Average Error: 0.0 → 0.0
Time: 2.1s
Precision: binary64
\[\sqrt{1 - x \cdot x}\]
↓
\[\sqrt{\frac{1 - {x}^{4}}{1 + x \cdot x}}\]
\sqrt{1 - x \cdot x}
↓
\sqrt{\frac{1 - {x}^{4}}{1 + x \cdot x}}
(FPCore (x) :precision binary64 (sqrt (- 1.0 (* x x))))
↓
(FPCore (x) :precision binary64 (sqrt (/ (- 1.0 (pow x 4.0)) (+ 1.0 (* x x)))))
double code(double x) {
	return sqrt(1.0 - (x * x));
}
↓
double code(double x) {
	return sqrt((1.0 - pow(x, 4.0)) / (1.0 + (x * x)));
}

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--_binary64_58500.0

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

  4. Applied sqrt-div_binary64_58920.0

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

  5. Simplified0.0

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

  7. Applied sqrt-undiv_binary64_58960.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2021019 
(FPCore (x)
  :name "Diagrams.TwoD.Ellipse:ellipse from diagrams-lib-1.3.0.3"
  :precision binary64
  (sqrt (- 1.0 (* x x))))