Average Error: 58.7 → 0.2
Time: 14.1s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(\frac{2}{5} \cdot {x}^{5} + \left(x \cdot 2 + x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{3}\right)\right)\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\left(\frac{2}{5} \cdot {x}^{5} + \left(x \cdot 2 + x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{3}\right)\right)\right) \cdot \frac{1}{2}
double f(double x) {
        double r1693302 = 1.0;
        double r1693303 = 2.0;
        double r1693304 = r1693302 / r1693303;
        double r1693305 = x;
        double r1693306 = r1693302 + r1693305;
        double r1693307 = r1693302 - r1693305;
        double r1693308 = r1693306 / r1693307;
        double r1693309 = log(r1693308);
        double r1693310 = r1693304 * r1693309;
        return r1693310;
}
↓
double f(double x) {
        double r1693311 = 0.4;
        double r1693312 = x;
        double r1693313 = 5.0;
        double r1693314 = pow(r1693312, r1693313);
        double r1693315 = r1693311 * r1693314;
        double r1693316 = 2.0;
        double r1693317 = r1693312 * r1693316;
        double r1693318 = r1693312 * r1693312;
        double r1693319 = 0.6666666666666666;
        double r1693320 = r1693318 * r1693319;
        double r1693321 = r1693312 * r1693320;
        double r1693322 = r1693317 + r1693321;
        double r1693323 = r1693315 + r1693322;
        double r1693324 = 0.5;
        double r1693325 = r1693323 * r1693324;
        return r1693325;
}

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

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

  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. Using strategy rm

  6. Applied distribute-rgt-in0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019156 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))