Average Error: 0.5 → 0.4
Time: 14.7s
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 r16845 = x;
        double r16846 = 1.0;
        double r16847 = r16845 - r16846;
        double r16848 = sqrt(r16847);
        double r16849 = sqrt(r16845);
        double r16850 = r16848 * r16849;
        return r16850;
}
↓
double f(double x) {
        double r16851 = x;
        double r16852 = 0.5;
        double r16853 = 0.125;
        double r16854 = r16853 / r16851;
        double r16855 = r16852 + r16854;
        double r16856 = r16851 - r16855;
        return r16856;
}

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 2019305 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))