Average Error: 58.7 → 0.2
Time: 16.1s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(\frac{2}{5} \cdot {x}^{5} + x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{3} + 2\right)\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\left(\frac{2}{5} \cdot {x}^{5} + x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{3} + 2\right)\right) \cdot \frac{1}{2}
double f(double x) {
        double r1516273 = 1.0;
        double r1516274 = 2.0;
        double r1516275 = r1516273 / r1516274;
        double r1516276 = x;
        double r1516277 = r1516273 + r1516276;
        double r1516278 = r1516273 - r1516276;
        double r1516279 = r1516277 / r1516278;
        double r1516280 = log(r1516279);
        double r1516281 = r1516275 * r1516280;
        return r1516281;
}
↓
double f(double x) {
        double r1516282 = 0.4;
        double r1516283 = x;
        double r1516284 = 5.0;
        double r1516285 = pow(r1516283, r1516284);
        double r1516286 = r1516282 * r1516285;
        double r1516287 = r1516283 * r1516283;
        double r1516288 = 0.6666666666666666;
        double r1516289 = r1516287 * r1516288;
        double r1516290 = 2.0;
        double r1516291 = r1516289 + r1516290;
        double r1516292 = r1516283 * r1516291;
        double r1516293 = r1516286 + r1516292;
        double r1516294 = 0.5;
        double r1516295 = r1516293 * r1516294;
        return r1516295;
}

Error

Bits error versus x

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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. Final simplification0.2

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

Reproduce

herbie shell --seed 2019152 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))