Average Error: 58.7 → 0.2
Time: 5.6s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left(\frac{2}{3} \cdot \frac{{x}^{3}}{{1}^{3}} + \left(2 \cdot x + \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \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)
double f(double x) {
        double r83254 = 1.0;
        double r83255 = 2.0;
        double r83256 = r83254 / r83255;
        double r83257 = x;
        double r83258 = r83254 + r83257;
        double r83259 = r83254 - r83257;
        double r83260 = r83258 / r83259;
        double r83261 = log(r83260);
        double r83262 = r83256 * r83261;
        return r83262;
}
↓
double f(double x) {
        double r83263 = 1.0;
        double r83264 = 2.0;
        double r83265 = r83263 / r83264;
        double r83266 = 0.6666666666666666;
        double r83267 = x;
        double r83268 = 3.0;
        double r83269 = pow(r83267, r83268);
        double r83270 = pow(r83263, r83268);
        double r83271 = r83269 / r83270;
        double r83272 = r83266 * r83271;
        double r83273 = r83264 * r83267;
        double r83274 = 0.4;
        double r83275 = 5.0;
        double r83276 = pow(r83267, r83275);
        double r83277 = pow(r83263, r83275);
        double r83278 = r83276 / r83277;
        double r83279 = r83274 * r83278;
        double r83280 = r83273 + r83279;
        double r83281 = r83272 + r83280;
        double r83282 = r83265 * r83281;
        return r83282;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Final simplification0.2

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

Reproduce

herbie shell --seed 2020003 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))