Average Error: 0.0 → 0.0
Time: 4.0s
Precision: 64
\[\sqrt{1 - x \cdot x}\]
↓
\[\frac{\sqrt{{1}^{3} - {x}^{6}}}{\sqrt{{x}^{4} + \mathsf{fma}\left(x, x, 1\right) \cdot 1}}\]
\sqrt{1 - x \cdot x}
↓
\frac{\sqrt{{1}^{3} - {x}^{6}}}{\sqrt{{x}^{4} + \mathsf{fma}\left(x, x, 1\right) \cdot 1}}
double f(double x) {
        double r157328 = 1.0;
        double r157329 = x;
        double r157330 = r157329 * r157329;
        double r157331 = r157328 - r157330;
        double r157332 = sqrt(r157331);
        return r157332;
}
↓
double f(double x) {
        double r157333 = 1.0;
        double r157334 = 3.0;
        double r157335 = pow(r157333, r157334);
        double r157336 = x;
        double r157337 = 6.0;
        double r157338 = pow(r157336, r157337);
        double r157339 = r157335 - r157338;
        double r157340 = sqrt(r157339);
        double r157341 = 4.0;
        double r157342 = pow(r157336, r157341);
        double r157343 = fma(r157336, r157336, r157333);
        double r157344 = r157343 * r157333;
        double r157345 = r157342 + r157344;
        double r157346 = sqrt(r157345);
        double r157347 = r157340 / r157346;
        return r157347;
}

Error

Bits error versus x

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. Simplified0.0

    \[\leadsto \frac{\color{blue}{\sqrt{{1}^{3} - {x}^{6}}}}{\sqrt{1 \cdot 1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + 1 \cdot \left(x \cdot x\right)\right)}}\]

  6. Simplified0.0

    \[\leadsto \frac{\sqrt{{1}^{3} - {x}^{6}}}{\color{blue}{\sqrt{{x}^{4} + \mathsf{fma}\left(x, x, 1\right) \cdot 1}}}\]

  7. Final simplification0.0

    \[\leadsto \frac{\sqrt{{1}^{3} - {x}^{6}}}{\sqrt{{x}^{4} + \mathsf{fma}\left(x, x, 1\right) \cdot 1}}\]

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))))