Average Error: 0.0 → 0.2
Time: 6.5s
Precision: 64
\[\sqrt{1.0 - x \cdot x}\]
↓
\[\sqrt{1.0} - \mathsf{fma}\left(\frac{x \cdot x}{\sqrt{1.0}}, \frac{1}{2}, \frac{x \cdot x}{\sqrt{1.0}} \cdot \frac{\left(x \cdot x\right) \cdot \frac{1}{8}}{1.0}\right)\]
\sqrt{1.0 - x \cdot x}
↓
\sqrt{1.0} - \mathsf{fma}\left(\frac{x \cdot x}{\sqrt{1.0}}, \frac{1}{2}, \frac{x \cdot x}{\sqrt{1.0}} \cdot \frac{\left(x \cdot x\right) \cdot \frac{1}{8}}{1.0}\right)
double f(double x) {
        double r8287144 = 1.0;
        double r8287145 = x;
        double r8287146 = r8287145 * r8287145;
        double r8287147 = r8287144 - r8287146;
        double r8287148 = sqrt(r8287147);
        return r8287148;
}
↓
double f(double x) {
        double r8287149 = 1.0;
        double r8287150 = sqrt(r8287149);
        double r8287151 = x;
        double r8287152 = r8287151 * r8287151;
        double r8287153 = r8287152 / r8287150;
        double r8287154 = 0.5;
        double r8287155 = 0.125;
        double r8287156 = r8287152 * r8287155;
        double r8287157 = r8287156 / r8287149;
        double r8287158 = r8287153 * r8287157;
        double r8287159 = fma(r8287153, r8287154, r8287158);
        double r8287160 = r8287150 - r8287159;
        return r8287160;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[\sqrt{1.0 - x \cdot x}\]

  2. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{\sqrt{1.0} - \left(\frac{1}{8} \cdot \frac{{x}^{4}}{{\left(\sqrt{1.0}\right)}^{3}} + \frac{1}{2} \cdot \frac{{x}^{2}}{\sqrt{1.0}}\right)}\]

  3. Simplified0.2

    \[\leadsto \color{blue}{\sqrt{1.0} - \mathsf{fma}\left(\frac{x \cdot x}{\sqrt{1.0}}, \frac{1}{2}, \frac{\left(x \cdot x\right) \cdot \frac{1}{8}}{1.0} \cdot \frac{x \cdot x}{\sqrt{1.0}}\right)}\]

  4. Final simplification0.2

    \[\leadsto \sqrt{1.0} - \mathsf{fma}\left(\frac{x \cdot x}{\sqrt{1.0}}, \frac{1}{2}, \frac{x \cdot x}{\sqrt{1.0}} \cdot \frac{\left(x \cdot x\right) \cdot \frac{1}{8}}{1.0}\right)\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x)
  :name "Diagrams.TwoD.Ellipse:ellipse from diagrams-lib-1.3.0.3"
  (sqrt (- 1.0 (* x x))))