Average Error: 0.0 → 0.0
Time: 3.6s
Precision: 64
\[\sqrt{1 - x \cdot x}\]
↓
\[\sqrt{\sqrt{1} + \sqrt{x \cdot x}} \cdot \sqrt{\sqrt{1} - \sqrt{x \cdot x}}\]
\sqrt{1 - x \cdot x}
↓
\sqrt{\sqrt{1} + \sqrt{x \cdot x}} \cdot \sqrt{\sqrt{1} - \sqrt{x \cdot x}}
double code(double x) {
	return sqrt((1.0 - (x * x)));
}
↓
double code(double x) {
	return (sqrt((sqrt(1.0) + sqrt((x * x)))) * sqrt((sqrt(1.0) - sqrt((x * x)))));
}

Error

Bits error versus x

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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 add-sqr-sqrt0.0

    \[\leadsto \sqrt{1 - \color{blue}{\sqrt{x \cdot x} \cdot \sqrt{x \cdot x}}}\]

  4. Applied add-sqr-sqrt0.0

    \[\leadsto \sqrt{\color{blue}{\sqrt{1} \cdot \sqrt{1}} - \sqrt{x \cdot x} \cdot \sqrt{x \cdot x}}\]

  5. Applied difference-of-squares0.0

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

  6. Applied sqrt-prod0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020066 
(FPCore (x)
  :name "Diagrams.TwoD.Ellipse:ellipse from diagrams-lib-1.3.0.3"
  :precision binary64
  (sqrt (- 1 (* x x))))