Average Error: 0.5 → 0.4
Time: 11.5s
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 r10114 = x;
        double r10115 = 1.0;
        double r10116 = r10114 - r10115;
        double r10117 = sqrt(r10116);
        double r10118 = sqrt(r10114);
        double r10119 = r10117 * r10118;
        return r10119;
}
↓
double f(double x) {
        double r10120 = x;
        double r10121 = 0.5;
        double r10122 = 0.125;
        double r10123 = r10122 / r10120;
        double r10124 = r10121 + r10123;
        double r10125 = r10120 - r10124;
        return r10125;
}

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)))