Average Error: 0.5 → 0.3
Time: 6.1s
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 r8991 = x;
        double r8992 = 1.0;
        double r8993 = r8991 - r8992;
        double r8994 = sqrt(r8993);
        double r8995 = sqrt(r8991);
        double r8996 = r8994 * r8995;
        return r8996;
}
↓
double f(double x) {
        double r8997 = x;
        double r8998 = 0.5;
        double r8999 = 0.125;
        double r9000 = r8999 / r8997;
        double r9001 = r8998 + r9000;
        double r9002 = r8997 - r9001;
        return r9002;
}

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 0.3

    \[\leadsto \color{blue}{x - \left(0.5 + 0.125 \cdot \frac{1}{x}\right)}\]

  3. Simplified0.3

    \[\leadsto \color{blue}{x - \left(0.5 + \frac{0.125}{x}\right)}\]

  4. Final simplification0.3

    \[\leadsto x - \left(0.5 + \frac{0.125}{x}\right)\]

Reproduce

herbie shell --seed 2020045 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))