Average Error: 58.5 → 0.2
Time: 29.1s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(\frac{2}{5}, \left({x}^{5}\right), \left(2 \cdot x + \log \left(e^{\frac{2}{3} \cdot \left(x \cdot x\right)}\right) \cdot x\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \mathsf{fma}\left(\frac{2}{5}, \left({x}^{5}\right), \left(2 \cdot x + \log \left(e^{\frac{2}{3} \cdot \left(x \cdot x\right)}\right) \cdot x\right)\right)
double f(double x) {
        double r5694652 = 1.0;
        double r5694653 = 2.0;
        double r5694654 = r5694652 / r5694653;
        double r5694655 = x;
        double r5694656 = r5694652 + r5694655;
        double r5694657 = r5694652 - r5694655;
        double r5694658 = r5694656 / r5694657;
        double r5694659 = log(r5694658);
        double r5694660 = r5694654 * r5694659;
        return r5694660;
}
↓
double f(double x) {
        double r5694661 = 0.5;
        double r5694662 = 0.4;
        double r5694663 = x;
        double r5694664 = 5.0;
        double r5694665 = pow(r5694663, r5694664);
        double r5694666 = 2.0;
        double r5694667 = r5694666 * r5694663;
        double r5694668 = 0.6666666666666666;
        double r5694669 = r5694663 * r5694663;
        double r5694670 = r5694668 * r5694669;
        double r5694671 = exp(r5694670);
        double r5694672 = log(r5694671);
        double r5694673 = r5694672 * r5694663;
        double r5694674 = r5694667 + r5694673;
        double r5694675 = fma(r5694662, r5694665, r5694674);
        double r5694676 = r5694661 * r5694675;
        return r5694676;
}

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

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

  9. Applied add-log-exp0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))