Average Error: 58.6 → 0.2
Time: 20.3s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[(\frac{2}{5} \cdot \left({x}^{5}\right) + \left(2 \cdot x + \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right))_* \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
(\frac{2}{5} \cdot \left({x}^{5}\right) + \left(2 \cdot x + \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right))_* \cdot \frac{1}{2}
double f(double x) {
        double r12085615 = 1.0;
        double r12085616 = 2.0;
        double r12085617 = r12085615 / r12085616;
        double r12085618 = x;
        double r12085619 = r12085615 + r12085618;
        double r12085620 = r12085615 - r12085618;
        double r12085621 = r12085619 / r12085620;
        double r12085622 = log(r12085621);
        double r12085623 = r12085617 * r12085622;
        return r12085623;
}
↓
double f(double x) {
        double r12085624 = 0.4;
        double r12085625 = x;
        double r12085626 = 5.0;
        double r12085627 = pow(r12085625, r12085626);
        double r12085628 = 2.0;
        double r12085629 = r12085628 * r12085625;
        double r12085630 = 0.6666666666666666;
        double r12085631 = r12085625 * r12085625;
        double r12085632 = r12085630 * r12085631;
        double r12085633 = r12085632 * r12085625;
        double r12085634 = r12085629 + r12085633;
        double r12085635 = fma(r12085624, r12085627, r12085634);
        double r12085636 = 0.5;
        double r12085637 = r12085635 * r12085636;
        return r12085637;
}

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

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

  6. Applied fma-udef0.2

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

  7. Applied distribute-rgt-in0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))