Average Error: 0.5 → 0.4
Time: 7.8s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\left(x - \frac{\frac{1}{8}}{x}\right) - \frac{1}{2}\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
\left(x - \frac{\frac{1}{8}}{x}\right) - \frac{1}{2}
double f(double x) {
        double r29884 = x;
        double r29885 = 1.0;
        double r29886 = r29884 - r29885;
        double r29887 = sqrt(r29886);
        double r29888 = sqrt(r29884);
        double r29889 = r29887 * r29888;
        return r29889;
}
↓
double f(double x) {
        double r29890 = x;
        double r29891 = 1.0;
        double r29892 = 8.0;
        double r29893 = r29891 / r29892;
        double r29894 = r29893 / r29890;
        double r29895 = r29890 - r29894;
        double r29896 = 2.0;
        double r29897 = r29891 / r29896;
        double r29898 = r29895 - r29897;
        return r29898;
}

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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019304 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))