Average Error: 58.5 → 0.2
Time: 19.4s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\mathsf{fma}\left(\frac{2}{5}, \left({x}^{5}\right), \left(2 \cdot x + \left(\left(x \cdot \frac{2}{3}\right) \cdot x\right) \cdot x\right)\right) \cdot \frac{1}{2}\]
\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 r2031750 = 1.0;
        double r2031751 = 2.0;
        double r2031752 = r2031750 / r2031751;
        double r2031753 = x;
        double r2031754 = r2031750 + r2031753;
        double r2031755 = r2031750 - r2031753;
        double r2031756 = r2031754 / r2031755;
        double r2031757 = log(r2031756);
        double r2031758 = r2031752 * r2031757;
        return r2031758;
}
↓
double f(double x) {
        double r2031759 = 0.4;
        double r2031760 = x;
        double r2031761 = 5.0;
        double r2031762 = pow(r2031760, r2031761);
        double r2031763 = 2.0;
        double r2031764 = r2031763 * r2031760;
        double r2031765 = 0.6666666666666666;
        double r2031766 = r2031760 * r2031765;
        double r2031767 = r2031766 * r2031760;
        double r2031768 = r2031767 * r2031760;
        double r2031769 = r2031764 + r2031768;
        double r2031770 = fma(r2031759, r2031762, r2031769);
        double r2031771 = 0.5;
        double r2031772 = r2031770 * r2031771;
        return r2031772;
}

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

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

  6. Applied distribute-lft-in0.2

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

  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 2019132 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))