Average Error: 0.5 → 0.4
Time: 2.1s
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 r1737 = x;
        double r1738 = 1.0;
        double r1739 = r1737 - r1738;
        double r1740 = sqrt(r1739);
        double r1741 = sqrt(r1737);
        double r1742 = r1740 * r1741;
        return r1742;
}
↓
double f(double x) {
        double r1743 = x;
        double r1744 = 0.5;
        double r1745 = 0.125;
        double r1746 = 1.0;
        double r1747 = r1746 / r1743;
        double r1748 = r1745 * r1747;
        double r1749 = r1744 + r1748;
        double r1750 = r1743 - r1749;
        return r1750;
}

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 2019354 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))