Average Error: 58.5 → 0.3
Time: 16.0s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(\frac{2}{3}, {\left(\frac{x}{1}\right)}^{3}, \mathsf{fma}\left(2, x, \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \mathsf{fma}\left(\frac{2}{3}, {\left(\frac{x}{1}\right)}^{3}, \mathsf{fma}\left(2, x, \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)
double f(double x) {
        double r42336 = 1.0;
        double r42337 = 2.0;
        double r42338 = r42336 / r42337;
        double r42339 = x;
        double r42340 = r42336 + r42339;
        double r42341 = r42336 - r42339;
        double r42342 = r42340 / r42341;
        double r42343 = log(r42342);
        double r42344 = r42338 * r42343;
        return r42344;
}
↓
double f(double x) {
        double r42345 = 1.0;
        double r42346 = 2.0;
        double r42347 = r42345 / r42346;
        double r42348 = 0.6666666666666666;
        double r42349 = x;
        double r42350 = r42349 / r42345;
        double r42351 = 3.0;
        double r42352 = pow(r42350, r42351);
        double r42353 = 0.4;
        double r42354 = 5.0;
        double r42355 = pow(r42349, r42354);
        double r42356 = pow(r42345, r42354);
        double r42357 = r42355 / r42356;
        double r42358 = r42353 * r42357;
        double r42359 = fma(r42346, r42349, r42358);
        double r42360 = fma(r42348, r42352, r42359);
        double r42361 = r42347 * r42360;
        return r42361;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.5

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

  3. Applied div-inv58.5

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

  4. Applied log-prod58.5

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

  5. Simplified58.5

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

  6. Taylor expanded around 0 0.3

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{2}{3} \cdot \frac{{x}^{3}}{{1}^{3}} + \left(2 \cdot x + \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)}\]

  7. Simplified0.3

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{2}{3}, {\left(\frac{x}{1}\right)}^{3}, \mathsf{fma}\left(2, x, \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)}\]

  8. Final simplification0.3

    \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(\frac{2}{3}, {\left(\frac{x}{1}\right)}^{3}, \mathsf{fma}\left(2, x, \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))