Average Error: 58.6 → 0.2
Time: 16.4s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\mathsf{fma}\left(\frac{2}{5}, {x}^{5}, x \cdot 2 + \left(\left(x \cdot x\right) \cdot \frac{2}{3}\right) \cdot x\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\mathsf{fma}\left(\frac{2}{5}, {x}^{5}, x \cdot 2 + \left(\left(x \cdot x\right) \cdot \frac{2}{3}\right) \cdot x\right) \cdot \frac{1}{2}
double f(double x) {
        double r2258816 = 1.0;
        double r2258817 = 2.0;
        double r2258818 = r2258816 / r2258817;
        double r2258819 = x;
        double r2258820 = r2258816 + r2258819;
        double r2258821 = r2258816 - r2258819;
        double r2258822 = r2258820 / r2258821;
        double r2258823 = log(r2258822);
        double r2258824 = r2258818 * r2258823;
        return r2258824;
}
↓
double f(double x) {
        double r2258825 = 0.4;
        double r2258826 = x;
        double r2258827 = 5.0;
        double r2258828 = pow(r2258826, r2258827);
        double r2258829 = 2.0;
        double r2258830 = r2258826 * r2258829;
        double r2258831 = r2258826 * r2258826;
        double r2258832 = 0.6666666666666666;
        double r2258833 = r2258831 * r2258832;
        double r2258834 = r2258833 * r2258826;
        double r2258835 = r2258830 + r2258834;
        double r2258836 = fma(r2258825, r2258828, r2258835);
        double r2258837 = 0.5;
        double r2258838 = r2258836 * r2258837;
        return r2258838;
}

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

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

  6. Applied distribute-lft-in0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))