Average Error: 0.5 → 0.4
Time: 4.4s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[x - \left(0.5 + \frac{0.125}{x}\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
x - \left(0.5 + \frac{0.125}{x}\right)
double f(double x) {
        double r2721 = x;
        double r2722 = 1.0;
        double r2723 = r2721 - r2722;
        double r2724 = sqrt(r2723);
        double r2725 = sqrt(r2721);
        double r2726 = r2724 * r2725;
        return r2726;
}
↓
double f(double x) {
        double r2727 = x;
        double r2728 = 0.5;
        double r2729 = 0.125;
        double r2730 = r2729 / r2727;
        double r2731 = r2728 + r2730;
        double r2732 = r2727 - r2731;
        return r2732;
}

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. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2020042 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))