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 r1780 = x;
        double r1781 = 1.0;
        double r1782 = r1780 - r1781;
        double r1783 = sqrt(r1782);
        double r1784 = sqrt(r1780);
        double r1785 = r1783 * r1784;
        return r1785;
}
↓
double f(double x) {
        double r1786 = x;
        double r1787 = 0.5;
        double r1788 = 0.125;
        double r1789 = 1.0;
        double r1790 = r1789 / r1786;
        double r1791 = r1788 * r1790;
        double r1792 = r1787 + r1791;
        double r1793 = r1786 - r1792;
        return r1793;
}

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 2019352 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))