Average Error: 0.5 → 0.4
Time: 9.7s
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 r15829 = x;
        double r15830 = 1.0;
        double r15831 = r15829 - r15830;
        double r15832 = sqrt(r15831);
        double r15833 = sqrt(r15829);
        double r15834 = r15832 * r15833;
        return r15834;
}
↓
double f(double x) {
        double r15835 = x;
        double r15836 = 0.5;
        double r15837 = 0.125;
        double r15838 = r15837 / r15835;
        double r15839 = r15836 + r15838;
        double r15840 = r15835 - r15839;
        return r15840;
}

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 2020043 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))