Average Error: 58.6 → 0.6
Time: 10.0s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(2, -\frac{{x}^{2}}{{1}^{2}}, \mathsf{fma}\left(2, \mathsf{fma}\left(x, x, x\right), \log 1\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \mathsf{fma}\left(2, -\frac{{x}^{2}}{{1}^{2}}, \mathsf{fma}\left(2, \mathsf{fma}\left(x, x, x\right), \log 1\right)\right)
double f(double x) {
        double r54437 = 1.0;
        double r54438 = 2.0;
        double r54439 = r54437 / r54438;
        double r54440 = x;
        double r54441 = r54437 + r54440;
        double r54442 = r54437 - r54440;
        double r54443 = r54441 / r54442;
        double r54444 = log(r54443);
        double r54445 = r54439 * r54444;
        return r54445;
}
↓
double f(double x) {
        double r54446 = 1.0;
        double r54447 = 2.0;
        double r54448 = r54446 / r54447;
        double r54449 = x;
        double r54450 = 2.0;
        double r54451 = pow(r54449, r54450);
        double r54452 = pow(r54446, r54450);
        double r54453 = r54451 / r54452;
        double r54454 = -r54453;
        double r54455 = fma(r54449, r54449, r54449);
        double r54456 = log(r54446);
        double r54457 = fma(r54447, r54455, r54456);
        double r54458 = fma(r54447, r54454, r54457);
        double r54459 = r54448 * r54458;
        return r54459;
}

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

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))