Average Error: 0.0 → 0.2
Time: 1.9s
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 r197109 = 1.0;
        double r197110 = x;
        double r197111 = r197110 * r197110;
        double r197112 = r197109 - r197111;
        double r197113 = sqrt(r197112);
        return r197113;
}
↓
double f(double x) {
        double r197114 = 1.0;
        double r197115 = sqrt(r197114);
        double r197116 = 0.125;
        double r197117 = x;
        double r197118 = 4.0;
        double r197119 = pow(r197117, r197118);
        double r197120 = 3.0;
        double r197121 = pow(r197115, r197120);
        double r197122 = r197119 / r197121;
        double r197123 = r197116 * r197122;
        double r197124 = 0.5;
        double r197125 = 2.0;
        double r197126 = pow(r197117, r197125);
        double r197127 = r197126 / r197115;
        double r197128 = r197124 * r197127;
        double r197129 = r197123 + r197128;
        double r197130 = r197115 - r197129;
        return r197130;
}

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))))