Average Error: 0.5 → 0.3
Time: 9.9s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[x - \left(0.5 + \frac{0.125}{x}\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
x - \left(0.5 + \frac{0.125}{x}\right)
double f(double x) {
        double r364596 = x;
        double r364597 = 1.0;
        double r364598 = r364596 - r364597;
        double r364599 = sqrt(r364598);
        double r364600 = sqrt(r364596);
        double r364601 = r364599 * r364600;
        return r364601;
}
↓
double f(double x) {
        double r364602 = x;
        double r364603 = 0.5;
        double r364604 = 0.125;
        double r364605 = r364604 / r364602;
        double r364606 = r364603 + r364605;
        double r364607 = r364602 - r364606;
        return r364607;
}

Error

Bits error versus x

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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 0.3

    \[\leadsto \color{blue}{x - \left(0.125 \cdot \frac{1}{x} + 0.5\right)}\]

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019170 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1.0)) (sqrt x)))