Average Error: 0.5 → 0.4
Time: 2.5s
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 r1522 = x;
        double r1523 = 1.0;
        double r1524 = r1522 - r1523;
        double r1525 = sqrt(r1524);
        double r1526 = sqrt(r1522);
        double r1527 = r1525 * r1526;
        return r1527;
}
↓
double f(double x) {
        double r1528 = x;
        double r1529 = 0.5;
        double r1530 = 0.125;
        double r1531 = 1.0;
        double r1532 = r1531 / r1528;
        double r1533 = r1530 * r1532;
        double r1534 = r1529 + r1533;
        double r1535 = r1528 - r1534;
        return r1535;
}

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 2020034 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))