Average Error: 0.0 → 0.0
Time: 1.6s
Precision: binary64
\[\sqrt{1 - x \cdot x} \]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{1 - x \cdot x}\right)\right) \]
\sqrt{1 - x \cdot x}
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{1 - x \cdot x}\right)\right)

(FPCore (x) :precision binary64 (sqrt (- 1.0 (* x x))))
↓
(FPCore (x) :precision binary64 (log1p (expm1 (sqrt (- 1.0 (* x x))))))
double code(double x) {
	return sqrt(1.0 - (x * x));
}
↓
double code(double x) {
	return log1p(expm1(sqrt(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. Applied add-cbrt-cube_binary640.0

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

  3. Simplified0.0

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

  4. Applied log1p-expm1-u_binary640.0

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{{\left(\sqrt{1 - x \cdot x}\right)}^{3}}\right)\right)} \]

  5. Simplified0.0

    \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\sqrt{1 - x \cdot x}\right)}\right) \]

  6. Final simplification0.0

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{1 - x \cdot x}\right)\right) \]

Reproduce

herbie shell --seed 2022088 
(FPCore (x)
  :name "Diagrams.TwoD.Ellipse:ellipse from diagrams-lib-1.3.0.3"
  :precision binary64
  (sqrt (- 1.0 (* x x))))