Average Error: 0.5 → 0.4
Time: 5.3s
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 r2699 = x;
        double r2700 = 1.0;
        double r2701 = r2699 - r2700;
        double r2702 = sqrt(r2701);
        double r2703 = sqrt(r2699);
        double r2704 = r2702 * r2703;
        return r2704;
}
↓
double f(double x) {
        double r2705 = x;
        double r2706 = 0.5;
        double r2707 = 0.125;
        double r2708 = r2707 / r2705;
        double r2709 = r2706 + r2708;
        double r2710 = r2705 - r2709;
        return r2710;
}

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 2019350 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))