Average Error: 0.5 → 0.4
Time: 4.4s
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 r21462 = x;
        double r21463 = 1.0;
        double r21464 = r21462 - r21463;
        double r21465 = sqrt(r21464);
        double r21466 = sqrt(r21462);
        double r21467 = r21465 * r21466;
        return r21467;
}
↓
double f(double x) {
        double r21468 = x;
        double r21469 = 0.5;
        double r21470 = 0.125;
        double r21471 = 1.0;
        double r21472 = r21471 / r21468;
        double r21473 = r21470 * r21472;
        double r21474 = r21469 + r21473;
        double r21475 = r21468 - r21474;
        return r21475;
}

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 2020003 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))