Average Error: 0.5 → 0.4
Time: 2.2s
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 r2387 = x;
        double r2388 = 1.0;
        double r2389 = r2387 - r2388;
        double r2390 = sqrt(r2389);
        double r2391 = sqrt(r2387);
        double r2392 = r2390 * r2391;
        return r2392;
}
↓
double f(double x) {
        double r2393 = x;
        double r2394 = 0.5;
        double r2395 = 0.125;
        double r2396 = 1.0;
        double r2397 = r2396 / r2393;
        double r2398 = r2395 * r2397;
        double r2399 = r2394 + r2398;
        double r2400 = r2393 - r2399;
        return r2400;
}

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 2020083 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))