Average Error: 0.5 → 0.5
Time: 2.9s
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 r8486 = x;
        double r8487 = 1.0;
        double r8488 = r8486 - r8487;
        double r8489 = sqrt(r8488);
        double r8490 = sqrt(r8486);
        double r8491 = r8489 * r8490;
        return r8491;
}
↓
double f(double x) {
        double r8492 = x;
        double r8493 = 0.5;
        double r8494 = 0.125;
        double r8495 = 1.0;
        double r8496 = r8495 / r8492;
        double r8497 = r8494 * r8496;
        double r8498 = r8493 + r8497;
        double r8499 = r8492 - r8498;
        return r8499;
}

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 0.5

    \[\leadsto \color{blue}{x - \left(0.5 + 0.125 \cdot \frac{1}{x}\right)}\]

  3. Final simplification0.5

    \[\leadsto x - \left(0.5 + 0.125 \cdot \frac{1}{x}\right)\]

Reproduce

herbie shell --seed 2020049 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))