Average Error: 58.5 → 0.7
Time: 6.6s
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 r78460 = 1.0;
        double r78461 = 2.0;
        double r78462 = r78460 / r78461;
        double r78463 = x;
        double r78464 = r78460 + r78463;
        double r78465 = r78460 - r78463;
        double r78466 = r78464 / r78465;
        double r78467 = log(r78466);
        double r78468 = r78462 * r78467;
        return r78468;
}
↓
double f(double x) {
        double r78469 = 1.0;
        double r78470 = 2.0;
        double r78471 = r78469 / r78470;
        double r78472 = x;
        double r78473 = fma(r78472, r78472, r78472);
        double r78474 = log(r78469);
        double r78475 = 2.0;
        double r78476 = pow(r78472, r78475);
        double r78477 = pow(r78469, r78475);
        double r78478 = r78476 / r78477;
        double r78479 = r78470 * r78478;
        double r78480 = r78474 - r78479;
        double r78481 = fma(r78473, r78470, r78480);
        double r78482 = r78471 * r78481;
        return r78482;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.5

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

  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 2020021 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))