Average Error: 0.5 → 0.4
Time: 7.3s
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 r110781 = x;
        double r110782 = 1.0;
        double r110783 = r110781 - r110782;
        double r110784 = sqrt(r110783);
        double r110785 = sqrt(r110781);
        double r110786 = r110784 * r110785;
        return r110786;
}
↓
double f(double x) {
        double r110787 = x;
        double r110788 = -0.5;
        double r110789 = 0.125;
        double r110790 = r110789 / r110787;
        double r110791 = r110788 - r110790;
        double r110792 = r110787 + r110791;
        return r110792;
}

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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019152 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))