Average Error: 58.6 → 0.2
Time: 5.6s
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 r63028 = 1.0;
        double r63029 = 2.0;
        double r63030 = r63028 / r63029;
        double r63031 = x;
        double r63032 = r63028 + r63031;
        double r63033 = r63028 - r63031;
        double r63034 = r63032 / r63033;
        double r63035 = log(r63034);
        double r63036 = r63030 * r63035;
        return r63036;
}
↓
double f(double x) {
        double r63037 = 1.0;
        double r63038 = 2.0;
        double r63039 = r63037 / r63038;
        double r63040 = 0.6666666666666666;
        double r63041 = x;
        double r63042 = 3.0;
        double r63043 = pow(r63041, r63042);
        double r63044 = pow(r63037, r63042);
        double r63045 = r63043 / r63044;
        double r63046 = 0.4;
        double r63047 = 5.0;
        double r63048 = pow(r63041, r63047);
        double r63049 = pow(r63037, r63047);
        double r63050 = r63048 / r63049;
        double r63051 = r63046 * r63050;
        double r63052 = fma(r63038, r63041, r63051);
        double r63053 = fma(r63040, r63045, r63052);
        double r63054 = r63039 * r63053;
        return r63054;
}

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 0.2

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

  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 2020018 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))