Average Error: 58.8 → 0.6
Time: 5.4s
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 r80746 = 1.0;
        double r80747 = 2.0;
        double r80748 = r80746 / r80747;
        double r80749 = x;
        double r80750 = r80746 + r80749;
        double r80751 = r80746 - r80749;
        double r80752 = r80750 / r80751;
        double r80753 = log(r80752);
        double r80754 = r80748 * r80753;
        return r80754;
}
↓
double f(double x) {
        double r80755 = 1.0;
        double r80756 = 2.0;
        double r80757 = r80755 / r80756;
        double r80758 = x;
        double r80759 = fma(r80758, r80758, r80758);
        double r80760 = log(r80755);
        double r80761 = 2.0;
        double r80762 = pow(r80758, r80761);
        double r80763 = pow(r80755, r80761);
        double r80764 = r80762 / r80763;
        double r80765 = r80756 * r80764;
        double r80766 = r80760 - r80765;
        double r80767 = fma(r80759, r80756, r80766);
        double r80768 = r80757 * r80767;
        return r80768;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.8

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

  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 2020001 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))