Average Error: 0.5 → 0.4
Time: 14.7s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[x + \left(\frac{-1}{2} - \frac{\frac{1}{8}}{x}\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
x + \left(\frac{-1}{2} - \frac{\frac{1}{8}}{x}\right)
double f(double x) {
        double r382667 = x;
        double r382668 = 1.0;
        double r382669 = r382667 - r382668;
        double r382670 = sqrt(r382669);
        double r382671 = sqrt(r382667);
        double r382672 = r382670 * r382671;
        return r382672;
}
↓
double f(double x) {
        double r382673 = x;
        double r382674 = -0.5;
        double r382675 = 0.125;
        double r382676 = r382675 / r382673;
        double r382677 = r382674 - r382676;
        double r382678 = r382673 + r382677;
        return r382678;
}

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

    \[\leadsto \color{blue}{x - \left(\frac{1}{8} \cdot \frac{1}{x} + \frac{1}{2}\right)}\]

  3. Simplified0.4

    \[\leadsto \color{blue}{x + \left(\frac{-1}{2} - \frac{\frac{1}{8}}{x}\right)}\]

  4. Final simplification0.4

    \[\leadsto x + \left(\frac{-1}{2} - \frac{\frac{1}{8}}{x}\right)\]

Reproduce

herbie shell --seed 2019151 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))