Average Error: 58.7 → 0.2
Time: 19.7s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left({x}^{5} \cdot \frac{2}{5} + \left(x \cdot 2 + x \cdot \left(\left(\frac{2}{3} \cdot x\right) \cdot x\right)\right)\right)\]
\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 r2135915 = 1.0;
        double r2135916 = 2.0;
        double r2135917 = r2135915 / r2135916;
        double r2135918 = x;
        double r2135919 = r2135915 + r2135918;
        double r2135920 = r2135915 - r2135918;
        double r2135921 = r2135919 / r2135920;
        double r2135922 = log(r2135921);
        double r2135923 = r2135917 * r2135922;
        return r2135923;
}
↓
double f(double x) {
        double r2135924 = 0.5;
        double r2135925 = x;
        double r2135926 = 5.0;
        double r2135927 = pow(r2135925, r2135926);
        double r2135928 = 0.4;
        double r2135929 = r2135927 * r2135928;
        double r2135930 = 2.0;
        double r2135931 = r2135925 * r2135930;
        double r2135932 = 0.6666666666666666;
        double r2135933 = r2135932 * r2135925;
        double r2135934 = r2135933 * r2135925;
        double r2135935 = r2135925 * r2135934;
        double r2135936 = r2135931 + r2135935;
        double r2135937 = r2135929 + r2135936;
        double r2135938 = r2135924 * r2135937;
        return r2135938;
}

Error

Bits error versus x

Try it out

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-rgt-in0.2

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

  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 2019144 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))