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 r11104 = x;
        double r11105 = 1.0;
        double r11106 = r11104 - r11105;
        double r11107 = sqrt(r11106);
        double r11108 = sqrt(r11104);
        double r11109 = r11107 * r11108;
        return r11109;
}
↓
double f(double x) {
        double r11110 = x;
        double r11111 = 0.5;
        double r11112 = 0.125;
        double r11113 = r11112 / r11110;
        double r11114 = r11111 + r11113;
        double r11115 = r11110 - r11114;
        return r11115;
}

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 2019322 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))