Average Error: 0.5 → 0.3
Time: 2.3s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[x - \left(0.5 + 0.125 \cdot \frac{1}{x}\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
x - \left(0.5 + 0.125 \cdot \frac{1}{x}\right)
double f(double x) {
        double r2839 = x;
        double r2840 = 1.0;
        double r2841 = r2839 - r2840;
        double r2842 = sqrt(r2841);
        double r2843 = sqrt(r2839);
        double r2844 = r2842 * r2843;
        return r2844;
}
↓
double f(double x) {
        double r2845 = x;
        double r2846 = 0.5;
        double r2847 = 0.125;
        double r2848 = 1.0;
        double r2849 = r2848 / r2845;
        double r2850 = r2847 * r2849;
        double r2851 = r2846 + r2850;
        double r2852 = r2845 - r2851;
        return r2852;
}

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. Final simplification0.3

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

Reproduce

herbie shell --seed 2019344 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))