Average Error: 58.5 → 0.2
Time: 18.0s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(\frac{2}{5} \cdot {x}^{5} + \left(x \cdot \left(x \cdot \left(x \cdot \frac{2}{3}\right)\right) + 2 \cdot x\right)\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\left(\frac{2}{5} \cdot {x}^{5} + \left(x \cdot \left(x \cdot \left(x \cdot \frac{2}{3}\right)\right) + 2 \cdot x\right)\right) \cdot \frac{1}{2}
double f(double x) {
        double r2423788 = 1.0;
        double r2423789 = 2.0;
        double r2423790 = r2423788 / r2423789;
        double r2423791 = x;
        double r2423792 = r2423788 + r2423791;
        double r2423793 = r2423788 - r2423791;
        double r2423794 = r2423792 / r2423793;
        double r2423795 = log(r2423794);
        double r2423796 = r2423790 * r2423795;
        return r2423796;
}
↓
double f(double x) {
        double r2423797 = 0.4;
        double r2423798 = x;
        double r2423799 = 5.0;
        double r2423800 = pow(r2423798, r2423799);
        double r2423801 = r2423797 * r2423800;
        double r2423802 = 0.6666666666666666;
        double r2423803 = r2423798 * r2423802;
        double r2423804 = r2423798 * r2423803;
        double r2423805 = r2423798 * r2423804;
        double r2423806 = 2.0;
        double r2423807 = r2423806 * r2423798;
        double r2423808 = r2423805 + r2423807;
        double r2423809 = r2423801 + r2423808;
        double r2423810 = 0.5;
        double r2423811 = r2423809 * r2423810;
        return r2423811;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.5

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

  2. Simplified58.5

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

  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

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

  6. Applied distribute-rgt-in0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019139 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))