Average Error: 0.5 → 0.4
Time: 8.5s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[x - \left(\frac{1}{2} + \frac{\frac{1}{8}}{x}\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
x - \left(\frac{1}{2} + \frac{\frac{1}{8}}{x}\right)
double f(double x) {
        double r158144 = x;
        double r158145 = 1.0;
        double r158146 = r158144 - r158145;
        double r158147 = sqrt(r158146);
        double r158148 = sqrt(r158144);
        double r158149 = r158147 * r158148;
        return r158149;
}
↓
double f(double x) {
        double r158150 = x;
        double r158151 = 0.5;
        double r158152 = 0.125;
        double r158153 = r158152 / r158150;
        double r158154 = r158151 + r158153;
        double r158155 = r158150 - r158154;
        return r158155;
}

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

    \[\leadsto \color{blue}{x - \left(\frac{1}{8} \cdot \frac{1}{x} + \frac{1}{2}\right)}\]

  3. Simplified0.4

    \[\leadsto \color{blue}{x - \left(\frac{1}{2} + \frac{\frac{1}{8}}{x}\right)}\]

  4. Final simplification0.4

    \[\leadsto x - \left(\frac{1}{2} + \frac{\frac{1}{8}}{x}\right)\]

Reproduce

herbie shell --seed 2019129 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))