Average Error: 0.0 → 0.0
Time: 2.1s
Precision: 64
\[\sqrt{1 - x \cdot x}\]
↓
\[\frac{\sqrt{{1}^{3} - {\left(x \cdot x\right)}^{3}}}{\sqrt{1 \cdot 1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + 1 \cdot \left(x \cdot x\right)\right)}}\]
\sqrt{1 - x \cdot x}
↓
\frac{\sqrt{{1}^{3} - {\left(x \cdot x\right)}^{3}}}{\sqrt{1 \cdot 1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + 1 \cdot \left(x \cdot x\right)\right)}}
double f(double x) {
        double r148618 = 1.0;
        double r148619 = x;
        double r148620 = r148619 * r148619;
        double r148621 = r148618 - r148620;
        double r148622 = sqrt(r148621);
        return r148622;
}
↓
double f(double x) {
        double r148623 = 1.0;
        double r148624 = 3.0;
        double r148625 = pow(r148623, r148624);
        double r148626 = x;
        double r148627 = r148626 * r148626;
        double r148628 = pow(r148627, r148624);
        double r148629 = r148625 - r148628;
        double r148630 = sqrt(r148629);
        double r148631 = r148623 * r148623;
        double r148632 = r148627 * r148627;
        double r148633 = r148623 * r148627;
        double r148634 = r148632 + r148633;
        double r148635 = r148631 + r148634;
        double r148636 = sqrt(r148635);
        double r148637 = r148630 / r148636;
        return r148637;
}

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. Using strategy rm

  3. Applied flip3--0.0

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

  4. Applied sqrt-div0.0

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

  5. Final simplification0.0

    \[\leadsto \frac{\sqrt{{1}^{3} - {\left(x \cdot x\right)}^{3}}}{\sqrt{1 \cdot 1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + 1 \cdot \left(x \cdot x\right)\right)}}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x)
  :name "Diagrams.TwoD.Ellipse:ellipse from diagrams-lib-1.3.0.3"
  :precision binary64
  (sqrt (- 1 (* x x))))