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 r7482 = x;
        double r7483 = 1.0;
        double r7484 = r7482 - r7483;
        double r7485 = sqrt(r7484);
        double r7486 = sqrt(r7482);
        double r7487 = r7485 * r7486;
        return r7487;
}
↓
double f(double x) {
        double r7488 = x;
        double r7489 = 0.5;
        double r7490 = 0.125;
        double r7491 = 1.0;
        double r7492 = r7491 / r7488;
        double r7493 = r7490 * r7492;
        double r7494 = r7489 + r7493;
        double r7495 = r7488 - r7494;
        return r7495;
}

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 2020059 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))