Average Error: 0.5 → 0.4
Time: 12.5s
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 r15114 = x;
        double r15115 = 1.0;
        double r15116 = r15114 - r15115;
        double r15117 = sqrt(r15116);
        double r15118 = sqrt(r15114);
        double r15119 = r15117 * r15118;
        return r15119;
}
↓
double f(double x) {
        double r15120 = x;
        double r15121 = 0.5;
        double r15122 = 0.125;
        double r15123 = r15122 / r15120;
        double r15124 = r15121 + r15123;
        double r15125 = r15120 - r15124;
        return r15125;
}

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 2019303 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))