Average Error: 58.5 → 0.7
Time: 23.8s
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 r95520 = 1.0;
        double r95521 = 2.0;
        double r95522 = r95520 / r95521;
        double r95523 = x;
        double r95524 = r95520 + r95523;
        double r95525 = r95520 - r95523;
        double r95526 = r95524 / r95525;
        double r95527 = log(r95526);
        double r95528 = r95522 * r95527;
        return r95528;
}
↓
double f(double x) {
        double r95529 = 2.0;
        double r95530 = x;
        double r95531 = r95529 * r95530;
        double r95532 = 1.0;
        double r95533 = log(r95532);
        double r95534 = r95531 + r95533;
        double r95535 = 2.0;
        double r95536 = pow(r95530, r95535);
        double r95537 = pow(r95532, r95535);
        double r95538 = r95529 / r95537;
        double r95539 = r95538 - r95529;
        double r95540 = r95536 * r95539;
        double r95541 = r95534 - r95540;
        double r95542 = r95541 * r95532;
        double r95543 = r95542 / r95529;
        return r95543;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.5

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019291 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))