Average Error: 58.7 → 0.2
Time: 13.7s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left(2 \cdot x + \left(x \cdot \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) + {x}^{5} \cdot \frac{2}{5}\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \left(2 \cdot x + \left(x \cdot \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) + {x}^{5} \cdot \frac{2}{5}\right)\right)
double f(double x) {
        double r991783 = 1.0;
        double r991784 = 2.0;
        double r991785 = r991783 / r991784;
        double r991786 = x;
        double r991787 = r991783 + r991786;
        double r991788 = r991783 - r991786;
        double r991789 = r991787 / r991788;
        double r991790 = log(r991789);
        double r991791 = r991785 * r991790;
        return r991791;
}
↓
double f(double x) {
        double r991792 = 0.5;
        double r991793 = 2.0;
        double r991794 = x;
        double r991795 = r991793 * r991794;
        double r991796 = 0.6666666666666666;
        double r991797 = r991794 * r991794;
        double r991798 = r991796 * r991797;
        double r991799 = r991794 * r991798;
        double r991800 = 5.0;
        double r991801 = pow(r991794, r991800);
        double r991802 = 0.4;
        double r991803 = r991801 * r991802;
        double r991804 = r991799 + r991803;
        double r991805 = r991795 + r991804;
        double r991806 = r991792 * r991805;
        return r991806;
}

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

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019128 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))