Average Error: 58.6 → 0.6
Time: 16.6s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(\frac{{x}^{2}}{{1}^{2}}, -2, \mathsf{fma}\left(2, \mathsf{fma}\left(x, x, x\right), \log 1\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \mathsf{fma}\left(\frac{{x}^{2}}{{1}^{2}}, -2, \mathsf{fma}\left(2, \mathsf{fma}\left(x, x, x\right), \log 1\right)\right)
double f(double x) {
        double r81374 = 1.0;
        double r81375 = 2.0;
        double r81376 = r81374 / r81375;
        double r81377 = x;
        double r81378 = r81374 + r81377;
        double r81379 = r81374 - r81377;
        double r81380 = r81378 / r81379;
        double r81381 = log(r81380);
        double r81382 = r81376 * r81381;
        return r81382;
}
↓
double f(double x) {
        double r81383 = 1.0;
        double r81384 = 2.0;
        double r81385 = r81383 / r81384;
        double r81386 = x;
        double r81387 = 2.0;
        double r81388 = pow(r81386, r81387);
        double r81389 = pow(r81383, r81387);
        double r81390 = r81388 / r81389;
        double r81391 = -r81384;
        double r81392 = fma(r81386, r81386, r81386);
        double r81393 = log(r81383);
        double r81394 = fma(r81384, r81392, r81393);
        double r81395 = fma(r81390, r81391, r81394);
        double r81396 = r81385 * r81395;
        return r81396;
}

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

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))