Average Error: 58.4 → 0.3
Time: 7.5s
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 r82989 = 1.0;
        double r82990 = 2.0;
        double r82991 = r82989 / r82990;
        double r82992 = x;
        double r82993 = r82989 + r82992;
        double r82994 = r82989 - r82992;
        double r82995 = r82993 / r82994;
        double r82996 = log(r82995);
        double r82997 = r82991 * r82996;
        return r82997;
}
↓
double f(double x) {
        double r82998 = 1.0;
        double r82999 = 2.0;
        double r83000 = r82998 / r82999;
        double r83001 = 0.6666666666666666;
        double r83002 = x;
        double r83003 = 3.0;
        double r83004 = pow(r83002, r83003);
        double r83005 = pow(r82998, r83003);
        double r83006 = r83004 / r83005;
        double r83007 = r83001 * r83006;
        double r83008 = r82999 * r83002;
        double r83009 = 0.4;
        double r83010 = 5.0;
        double r83011 = pow(r83002, r83010);
        double r83012 = pow(r82998, r83010);
        double r83013 = r83011 / r83012;
        double r83014 = r83009 * r83013;
        double r83015 = r83008 + r83014;
        double r83016 = r83007 + r83015;
        double r83017 = r83000 * r83016;
        return r83017;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.4

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

  3. Applied log-div58.4

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

  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. Final simplification0.3

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

Reproduce

herbie shell --seed 2020034 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))