Average Error: 58.5 → 0.3
Time: 14.3s
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 r48105 = 1.0;
        double r48106 = 2.0;
        double r48107 = r48105 / r48106;
        double r48108 = x;
        double r48109 = r48105 + r48108;
        double r48110 = r48105 - r48108;
        double r48111 = r48109 / r48110;
        double r48112 = log(r48111);
        double r48113 = r48107 * r48112;
        return r48113;
}
↓
double f(double x) {
        double r48114 = 1.0;
        double r48115 = 2.0;
        double r48116 = r48114 / r48115;
        double r48117 = 0.6666666666666666;
        double r48118 = x;
        double r48119 = r48118 / r48114;
        double r48120 = 3.0;
        double r48121 = pow(r48119, r48120);
        double r48122 = 0.4;
        double r48123 = 5.0;
        double r48124 = pow(r48118, r48123);
        double r48125 = pow(r48114, r48123);
        double r48126 = r48124 / r48125;
        double r48127 = r48122 * r48126;
        double r48128 = fma(r48115, r48118, r48127);
        double r48129 = fma(r48117, r48121, r48128);
        double r48130 = r48116 * r48129;
        return r48130;
}

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)))))