Average Error: 0.5 → 0.4
Time: 13.1s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[x - \left(0.5 + \frac{0.125}{x}\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
x - \left(0.5 + \frac{0.125}{x}\right)
double f(double x) {
        double r17006 = x;
        double r17007 = 1.0;
        double r17008 = r17006 - r17007;
        double r17009 = sqrt(r17008);
        double r17010 = sqrt(r17006);
        double r17011 = r17009 * r17010;
        return r17011;
}
↓
double f(double x) {
        double r17012 = x;
        double r17013 = 0.5;
        double r17014 = 0.125;
        double r17015 = r17014 / r17012;
        double r17016 = r17013 + r17015;
        double r17017 = r17012 - r17016;
        return r17017;
}

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 around inf 0.4

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019303 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))