Average Error: 58.4 → 0.8
Time: 17.9s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{\left(\left(2 \cdot x + \log 1\right) - {x}^{2} \cdot \left(\frac{2}{{1}^{2}} - 2\right)\right) \cdot 1}{2}\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{\left(\left(2 \cdot x + \log 1\right) - {x}^{2} \cdot \left(\frac{2}{{1}^{2}} - 2\right)\right) \cdot 1}{2}
double f(double x) {
        double r66547 = 1.0;
        double r66548 = 2.0;
        double r66549 = r66547 / r66548;
        double r66550 = x;
        double r66551 = r66547 + r66550;
        double r66552 = r66547 - r66550;
        double r66553 = r66551 / r66552;
        double r66554 = log(r66553);
        double r66555 = r66549 * r66554;
        return r66555;
}
↓
double f(double x) {
        double r66556 = 2.0;
        double r66557 = x;
        double r66558 = r66556 * r66557;
        double r66559 = 1.0;
        double r66560 = log(r66559);
        double r66561 = r66558 + r66560;
        double r66562 = 2.0;
        double r66563 = pow(r66557, r66562);
        double r66564 = pow(r66559, r66562);
        double r66565 = r66556 / r66564;
        double r66566 = r66565 - r66556;
        double r66567 = r66563 * r66566;
        double r66568 = r66561 - r66567;
        double r66569 = r66568 * r66559;
        double r66570 = r66569 / r66556;
        return r66570;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.4

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019297 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))