Average Error: 58.5 → 0.2
Time: 19.6s
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 r2656252 = 1.0;
        double r2656253 = 2.0;
        double r2656254 = r2656252 / r2656253;
        double r2656255 = x;
        double r2656256 = r2656252 + r2656255;
        double r2656257 = r2656252 - r2656255;
        double r2656258 = r2656256 / r2656257;
        double r2656259 = log(r2656258);
        double r2656260 = r2656254 * r2656259;
        return r2656260;
}
↓
double f(double x) {
        double r2656261 = 0.5;
        double r2656262 = x;
        double r2656263 = 5.0;
        double r2656264 = pow(r2656262, r2656263);
        double r2656265 = 0.4;
        double r2656266 = r2656264 * r2656265;
        double r2656267 = 2.0;
        double r2656268 = r2656262 * r2656267;
        double r2656269 = 0.6666666666666666;
        double r2656270 = r2656269 * r2656262;
        double r2656271 = r2656270 * r2656262;
        double r2656272 = r2656262 * r2656271;
        double r2656273 = r2656268 + r2656272;
        double r2656274 = r2656266 + r2656273;
        double r2656275 = r2656261 * r2656274;
        return r2656275;
}

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

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

  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-lft-in0.2

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

  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 2019146 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))