0;
        double r71262 = 2.0;
        double r71263 = r71261 / r71262;
        double r71264 = x;
        double r71265 = fma(r71264, r71264, r71264);
        double r71266 = log(r71261);
        double r71267 = 2.0;
        double r71268 = pow(r71264, r71267);
        double r71269 = pow(r71261, r71267);
        double r71270 = r71268 / r71269;
        double r71271 = r71262 * r71270;
        double r71272 = r71266 - r71271;
        double r71273 = fma(r71265, r71262, r71272);
        double r71274 = r71263 * r71273;
        return r71274;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.4

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

  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 2020057 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))