Average Error: 0.5 → 0.4
Time: 3.2s
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 r10456 = x;
        double r10457 = 1.0;
        double r10458 = r10456 - r10457;
        double r10459 = sqrt(r10458);
        double r10460 = sqrt(r10456);
        double r10461 = r10459 * r10460;
        return r10461;
}
↓
double f(double x) {
        double r10462 = x;
        double r10463 = 0.5;
        double r10464 = 0.125;
        double r10465 = 1.0;
        double r10466 = r10465 / r10462;
        double r10467 = r10464 * r10466;
        double r10468 = r10463 + r10467;
        double r10469 = r10462 - r10468;
        return r10469;
}

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 2019322 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))