Average Error: 58.7 → 0.2
Time: 19.4s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\mathsf{fma}\left(\frac{2}{5}, {x}^{5}, 2 \cdot x + \left(\left(x \cdot \frac{2}{3}\right) \cdot x\right) \cdot x\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\mathsf{fma}\left(\frac{2}{5}, {x}^{5}, 2 \cdot x + \left(\left(x \cdot \frac{2}{3}\right) \cdot x\right) \cdot x\right) \cdot \frac{1}{2}
double f(double x) {
        double r2339415 = 1.0;
        double r2339416 = 2.0;
        double r2339417 = r2339415 / r2339416;
        double r2339418 = x;
        double r2339419 = r2339415 + r2339418;
        double r2339420 = r2339415 - r2339418;
        double r2339421 = r2339419 / r2339420;
        double r2339422 = log(r2339421);
        double r2339423 = r2339417 * r2339422;
        return r2339423;
}
↓
double f(double x) {
        double r2339424 = 0.4;
        double r2339425 = x;
        double r2339426 = 5.0;
        double r2339427 = pow(r2339425, r2339426);
        double r2339428 = 2.0;
        double r2339429 = r2339428 * r2339425;
        double r2339430 = 0.6666666666666666;
        double r2339431 = r2339425 * r2339430;
        double r2339432 = r2339431 * r2339425;
        double r2339433 = r2339432 * r2339425;
        double r2339434 = r2339429 + r2339433;
        double r2339435 = fma(r2339424, r2339427, r2339434);
        double r2339436 = 0.5;
        double r2339437 = r2339435 * r2339436;
        return r2339437;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.7

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

  2. Simplified58.7

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

  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

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

  6. Applied distribute-lft-in0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))