Average Error: 58.7 → 0.2
Time: 26.5s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left({x}^{5} \cdot \frac{2}{5} + \left(x \cdot 2 + x \cdot \left(\left(\frac{2}{3} \cdot x\right) \cdot x\right)\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \left({x}^{5} \cdot \frac{2}{5} + \left(x \cdot 2 + x \cdot \left(\left(\frac{2}{3} \cdot x\right) \cdot x\right)\right)\right)
double f(double x) {
        double r2604978 = 1.0;
        double r2604979 = 2.0;
        double r2604980 = r2604978 / r2604979;
        double r2604981 = x;
        double r2604982 = r2604978 + r2604981;
        double r2604983 = r2604978 - r2604981;
        double r2604984 = r2604982 / r2604983;
        double r2604985 = log(r2604984);
        double r2604986 = r2604980 * r2604985;
        return r2604986;
}
↓
double f(double x) {
        double r2604987 = 0.5;
        double r2604988 = x;
        double r2604989 = 5.0;
        double r2604990 = pow(r2604988, r2604989);
        double r2604991 = 0.4;
        double r2604992 = r2604990 * r2604991;
        double r2604993 = 2.0;
        double r2604994 = r2604988 * r2604993;
        double r2604995 = 0.6666666666666666;
        double r2604996 = r2604995 * r2604988;
        double r2604997 = r2604996 * r2604988;
        double r2604998 = r2604988 * r2604997;
        double r2604999 = r2604994 + r2604998;
        double r2605000 = r2604992 + r2604999;
        double r2605001 = r2604987 * r2605000;
        return r2605001;
}

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. Simplified58.7

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

  3. Taylor expanded around 0 0.2

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

  4. Simplified0.2

    \[\leadsto \color{blue}{\left(\frac{2}{5} \cdot {x}^{5} + x \cdot \left(\left(x \cdot \frac{2}{3}\right) \cdot x + 2\right)\right)} \cdot \frac{1}{2}\]
  5. Using strategy rm

  6. Applied distribute-lft-in0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019142 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))