Average Error: 58.7 → 0.6
Time: 18.9s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left(2 \cdot \left(\mathsf{fma}\left(x, x, x\right) - \frac{x}{1} \cdot \frac{x}{1}\right) + \log 1\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \left(2 \cdot \left(\mathsf{fma}\left(x, x, x\right) - \frac{x}{1} \cdot \frac{x}{1}\right) + \log 1\right)
double f(double x) {
        double r2597252 = 1.0;
        double r2597253 = 2.0;
        double r2597254 = r2597252 / r2597253;
        double r2597255 = x;
        double r2597256 = r2597252 + r2597255;
        double r2597257 = r2597252 - r2597255;
        double r2597258 = r2597256 / r2597257;
        double r2597259 = log(r2597258);
        double r2597260 = r2597254 * r2597259;
        return r2597260;
}
↓
double f(double x) {
        double r2597261 = 1.0;
        double r2597262 = 2.0;
        double r2597263 = r2597261 / r2597262;
        double r2597264 = x;
        double r2597265 = fma(r2597264, r2597264, r2597264);
        double r2597266 = r2597264 / r2597261;
        double r2597267 = r2597266 * r2597266;
        double r2597268 = r2597265 - r2597267;
        double r2597269 = r2597262 * r2597268;
        double r2597270 = log(r2597261);
        double r2597271 = r2597269 + r2597270;
        double r2597272 = r2597263 * r2597271;
        return r2597272;
}

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

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))