Average Error: 0.0 → 0.2
Time: 1.5s
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 r170572 = 1.0;
        double r170573 = x;
        double r170574 = r170573 * r170573;
        double r170575 = r170572 - r170574;
        double r170576 = sqrt(r170575);
        return r170576;
}
↓
double f(double x) {
        double r170577 = 1.0;
        double r170578 = sqrt(r170577);
        double r170579 = 0.125;
        double r170580 = x;
        double r170581 = 4.0;
        double r170582 = pow(r170580, r170581);
        double r170583 = 3.0;
        double r170584 = pow(r170578, r170583);
        double r170585 = r170582 / r170584;
        double r170586 = r170579 * r170585;
        double r170587 = 0.5;
        double r170588 = 2.0;
        double r170589 = pow(r170580, r170588);
        double r170590 = r170589 / r170578;
        double r170591 = r170587 * r170590;
        double r170592 = r170586 + r170591;
        double r170593 = r170578 - r170592;
        return r170593;
}

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 2020046 +o rules:numerics
(FPCore (x)
  :name "Diagrams.TwoD.Ellipse:ellipse from diagrams-lib-1.3.0.3"
  :precision binary64
  (sqrt (- 1 (* x x))))