Average Error: 58.5 → 0.2
Time: 19.2s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(x \cdot \left(2 + \frac{2}{3} \cdot \left(x \cdot x\right)\right) + {x}^{5} \cdot \frac{2}{5}\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\left(x \cdot \left(2 + \frac{2}{3} \cdot \left(x \cdot x\right)\right) + {x}^{5} \cdot \frac{2}{5}\right) \cdot \frac{1}{2}
double f(double x) {
        double r1846298 = 1.0;
        double r1846299 = 2.0;
        double r1846300 = r1846298 / r1846299;
        double r1846301 = x;
        double r1846302 = r1846298 + r1846301;
        double r1846303 = r1846298 - r1846301;
        double r1846304 = r1846302 / r1846303;
        double r1846305 = log(r1846304);
        double r1846306 = r1846300 * r1846305;
        return r1846306;
}
↓
double f(double x) {
        double r1846307 = x;
        double r1846308 = 2.0;
        double r1846309 = 0.6666666666666666;
        double r1846310 = r1846307 * r1846307;
        double r1846311 = r1846309 * r1846310;
        double r1846312 = r1846308 + r1846311;
        double r1846313 = r1846307 * r1846312;
        double r1846314 = 5.0;
        double r1846315 = pow(r1846307, r1846314);
        double r1846316 = 0.4;
        double r1846317 = r1846315 * r1846316;
        double r1846318 = r1846313 + r1846317;
        double r1846319 = 0.5;
        double r1846320 = r1846318 * r1846319;
        return r1846320;
}

Error

Bits error versus x

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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

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019129 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))