Average Error: 58.7 → 0.6
Time: 6.6s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, x\right), 2, \log 1 - 2 \cdot \frac{{x}^{2}}{{1}^{2}}\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, x\right), 2, \log 1 - 2 \cdot \frac{{x}^{2}}{{1}^{2}}\right)
double f(double x) {
        double r61719 = 1.0;
        double r61720 = 2.0;
        double r61721 = r61719 / r61720;
        double r61722 = x;
        double r61723 = r61719 + r61722;
        double r61724 = r61719 - r61722;
        double r61725 = r61723 / r61724;
        double r61726 = log(r61725);
        double r61727 = r61721 * r61726;
        return r61727;
}
↓
double f(double x) {
        double r61728 = 1.0;
        double r61729 = 2.0;
        double r61730 = r61728 / r61729;
        double r61731 = x;
        double r61732 = fma(r61731, r61731, r61731);
        double r61733 = log(r61728);
        double r61734 = 2.0;
        double r61735 = pow(r61731, r61734);
        double r61736 = pow(r61728, r61734);
        double r61737 = r61735 / r61736;
        double r61738 = r61729 * r61737;
        double r61739 = r61733 - r61738;
        double r61740 = fma(r61732, r61729, r61739);
        double r61741 = r61730 * r61740;
        return r61741;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.7

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))