Average Error: 0.5 → 0.4
Time: 10.8s
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 r10850 = x;
        double r10851 = 1.0;
        double r10852 = r10850 - r10851;
        double r10853 = sqrt(r10852);
        double r10854 = sqrt(r10850);
        double r10855 = r10853 * r10854;
        return r10855;
}
↓
double f(double x) {
        double r10856 = x;
        double r10857 = 0.5;
        double r10858 = 0.125;
        double r10859 = r10858 / r10856;
        double r10860 = r10857 + r10859;
        double r10861 = r10856 - r10860;
        return r10861;
}

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 2019323 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))