Average Error: 0.5 → 0.4
Time: 2.1s
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 r2491 = x;
        double r2492 = 1.0;
        double r2493 = r2491 - r2492;
        double r2494 = sqrt(r2493);
        double r2495 = sqrt(r2491);
        double r2496 = r2494 * r2495;
        return r2496;
}
↓
double f(double x) {
        double r2497 = x;
        double r2498 = 0.5;
        double r2499 = 0.125;
        double r2500 = 1.0;
        double r2501 = r2500 / r2497;
        double r2502 = r2499 * r2501;
        double r2503 = r2498 + r2502;
        double r2504 = r2497 - r2503;
        return r2504;
}

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 2020027 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))