Average Error: 0.0 → 0.2
Time: 1.2s
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 r193348 = 1.0;
        double r193349 = x;
        double r193350 = r193349 * r193349;
        double r193351 = r193348 - r193350;
        double r193352 = sqrt(r193351);
        return r193352;
}
↓
double f(double x) {
        double r193353 = 1.0;
        double r193354 = sqrt(r193353);
        double r193355 = 0.125;
        double r193356 = x;
        double r193357 = 4.0;
        double r193358 = pow(r193356, r193357);
        double r193359 = 3.0;
        double r193360 = pow(r193354, r193359);
        double r193361 = r193358 / r193360;
        double r193362 = r193355 * r193361;
        double r193363 = 0.5;
        double r193364 = 2.0;
        double r193365 = pow(r193356, r193364);
        double r193366 = r193365 / r193354;
        double r193367 = r193363 * r193366;
        double r193368 = r193362 + r193367;
        double r193369 = r193354 - r193368;
        return r193369;
}

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 2020018 
(FPCore (x)
  :name "Diagrams.TwoD.Ellipse:ellipse from diagrams-lib-1.3.0.3"
  :precision binary64
  (sqrt (- 1 (* x x))))