Average Error: 0.0 → 0.0
Time: 1.8s
Precision: binary64
\[\sqrt{1 - x \cdot x} \]
↓
\[e^{0.5 \cdot \mathsf{log1p}\left(-x \cdot x\right)} \]
\sqrt{1 - x \cdot x}
↓
e^{0.5 \cdot \mathsf{log1p}\left(-x \cdot x\right)}

(FPCore (x) :precision binary64 (sqrt (- 1.0 (* x x))))
↓
(FPCore (x) :precision binary64 (exp (* 0.5 (log1p (- (* x x))))))
double code(double x) {
	return sqrt(1.0 - (x * x));
}
↓
double code(double x) {
	return exp(0.5 * log1p(-(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

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

  3. Simplified0.0

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

  4. Applied add-exp-log_binary640.0

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

  5. Simplified0.0

    \[\leadsto e^{\color{blue}{0.5 \cdot \mathsf{log1p}\left(-x \cdot x\right)}} \]

  6. Final simplification0.0

    \[\leadsto e^{0.5 \cdot \mathsf{log1p}\left(-x \cdot x\right)} \]

Reproduce

herbie shell --seed 2022067 
(FPCore (x)
  :name "Diagrams.TwoD.Ellipse:ellipse from diagrams-lib-1.3.0.3"
  :precision binary64
  (sqrt (- 1.0 (* x x))))