Average Error: 58.4 → 0.7
Time: 15.8s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(2, \mathsf{fma}\left(x, x, x\right) - \frac{x}{1} \cdot \frac{x}{1}, \log 1\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \mathsf{fma}\left(2, \mathsf{fma}\left(x, x, x\right) - \frac{x}{1} \cdot \frac{x}{1}, \log 1\right)
double f(double x) {
        double r4931763 = 1.0;
        double r4931764 = 2.0;
        double r4931765 = r4931763 / r4931764;
        double r4931766 = x;
        double r4931767 = r4931763 + r4931766;
        double r4931768 = r4931763 - r4931766;
        double r4931769 = r4931767 / r4931768;
        double r4931770 = log(r4931769);
        double r4931771 = r4931765 * r4931770;
        return r4931771;
}
↓
double f(double x) {
        double r4931772 = 1.0;
        double r4931773 = 2.0;
        double r4931774 = r4931772 / r4931773;
        double r4931775 = x;
        double r4931776 = fma(r4931775, r4931775, r4931775);
        double r4931777 = r4931775 / r4931772;
        double r4931778 = r4931777 * r4931777;
        double r4931779 = r4931776 - r4931778;
        double r4931780 = log(r4931772);
        double r4931781 = fma(r4931773, r4931779, r4931780);
        double r4931782 = r4931774 * r4931781;
        return r4931782;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.4

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

    \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(2, \mathsf{fma}\left(x, x, x\right) - \frac{x}{1} \cdot \frac{x}{1}, \log 1\right)\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))