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 r254382 = 1.0;
        double r254383 = x;
        double r254384 = r254383 * r254383;
        double r254385 = r254382 - r254384;
        double r254386 = sqrt(r254385);
        return r254386;
}
↓
double f(double x) {
        double r254387 = 1.0;
        double r254388 = sqrt(r254387);
        double r254389 = 0.125;
        double r254390 = x;
        double r254391 = 4.0;
        double r254392 = pow(r254390, r254391);
        double r254393 = 3.0;
        double r254394 = pow(r254388, r254393);
        double r254395 = r254392 / r254394;
        double r254396 = r254389 * r254395;
        double r254397 = 0.5;
        double r254398 = 2.0;
        double r254399 = pow(r254390, r254398);
        double r254400 = r254399 / r254388;
        double r254401 = r254397 * r254400;
        double r254402 = r254396 + r254401;
        double r254403 = r254388 - r254402;
        return r254403;
}

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