Average Error: 0.0 → 0.0
Time: 2.3s
Precision: 64
\[\sqrt{1 - x \cdot x}\]
↓
\[\frac{\sqrt{1 \cdot 1 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{\mathsf{hypot}\left(x, \sqrt{1}\right)}\]
\sqrt{1 - x \cdot x}
↓
\frac{\sqrt{1 \cdot 1 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{\mathsf{hypot}\left(x, \sqrt{1}\right)}
double code(double x) {
	return sqrt((1.0 - (x * x)));
}
↓
double code(double x) {
	return (sqrt(((1.0 * 1.0) - ((x * x) * (x * x)))) / hypot(x, sqrt(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 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. Applied sqrt-div0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

    \[\leadsto \frac{\sqrt{1 \cdot 1 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{\mathsf{hypot}\left(x, \sqrt{1}\right)}\]

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))))