Average Error: 0.5 → 0.4
Time: 11.3s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\left(x - \frac{0.125}{x}\right) - 0.5\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
\left(x - \frac{0.125}{x}\right) - 0.5
double f(double x) {
        double r10667 = x;
        double r10668 = 1.0;
        double r10669 = r10667 - r10668;
        double r10670 = sqrt(r10669);
        double r10671 = sqrt(r10667);
        double r10672 = r10670 * r10671;
        return r10672;
}
↓
double f(double x) {
        double r10673 = x;
        double r10674 = 0.125;
        double r10675 = r10674 / r10673;
        double r10676 = r10673 - r10675;
        double r10677 = 0.5;
        double r10678 = r10676 - r10677;
        return r10678;
}

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

    \[\leadsto \left(x - \frac{0.125}{x}\right) - 0.5\]

Reproduce

herbie shell --seed 2019297 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))