Average Error: 0.0 → 0.0
Time: 2.2s
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 r163292 = 1.0;
        double r163293 = x;
        double r163294 = r163293 * r163293;
        double r163295 = r163292 - r163294;
        double r163296 = sqrt(r163295);
        return r163296;
}
↓
double f(double x) {
        double r163297 = 1.0;
        double r163298 = 3.0;
        double r163299 = pow(r163297, r163298);
        double r163300 = x;
        double r163301 = r163300 * r163300;
        double r163302 = pow(r163301, r163298);
        double r163303 = r163299 - r163302;
        double r163304 = sqrt(r163303);
        double r163305 = r163297 * r163297;
        double r163306 = r163301 * r163301;
        double r163307 = r163297 * r163301;
        double r163308 = r163306 + r163307;
        double r163309 = r163305 + r163308;
        double r163310 = sqrt(r163309);
        double r163311 = r163304 / r163310;
        return r163311;
}

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 2020001 +o rules:numerics
(FPCore (x)
  :name "Diagrams.TwoD.Ellipse:ellipse from diagrams-lib-1.3.0.3"
  :precision binary64
  (sqrt (- 1 (* x x))))