Average Error: 0.5 → 0.4
Time: 4.6s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\left(x - 0.5\right) - \frac{0.125}{x}\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
\left(x - 0.5\right) - \frac{0.125}{x}
double f(double x) {
        double r9907 = x;
        double r9908 = 1.0;
        double r9909 = r9907 - r9908;
        double r9910 = sqrt(r9909);
        double r9911 = sqrt(r9907);
        double r9912 = r9910 * r9911;
        return r9912;
}
↓
double f(double x) {
        double r9913 = x;
        double r9914 = 0.5;
        double r9915 = r9913 - r9914;
        double r9916 = 0.125;
        double r9917 = r9916 / r9913;
        double r9918 = r9915 - r9917;
        return r9918;
}

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. Final simplification0.4

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

Reproduce

herbie shell --seed 2019308 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))