Average Error: 58.7 → 0.2
Time: 7.2s
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 r56230 = 1.0;
        double r56231 = 2.0;
        double r56232 = r56230 / r56231;
        double r56233 = x;
        double r56234 = r56230 + r56233;
        double r56235 = r56230 - r56233;
        double r56236 = r56234 / r56235;
        double r56237 = log(r56236);
        double r56238 = r56232 * r56237;
        return r56238;
}
↓
double f(double x) {
        double r56239 = 1.0;
        double r56240 = 2.0;
        double r56241 = r56239 / r56240;
        double r56242 = 0.6666666666666666;
        double r56243 = x;
        double r56244 = 3.0;
        double r56245 = pow(r56243, r56244);
        double r56246 = pow(r56239, r56244);
        double r56247 = r56245 / r56246;
        double r56248 = r56242 * r56247;
        double r56249 = r56240 * r56243;
        double r56250 = 0.4;
        double r56251 = 5.0;
        double r56252 = pow(r56243, r56251);
        double r56253 = pow(r56239, r56251);
        double r56254 = r56252 / r56253;
        double r56255 = r56250 * r56254;
        double r56256 = r56249 + r56255;
        double r56257 = r56248 + r56256;
        double r56258 = r56241 * r56257;
        return r56258;
}

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 2020035 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))