Average Error: 0.0 → 0.2
Time: 3.5s
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 r233907 = 1.0;
        double r233908 = x;
        double r233909 = r233908 * r233908;
        double r233910 = r233907 - r233909;
        double r233911 = sqrt(r233910);
        return r233911;
}
↓
double f(double x) {
        double r233912 = 1.0;
        double r233913 = sqrt(r233912);
        double r233914 = 0.125;
        double r233915 = x;
        double r233916 = 4.0;
        double r233917 = pow(r233915, r233916);
        double r233918 = 3.0;
        double r233919 = pow(r233913, r233918);
        double r233920 = r233917 / r233919;
        double r233921 = r233914 * r233920;
        double r233922 = 0.5;
        double r233923 = 2.0;
        double r233924 = pow(r233915, r233923);
        double r233925 = r233924 / r233913;
        double r233926 = r233922 * r233925;
        double r233927 = r233921 + r233926;
        double r233928 = r233913 - r233927;
        return r233928;
}

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))))