Average Error: 58.7 → 0.2
Time: 5.7s
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 r60754 = 1.0;
        double r60755 = 2.0;
        double r60756 = r60754 / r60755;
        double r60757 = x;
        double r60758 = r60754 + r60757;
        double r60759 = r60754 - r60757;
        double r60760 = r60758 / r60759;
        double r60761 = log(r60760);
        double r60762 = r60756 * r60761;
        return r60762;
}
↓
double f(double x) {
        double r60763 = 1.0;
        double r60764 = 2.0;
        double r60765 = r60763 / r60764;
        double r60766 = 0.6666666666666666;
        double r60767 = x;
        double r60768 = 3.0;
        double r60769 = pow(r60767, r60768);
        double r60770 = pow(r60763, r60768);
        double r60771 = r60769 / r60770;
        double r60772 = r60766 * r60771;
        double r60773 = r60764 * r60767;
        double r60774 = 0.4;
        double r60775 = 5.0;
        double r60776 = pow(r60767, r60775);
        double r60777 = pow(r60763, r60775);
        double r60778 = r60776 / r60777;
        double r60779 = r60774 * r60778;
        double r60780 = r60773 + r60779;
        double r60781 = r60772 + r60780;
        double r60782 = r60765 * r60781;
        return r60782;
}

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. Using strategy rm

  3. Applied flip--58.7

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

  4. Applied associate-/r/58.7

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

  5. Applied log-prod58.7

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

  6. Simplified58.6

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

  8. Applied add-cbrt-cube58.7

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt[3]{\left(\left(\log \left(\frac{1}{1 - x}\right) + \log \left(1 + x\right)\right) \cdot \left(\log \left(\frac{1}{1 - x}\right) + \log \left(1 + x\right)\right)\right) \cdot \left(\log \left(\frac{1}{1 - x}\right) + \log \left(1 + x\right)\right)}}\]

  9. Simplified58.6

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

  10. Taylor expanded around 0 0.2

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

  11. 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 2020002 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))