Average Error: 58.4 → 0.2
Time: 10.4s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(\frac{2}{5} \cdot {x}^{5} + x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{3} + 2\right)\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\left(\frac{2}{5} \cdot {x}^{5} + x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{3} + 2\right)\right) \cdot \frac{1}{2}
double f(double x) {
        double r938721 = 1.0;
        double r938722 = 2.0;
        double r938723 = r938721 / r938722;
        double r938724 = x;
        double r938725 = r938721 + r938724;
        double r938726 = r938721 - r938724;
        double r938727 = r938725 / r938726;
        double r938728 = log(r938727);
        double r938729 = r938723 * r938728;
        return r938729;
}
↓
double f(double x) {
        double r938730 = 0.4;
        double r938731 = x;
        double r938732 = 5.0;
        double r938733 = pow(r938731, r938732);
        double r938734 = r938730 * r938733;
        double r938735 = r938731 * r938731;
        double r938736 = 0.6666666666666666;
        double r938737 = r938735 * r938736;
        double r938738 = 2.0;
        double r938739 = r938737 + r938738;
        double r938740 = r938731 * r938739;
        double r938741 = r938734 + r938740;
        double r938742 = 0.5;
        double r938743 = r938741 * r938742;
        return r938743;
}

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. Simplified58.4

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

  3. Taylor expanded around 0 0.2

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019154 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))