Average Error: 0.5 → 0.4
Time: 4.0s
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 r15164 = x;
        double r15165 = 1.0;
        double r15166 = r15164 - r15165;
        double r15167 = sqrt(r15166);
        double r15168 = sqrt(r15164);
        double r15169 = r15167 * r15168;
        return r15169;
}
↓
double f(double x) {
        double r15170 = x;
        double r15171 = 0.5;
        double r15172 = 0.125;
        double r15173 = 1.0;
        double r15174 = r15173 / r15170;
        double r15175 = r15172 * r15174;
        double r15176 = r15171 + r15175;
        double r15177 = r15170 - r15176;
        return r15177;
}

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

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

Reproduce

herbie shell --seed 2019353 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))