Average Error: 0.5 → 0.4
Time: 5.1s
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 r3460 = x;
        double r3461 = 1.0;
        double r3462 = r3460 - r3461;
        double r3463 = sqrt(r3462);
        double r3464 = sqrt(r3460);
        double r3465 = r3463 * r3464;
        return r3465;
}
↓
double f(double x) {
        double r3466 = x;
        double r3467 = 0.5;
        double r3468 = 0.125;
        double r3469 = r3468 / r3466;
        double r3470 = r3467 + r3469;
        double r3471 = r3466 - r3470;
        return r3471;
}

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 2019351 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))