Average Error: 58.7 → 0.2
Time: 6.8s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(\frac{2}{3}, \frac{{x}^{3}}{{1}^{3}}, \mathsf{fma}\left(2, x, \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \mathsf{fma}\left(\frac{2}{3}, \frac{{x}^{3}}{{1}^{3}}, \mathsf{fma}\left(2, x, \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)
double f(double x) {
        double r51297 = 1.0;
        double r51298 = 2.0;
        double r51299 = r51297 / r51298;
        double r51300 = x;
        double r51301 = r51297 + r51300;
        double r51302 = r51297 - r51300;
        double r51303 = r51301 / r51302;
        double r51304 = log(r51303);
        double r51305 = r51299 * r51304;
        return r51305;
}
↓
double f(double x) {
        double r51306 = 1.0;
        double r51307 = 2.0;
        double r51308 = r51306 / r51307;
        double r51309 = 0.6666666666666666;
        double r51310 = x;
        double r51311 = 3.0;
        double r51312 = pow(r51310, r51311);
        double r51313 = pow(r51306, r51311);
        double r51314 = r51312 / r51313;
        double r51315 = 0.4;
        double r51316 = 5.0;
        double r51317 = pow(r51310, r51316);
        double r51318 = pow(r51306, r51316);
        double r51319 = r51317 / r51318;
        double r51320 = r51315 * r51319;
        double r51321 = fma(r51307, r51310, r51320);
        double r51322 = fma(r51309, r51314, r51321);
        double r51323 = r51308 * r51322;
        return r51323;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.7

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

  3. Applied log-div58.7

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

  4. Taylor expanded around 0 0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))