Average Error: 0.5 → 0.4
Time: 11.7s
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 r10128 = x;
        double r10129 = 1.0;
        double r10130 = r10128 - r10129;
        double r10131 = sqrt(r10130);
        double r10132 = sqrt(r10128);
        double r10133 = r10131 * r10132;
        return r10133;
}
↓
double f(double x) {
        double r10134 = x;
        double r10135 = 0.5;
        double r10136 = 0.125;
        double r10137 = r10136 / r10134;
        double r10138 = r10135 + r10137;
        double r10139 = r10134 - r10138;
        return r10139;
}

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 2019326 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))