Average Error: 58.4 → 0.2
Time: 26.7s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\mathsf{fma}\left(\frac{2}{5}, \left({x}^{5}\right), \left(2 \cdot x + \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right)\right) \cdot \frac{1}{2}\]
\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(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right)\right) \cdot \frac{1}{2}
double f(double x) {
        double r12469263 = 1.0;
        double r12469264 = 2.0;
        double r12469265 = r12469263 / r12469264;
        double r12469266 = x;
        double r12469267 = r12469263 + r12469266;
        double r12469268 = r12469263 - r12469266;
        double r12469269 = r12469267 / r12469268;
        double r12469270 = log(r12469269);
        double r12469271 = r12469265 * r12469270;
        return r12469271;
}
↓
double f(double x) {
        double r12469272 = 0.4;
        double r12469273 = x;
        double r12469274 = 5.0;
        double r12469275 = pow(r12469273, r12469274);
        double r12469276 = 2.0;
        double r12469277 = r12469276 * r12469273;
        double r12469278 = 0.6666666666666666;
        double r12469279 = r12469273 * r12469273;
        double r12469280 = r12469278 * r12469279;
        double r12469281 = r12469280 * r12469273;
        double r12469282 = r12469277 + r12469281;
        double r12469283 = fma(r12469272, r12469275, r12469282);
        double r12469284 = 0.5;
        double r12469285 = r12469283 * r12469284;
        return r12469285;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.4

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

  2. Simplified58.4

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

  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

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

  6. Applied fma-udef0.2

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

  7. Applied distribute-lft-in0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))