Average Error: 0.5 → 0.4
Time: 2.2s
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 r1636 = x;
        double r1637 = 1.0;
        double r1638 = r1636 - r1637;
        double r1639 = sqrt(r1638);
        double r1640 = sqrt(r1636);
        double r1641 = r1639 * r1640;
        return r1641;
}
↓
double f(double x) {
        double r1642 = x;
        double r1643 = 0.5;
        double r1644 = 0.125;
        double r1645 = 1.0;
        double r1646 = r1645 / r1642;
        double r1647 = r1644 * r1646;
        double r1648 = r1643 + r1647;
        double r1649 = r1642 - r1648;
        return r1649;
}

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 2020100 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))