Average Error: 58.5 → 0.2
Time: 19.0s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\mathsf{fma}\left(\frac{2}{5}, \left({x}^{5}\right), \left(2 \cdot x + \left(\left(x \cdot \frac{2}{3}\right) \cdot x\right) \cdot x\right)\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\mathsf{fma}\left(\frac{2}{5}, \left({x}^{5}\right), \left(2 \cdot x + \left(\left(x \cdot \frac{2}{3}\right) \cdot x\right) \cdot x\right)\right) \cdot \frac{1}{2}
double f(double x) {
        double r2942289 = 1.0;
        double r2942290 = 2.0;
        double r2942291 = r2942289 / r2942290;
        double r2942292 = x;
        double r2942293 = r2942289 + r2942292;
        double r2942294 = r2942289 - r2942292;
        double r2942295 = r2942293 / r2942294;
        double r2942296 = log(r2942295);
        double r2942297 = r2942291 * r2942296;
        return r2942297;
}
↓
double f(double x) {
        double r2942298 = 0.4;
        double r2942299 = x;
        double r2942300 = 5.0;
        double r2942301 = pow(r2942299, r2942300);
        double r2942302 = 2.0;
        double r2942303 = r2942302 * r2942299;
        double r2942304 = 0.6666666666666666;
        double r2942305 = r2942299 * r2942304;
        double r2942306 = r2942305 * r2942299;
        double r2942307 = r2942306 * r2942299;
        double r2942308 = r2942303 + r2942307;
        double r2942309 = fma(r2942298, r2942301, r2942308);
        double r2942310 = 0.5;
        double r2942311 = r2942309 * r2942310;
        return r2942311;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.5

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

  2. Simplified58.5

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

  3. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{\left(2 \cdot x + \left(\frac{2}{3} \cdot {x}^{3} + \frac{2}{5} \cdot {x}^{5}\right)\right)} \cdot \frac{1}{2}\]

  4. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{5}, \left({x}^{5}\right), \left(x \cdot \left(\left(\frac{2}{3} \cdot x\right) \cdot x + 2\right)\right)\right)} \cdot \frac{1}{2}\]
  5. Using strategy rm

  6. Applied distribute-rgt-in0.2

    \[\leadsto \mathsf{fma}\left(\frac{2}{5}, \left({x}^{5}\right), \color{blue}{\left(\left(\left(\frac{2}{3} \cdot x\right) \cdot x\right) \cdot x + 2 \cdot x\right)}\right) \cdot \frac{1}{2}\]

  7. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(\frac{2}{5}, \left({x}^{5}\right), \left(2 \cdot x + \left(\left(x \cdot \frac{2}{3}\right) \cdot x\right) \cdot x\right)\right) \cdot \frac{1}{2}\]

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))