Average Error: 0.5 → 0.2
Time: 2.4s
Precision: binary64
\[\sqrt{x - 1} \cdot \sqrt{x} \]
↓
\[x - \left(0.5 + \left(\frac{0.125}{x} + \frac{0.0625}{x \cdot x}\right)\right) \]
\sqrt{x - 1} \cdot \sqrt{x}
↓
x - \left(0.5 + \left(\frac{0.125}{x} + \frac{0.0625}{x \cdot x}\right)\right)

(FPCore (x) :precision binary64 (* (sqrt (- x 1.0)) (sqrt x)))
↓
(FPCore (x)
 :precision binary64
 (- x (+ 0.5 (+ (/ 0.125 x) (/ 0.0625 (* x x))))))
double code(double x) {
	return sqrt(x - 1.0) * sqrt(x);
}
↓
double code(double x) {
	return x - (0.5 + ((0.125 / x) + (0.0625 / (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 0.5

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

  2. Taylor expanded in x around inf 0.2

    \[\leadsto \color{blue}{x - \left(0.5 + \left(0.125 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{2}}\right)\right)} \]

  3. Simplified0.2

    \[\leadsto \color{blue}{x - \left(0.5 + \left(\frac{0.125}{x} + \frac{0.0625}{x \cdot x}\right)\right)} \]

  4. Final simplification0.2

    \[\leadsto x - \left(0.5 + \left(\frac{0.125}{x} + \frac{0.0625}{x \cdot x}\right)\right) \]

Reproduce

herbie shell --seed 2021280 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))