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 r11266 = x;
        double r11267 = 1.0;
        double r11268 = r11266 - r11267;
        double r11269 = sqrt(r11268);
        double r11270 = sqrt(r11266);
        double r11271 = r11269 * r11270;
        return r11271;
}
↓
double f(double x) {
        double r11272 = x;
        double r11273 = 0.5;
        double r11274 = 0.125;
        double r11275 = r11274 / r11272;
        double r11276 = r11273 + r11275;
        double r11277 = r11272 - r11276;
        return r11277;
}

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 2019306 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))