Average Error: 58.6 → 0.2
Time: 27.7s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left({x}^{5} \cdot \frac{2}{5} + \left(2 \cdot x + \left(x \cdot \left(x \cdot \frac{2}{3}\right)\right) \cdot x\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \left({x}^{5} \cdot \frac{2}{5} + \left(2 \cdot x + \left(x \cdot \left(x \cdot \frac{2}{3}\right)\right) \cdot x\right)\right)
double f(double x) {
        double r9212734 = 1.0;
        double r9212735 = 2.0;
        double r9212736 = r9212734 / r9212735;
        double r9212737 = x;
        double r9212738 = r9212734 + r9212737;
        double r9212739 = r9212734 - r9212737;
        double r9212740 = r9212738 / r9212739;
        double r9212741 = log(r9212740);
        double r9212742 = r9212736 * r9212741;
        return r9212742;
}
↓
double f(double x) {
        double r9212743 = 0.5;
        double r9212744 = x;
        double r9212745 = 5.0;
        double r9212746 = pow(r9212744, r9212745);
        double r9212747 = 0.4;
        double r9212748 = r9212746 * r9212747;
        double r9212749 = 2.0;
        double r9212750 = r9212749 * r9212744;
        double r9212751 = 0.6666666666666666;
        double r9212752 = r9212744 * r9212751;
        double r9212753 = r9212744 * r9212752;
        double r9212754 = r9212753 * r9212744;
        double r9212755 = r9212750 + r9212754;
        double r9212756 = r9212748 + r9212755;
        double r9212757 = r9212743 * r9212756;
        return r9212757;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.6

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

  2. Simplified58.6

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

  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

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

  6. Applied distribute-rgt-in0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019124 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))