Average Error: 0.0 → 0.2
Time: 2.0s
Precision: 64
\[\sqrt{1 - x \cdot x}\]
↓
\[\sqrt{1} - \left(\frac{1}{8} \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{3}} + \frac{1}{2} \cdot \frac{{x}^{2}}{\sqrt{1}}\right)\]
\sqrt{1 - x \cdot x}
↓
\sqrt{1} - \left(\frac{1}{8} \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{3}} + \frac{1}{2} \cdot \frac{{x}^{2}}{\sqrt{1}}\right)
double f(double x) {
        double r170507 = 1.0;
        double r170508 = x;
        double r170509 = r170508 * r170508;
        double r170510 = r170507 - r170509;
        double r170511 = sqrt(r170510);
        return r170511;
}
↓
double f(double x) {
        double r170512 = 1.0;
        double r170513 = sqrt(r170512);
        double r170514 = 0.125;
        double r170515 = x;
        double r170516 = 4.0;
        double r170517 = pow(r170515, r170516);
        double r170518 = 3.0;
        double r170519 = pow(r170513, r170518);
        double r170520 = r170517 / r170519;
        double r170521 = r170514 * r170520;
        double r170522 = 0.5;
        double r170523 = 2.0;
        double r170524 = pow(r170515, r170523);
        double r170525 = r170524 / r170513;
        double r170526 = r170522 * r170525;
        double r170527 = r170521 + r170526;
        double r170528 = r170513 - r170527;
        return r170528;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[\sqrt{1 - x \cdot x}\]

  2. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{\sqrt{1} - \left(\frac{1}{8} \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{3}} + \frac{1}{2} \cdot \frac{{x}^{2}}{\sqrt{1}}\right)}\]

  3. Final simplification0.2

    \[\leadsto \sqrt{1} - \left(\frac{1}{8} \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{3}} + \frac{1}{2} \cdot \frac{{x}^{2}}{\sqrt{1}}\right)\]

Reproduce

herbie shell --seed 2020060 
(FPCore (x)
  :name "Diagrams.TwoD.Ellipse:ellipse from diagrams-lib-1.3.0.3"
  :precision binary64
  (sqrt (- 1 (* x x))))