Average Error: 0.5 → 0.4
Time: 16.1s
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 r444656 = x;
        double r444657 = 1.0;
        double r444658 = r444656 - r444657;
        double r444659 = sqrt(r444658);
        double r444660 = sqrt(r444656);
        double r444661 = r444659 * r444660;
        return r444661;
}
↓
double f(double x) {
        double r444662 = x;
        double r444663 = -0.5;
        double r444664 = 0.125;
        double r444665 = r444664 / r444662;
        double r444666 = r444663 - r444665;
        double r444667 = r444662 + r444666;
        return r444667;
}

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 2019142 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))