Average Error: 0.0 → 0.0
Time: 16.6s
Precision: 64
\[\sqrt{1 - x \cdot x}\]
↓
\[\sqrt{\frac{\sqrt{1 \cdot 1 - {x}^{4}}}{\frac{1 + x \cdot x}{\sqrt{1 \cdot 1 - {x}^{4}}}}}\]
\sqrt{1 - x \cdot x}
↓
\sqrt{\frac{\sqrt{1 \cdot 1 - {x}^{4}}}{\frac{1 + x \cdot x}{\sqrt{1 \cdot 1 - {x}^{4}}}}}
double f(double x) {
        double r207258 = 1.0;
        double r207259 = x;
        double r207260 = r207259 * r207259;
        double r207261 = r207258 - r207260;
        double r207262 = sqrt(r207261);
        return r207262;
}
↓
double f(double x) {
        double r207263 = 1.0;
        double r207264 = r207263 * r207263;
        double r207265 = x;
        double r207266 = 4.0;
        double r207267 = pow(r207265, r207266);
        double r207268 = r207264 - r207267;
        double r207269 = sqrt(r207268);
        double r207270 = r207265 * r207265;
        double r207271 = r207263 + r207270;
        double r207272 = r207271 / r207269;
        double r207273 = r207269 / r207272;
        double r207274 = sqrt(r207273);
        return r207274;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 0.0

    \[\sqrt{1 - x \cdot x}\]
  2. Using strategy rm

  3. Applied flip--0.0

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

  4. Simplified0.0

    \[\leadsto \sqrt{\frac{\color{blue}{1 \cdot 1 - {x}^{4}}}{1 + x \cdot x}}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.0

    \[\leadsto \sqrt{\frac{\color{blue}{\sqrt{1 \cdot 1 - {x}^{4}} \cdot \sqrt{1 \cdot 1 - {x}^{4}}}}{1 + x \cdot x}}\]

  7. Applied associate-/l*0.0

    \[\leadsto \sqrt{\color{blue}{\frac{\sqrt{1 \cdot 1 - {x}^{4}}}{\frac{1 + x \cdot x}{\sqrt{1 \cdot 1 - {x}^{4}}}}}}\]

  8. Final simplification0.0

    \[\leadsto \sqrt{\frac{\sqrt{1 \cdot 1 - {x}^{4}}}{\frac{1 + x \cdot x}{\sqrt{1 \cdot 1 - {x}^{4}}}}}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (x)
  :name "Diagrams.TwoD.Ellipse:ellipse from diagrams-lib-1.3.0.3"
  :precision binary64
  (sqrt (- 1 (* x x))))