Average Error: 58.6 → 0.2
Time: 15.5s
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 r2754480 = 1.0;
        double r2754481 = 2.0;
        double r2754482 = r2754480 / r2754481;
        double r2754483 = x;
        double r2754484 = r2754480 + r2754483;
        double r2754485 = r2754480 - r2754483;
        double r2754486 = r2754484 / r2754485;
        double r2754487 = log(r2754486);
        double r2754488 = r2754482 * r2754487;
        return r2754488;
}
↓
double f(double x) {
        double r2754489 = 0.4;
        double r2754490 = x;
        double r2754491 = 5.0;
        double r2754492 = pow(r2754490, r2754491);
        double r2754493 = 2.0;
        double r2754494 = r2754493 * r2754490;
        double r2754495 = 0.6666666666666666;
        double r2754496 = r2754490 * r2754495;
        double r2754497 = r2754496 * r2754490;
        double r2754498 = r2754497 * r2754490;
        double r2754499 = r2754494 + r2754498;
        double r2754500 = fma(r2754489, r2754492, r2754499);
        double r2754501 = 0.5;
        double r2754502 = r2754500 * r2754501;
        return r2754502;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.6

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

  2. Simplified58.6

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

  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 2019133 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))