Average Error: 58.6 → 0.7
Time: 7.9s
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 r89415 = 1.0;
        double r89416 = 2.0;
        double r89417 = r89415 / r89416;
        double r89418 = x;
        double r89419 = r89415 + r89418;
        double r89420 = r89415 - r89418;
        double r89421 = r89419 / r89420;
        double r89422 = log(r89421);
        double r89423 = r89417 * r89422;
        return r89423;
}
↓
double f(double x) {
        double r89424 = 1.0;
        double r89425 = 2.0;
        double r89426 = r89424 / r89425;
        double r89427 = x;
        double r89428 = fma(r89427, r89427, r89427);
        double r89429 = log(r89424);
        double r89430 = 2.0;
        double r89431 = pow(r89427, r89430);
        double r89432 = pow(r89424, r89430);
        double r89433 = r89431 / r89432;
        double r89434 = r89425 * r89433;
        double r89435 = r89429 - r89434;
        double r89436 = fma(r89428, r89425, r89435);
        double r89437 = r89426 * r89436;
        return r89437;
}

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 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))