Average Error: 58.5 → 0.2
Time: 10.4s
Precision: 64
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \left(2 \cdot x + \left(\frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}} + {\left(\frac{x}{1}\right)}^{3} \cdot \frac{2}{3}\right)\right)\]
\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)
↓
\frac{1}{2} \cdot \left(2 \cdot x + \left(\frac{2}{5} \cdot \frac{{x}^{5}}{{1}^{5}} + {\left(\frac{x}{1}\right)}^{3} \cdot \frac{2}{3}\right)\right)
double f(double x) {
        double r61687 = 1.0;
        double r61688 = 2.0;
        double r61689 = r61687 / r61688;
        double r61690 = x;
        double r61691 = r61687 + r61690;
        double r61692 = r61687 - r61690;
        double r61693 = r61691 / r61692;
        double r61694 = log(r61693);
        double r61695 = r61689 * r61694;
        return r61695;
}
↓
double f(double x) {
        double r61696 = 1.0;
        double r61697 = 2.0;
        double r61698 = r61696 / r61697;
        double r61699 = x;
        double r61700 = r61697 * r61699;
        double r61701 = 0.4;
        double r61702 = 5.0;
        double r61703 = pow(r61699, r61702);
        double r61704 = pow(r61696, r61702);
        double r61705 = r61703 / r61704;
        double r61706 = r61701 * r61705;
        double r61707 = r61699 / r61696;
        double r61708 = 3.0;
        double r61709 = pow(r61707, r61708);
        double r61710 = 0.6666666666666666;
        double r61711 = r61709 * r61710;
        double r61712 = r61706 + r61711;
        double r61713 = r61700 + r61712;
        double r61714 = r61698 * r61713;
        return r61714;
}

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 div-inv58.5

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

  4. Applied log-prod58.5

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

  5. Simplified58.5

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

  6. Taylor expanded around 0 0.2

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

  7. Simplified0.2

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

  9. Applied associate-+l+0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2020046 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))