Average Error: 58.6 → 0.6
Time: 10.1s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(-\frac{{x}^{2}}{{1}^{2}}, 2, \mathsf{fma}\left(2, \mathsf{fma}\left(x, x, x\right), \log 1\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \mathsf{fma}\left(-\frac{{x}^{2}}{{1}^{2}}, 2, \mathsf{fma}\left(2, \mathsf{fma}\left(x, x, x\right), \log 1\right)\right)
double f(double x) {
        double r99698 = 1.0;
        double r99699 = 2.0;
        double r99700 = r99698 / r99699;
        double r99701 = x;
        double r99702 = r99698 + r99701;
        double r99703 = r99698 - r99701;
        double r99704 = r99702 / r99703;
        double r99705 = log(r99704);
        double r99706 = r99700 * r99705;
        return r99706;
}
↓
double f(double x) {
        double r99707 = 1.0;
        double r99708 = 2.0;
        double r99709 = r99707 / r99708;
        double r99710 = x;
        double r99711 = 2.0;
        double r99712 = pow(r99710, r99711);
        double r99713 = pow(r99707, r99711);
        double r99714 = r99712 / r99713;
        double r99715 = -r99714;
        double r99716 = fma(r99710, r99710, r99710);
        double r99717 = log(r99707);
        double r99718 = fma(r99708, r99716, r99717);
        double r99719 = fma(r99715, r99708, r99718);
        double r99720 = r99709 * r99719;
        return r99720;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.6

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

  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

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))