Average Error: 0.5 → 0.3
Time: 8.1s
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 r10659 = x;
        double r10660 = 1.0;
        double r10661 = r10659 - r10660;
        double r10662 = sqrt(r10661);
        double r10663 = sqrt(r10659);
        double r10664 = r10662 * r10663;
        return r10664;
}
↓
double f(double x) {
        double r10665 = x;
        double r10666 = 0.5;
        double r10667 = 0.125;
        double r10668 = r10667 / r10665;
        double r10669 = r10666 + r10668;
        double r10670 = r10665 - r10669;
        return r10670;
}

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 0.3

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019209 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))