Average Error: 0.5 → 0.4
Time: 1.9s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[x - \left(0.5 + 0.125 \cdot \frac{1}{x}\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
x - \left(0.5 + 0.125 \cdot \frac{1}{x}\right)
double f(double x) {
        double r1726 = x;
        double r1727 = 1.0;
        double r1728 = r1726 - r1727;
        double r1729 = sqrt(r1728);
        double r1730 = sqrt(r1726);
        double r1731 = r1729 * r1730;
        return r1731;
}
↓
double f(double x) {
        double r1732 = x;
        double r1733 = 0.5;
        double r1734 = 0.125;
        double r1735 = 1.0;
        double r1736 = r1735 / r1732;
        double r1737 = r1734 * r1736;
        double r1738 = r1733 + r1737;
        double r1739 = r1732 - r1738;
        return r1739;
}

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. Final simplification0.4

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

Reproduce

herbie shell --seed 2020057 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))