Average Error: 58.6 → 0.6
Time: 8.5s
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 r111594 = 1.0;
        double r111595 = 2.0;
        double r111596 = r111594 / r111595;
        double r111597 = x;
        double r111598 = r111594 + r111597;
        double r111599 = r111594 - r111597;
        double r111600 = r111598 / r111599;
        double r111601 = log(r111600);
        double r111602 = r111596 * r111601;
        return r111602;
}
↓
double f(double x) {
        double r111603 = 1.0;
        double r111604 = 2.0;
        double r111605 = r111603 / r111604;
        double r111606 = x;
        double r111607 = fma(r111606, r111606, r111606);
        double r111608 = log(r111603);
        double r111609 = 2.0;
        double r111610 = pow(r111606, r111609);
        double r111611 = pow(r111603, r111609);
        double r111612 = r111610 / r111611;
        double r111613 = r111604 * r111612;
        double r111614 = r111608 - r111613;
        double r111615 = fma(r111607, r111604, r111614);
        double r111616 = r111605 * r111615;
        return r111616;
}

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

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

  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 2020024 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))