Average Error: 58.5 → 0.2
Time: 6.0s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left(\frac{2}{3} \cdot \frac{{x}^{3}}{{1}^{3}} + \left(2 \cdot x + \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \left(\frac{2}{3} \cdot \frac{{x}^{3}}{{1}^{3}} + \left(2 \cdot x + \frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}}\right)\right)
double f(double x) {
        double r53573 = 1.0;
        double r53574 = 2.0;
        double r53575 = r53573 / r53574;
        double r53576 = x;
        double r53577 = r53573 + r53576;
        double r53578 = r53573 - r53576;
        double r53579 = r53577 / r53578;
        double r53580 = log(r53579);
        double r53581 = r53575 * r53580;
        return r53581;
}
↓
double f(double x) {
        double r53582 = 1.0;
        double r53583 = 2.0;
        double r53584 = r53582 / r53583;
        double r53585 = 0.6666666666666666;
        double r53586 = x;
        double r53587 = 3.0;
        double r53588 = pow(r53586, r53587);
        double r53589 = pow(r53582, r53587);
        double r53590 = r53588 / r53589;
        double r53591 = r53585 * r53590;
        double r53592 = r53583 * r53586;
        double r53593 = 0.4;
        double r53594 = 5.0;
        double r53595 = pow(r53586, r53594);
        double r53596 = pow(r53582, r53594);
        double r53597 = r53595 / r53596;
        double r53598 = r53593 * r53597;
        double r53599 = r53592 + r53598;
        double r53600 = r53591 + r53599;
        double r53601 = r53584 * r53600;
        return r53601;
}

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. Using strategy rm

  3. Applied log-div58.5

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

  4. Taylor expanded around 0 0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2020025 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))