Average Error: 58.6 → 0.6
Time: 14.9s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{\mathsf{fma}\left(2, \mathsf{fma}\left(x, x, x\right) - \frac{x}{1} \cdot \frac{x}{1}, \log 1\right)}{\frac{2}{1}}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{\mathsf{fma}\left(2, \mathsf{fma}\left(x, x, x\right) - \frac{x}{1} \cdot \frac{x}{1}, \log 1\right)}{\frac{2}{1}}
double f(double x) {
        double r78643 = 1.0;
        double r78644 = 2.0;
        double r78645 = r78643 / r78644;
        double r78646 = x;
        double r78647 = r78643 + r78646;
        double r78648 = r78643 - r78646;
        double r78649 = r78647 / r78648;
        double r78650 = log(r78649);
        double r78651 = r78645 * r78650;
        return r78651;
}
↓
double f(double x) {
        double r78652 = 2.0;
        double r78653 = x;
        double r78654 = fma(r78653, r78653, r78653);
        double r78655 = 1.0;
        double r78656 = r78653 / r78655;
        double r78657 = r78656 * r78656;
        double r78658 = r78654 - r78657;
        double r78659 = log(r78655);
        double r78660 = fma(r78652, r78658, r78659);
        double r78661 = r78652 / r78655;
        double r78662 = r78660 / r78661;
        return r78662;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.6

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

  2. Simplified58.6

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

  3. Taylor expanded around 0 0.6

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

  4. Simplified0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))