Average Error: 58.7 → 0.2
Time: 16.1s
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 r40017 = 1.0;
        double r40018 = 2.0;
        double r40019 = r40017 / r40018;
        double r40020 = x;
        double r40021 = r40017 + r40020;
        double r40022 = r40017 - r40020;
        double r40023 = r40021 / r40022;
        double r40024 = log(r40023);
        double r40025 = r40019 * r40024;
        return r40025;
}
↓
double f(double x) {
        double r40026 = 1.0;
        double r40027 = 2.0;
        double r40028 = r40026 / r40027;
        double r40029 = 0.6666666666666666;
        double r40030 = x;
        double r40031 = 3.0;
        double r40032 = pow(r40030, r40031);
        double r40033 = pow(r40026, r40031);
        double r40034 = r40032 / r40033;
        double r40035 = r40029 * r40034;
        double r40036 = r40027 * r40030;
        double r40037 = 0.4;
        double r40038 = 5.0;
        double r40039 = pow(r40030, r40038);
        double r40040 = pow(r40026, r40038);
        double r40041 = r40039 / r40040;
        double r40042 = r40037 * r40041;
        double r40043 = r40036 + r40042;
        double r40044 = r40035 + r40043;
        double r40045 = r40028 * r40044;
        return r40045;
}

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 2019208 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))