Average Error: 58.6 → 0.3
Time: 6.0s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(\frac{2}{3}, \frac{{x}^{3}}{{1}^{3}}, \mathsf{fma}\left(2, x, \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)\]
\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 r84000 = 1.0;
        double r84001 = 2.0;
        double r84002 = r84000 / r84001;
        double r84003 = x;
        double r84004 = r84000 + r84003;
        double r84005 = r84000 - r84003;
        double r84006 = r84004 / r84005;
        double r84007 = log(r84006);
        double r84008 = r84002 * r84007;
        return r84008;
}
↓
double f(double x) {
        double r84009 = 1.0;
        double r84010 = 2.0;
        double r84011 = r84009 / r84010;
        double r84012 = 0.6666666666666666;
        double r84013 = x;
        double r84014 = 3.0;
        double r84015 = pow(r84013, r84014);
        double r84016 = pow(r84009, r84014);
        double r84017 = r84015 / r84016;
        double r84018 = 0.4;
        double r84019 = 5.0;
        double r84020 = pow(r84013, r84019);
        double r84021 = pow(r84009, r84019);
        double r84022 = r84020 / r84021;
        double r84023 = r84018 * r84022;
        double r84024 = fma(r84010, r84013, r84023);
        double r84025 = fma(r84012, r84017, r84024);
        double r84026 = r84011 * r84025;
        return r84026;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.6

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

  3. Applied log-div58.6

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

  4. Taylor expanded around 0 0.3

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019356 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))