Average Error: 0.5 → 0.4
Time: 16.4s
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 r28698 = x;
        double r28699 = 1.0;
        double r28700 = r28698 - r28699;
        double r28701 = sqrt(r28700);
        double r28702 = sqrt(r28698);
        double r28703 = r28701 * r28702;
        return r28703;
}
↓
double f(double x) {
        double r28704 = x;
        double r28705 = 0.5;
        double r28706 = 0.125;
        double r28707 = r28706 / r28704;
        double r28708 = r28705 + r28707;
        double r28709 = r28704 - r28708;
        return r28709;
}

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 2019304 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))