Average Error: 0.5 → 0.4
Time: 6.6s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\left(x - 0.5\right) - \frac{0.125}{x}\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
\left(x - 0.5\right) - \frac{0.125}{x}
double f(double x) {
        double r3539 = x;
        double r3540 = 1.0;
        double r3541 = r3539 - r3540;
        double r3542 = sqrt(r3541);
        double r3543 = sqrt(r3539);
        double r3544 = r3542 * r3543;
        return r3544;
}
↓
double f(double x) {
        double r3545 = x;
        double r3546 = 0.5;
        double r3547 = r3545 - r3546;
        double r3548 = 0.125;
        double r3549 = r3548 / r3545;
        double r3550 = r3547 - r3549;
        return r3550;
}

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. Simplified0.4

    \[\leadsto \color{blue}{\left(x - 0.5\right) - \frac{0.125}{x}}\]

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))