Average Error: 0.5 → 0.4
Time: 1.9s
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 r1196 = x;
        double r1197 = 1.0;
        double r1198 = r1196 - r1197;
        double r1199 = sqrt(r1198);
        double r1200 = sqrt(r1196);
        double r1201 = r1199 * r1200;
        return r1201;
}
↓
double f(double x) {
        double r1202 = x;
        double r1203 = 0.5;
        double r1204 = 0.125;
        double r1205 = 1.0;
        double r1206 = r1205 / r1202;
        double r1207 = r1204 * r1206;
        double r1208 = r1203 + r1207;
        double r1209 = r1202 - r1208;
        return r1209;
}

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 2019356 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))