Average Error: 0.0 → 0.0
Time: 3.1s
Precision: 64
\[\sqrt{1 - x \cdot x}\]
↓
\[\sqrt{\frac{{1}^{3} - {\left(x \cdot x\right)}^{3}}{\mathsf{fma}\left({x}^{2}, {x}^{2} + 1, 1 \cdot 1\right)}}\]
\sqrt{1 - x \cdot x}
↓
\sqrt{\frac{{1}^{3} - {\left(x \cdot x\right)}^{3}}{\mathsf{fma}\left({x}^{2}, {x}^{2} + 1, 1 \cdot 1\right)}}
double code(double x) {
	return sqrt((1.0 - (x * x)));
}
↓
double code(double x) {
	return sqrt(((pow(1.0, 3.0) - pow((x * x), 3.0)) / fma(pow(x, 2.0), (pow(x, 2.0) + 1.0), (1.0 * 1.0))));
}

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. Using strategy rm

  6. Applied sqrt-undiv0.0

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

  7. Simplified0.0

    \[\leadsto \sqrt{\color{blue}{\frac{{1}^{3} - {\left(x \cdot x\right)}^{3}}{\mathsf{fma}\left({x}^{2}, {x}^{2} + 1, 1 \cdot 1\right)}}}\]

  8. Final simplification0.0

    \[\leadsto \sqrt{\frac{{1}^{3} - {\left(x \cdot x\right)}^{3}}{\mathsf{fma}\left({x}^{2}, {x}^{2} + 1, 1 \cdot 1\right)}}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (x)
  :name "Diagrams.TwoD.Ellipse:ellipse from diagrams-lib-1.3.0.3"
  :precision binary64
  (sqrt (- 1 (* x x))))