Average Error: 0.5 → 0.4
Time: 4.3s
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 r2579 = x;
        double r2580 = 1.0;
        double r2581 = r2579 - r2580;
        double r2582 = sqrt(r2581);
        double r2583 = sqrt(r2579);
        double r2584 = r2582 * r2583;
        return r2584;
}
↓
double f(double x) {
        double r2585 = x;
        double r2586 = 0.5;
        double r2587 = 0.125;
        double r2588 = r2587 / r2585;
        double r2589 = r2586 + r2588;
        double r2590 = r2585 - r2589;
        return r2590;
}

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)))