Average Error: 0.5 → 0.3
Time: 9.1s
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 r357593 = x;
        double r357594 = 1.0;
        double r357595 = r357593 - r357594;
        double r357596 = sqrt(r357595);
        double r357597 = sqrt(r357593);
        double r357598 = r357596 * r357597;
        return r357598;
}
↓
double f(double x) {
        double r357599 = x;
        double r357600 = 0.5;
        double r357601 = 0.125;
        double r357602 = r357601 / r357599;
        double r357603 = r357600 + r357602;
        double r357604 = r357599 - r357603;
        return r357604;
}

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 2019171 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1.0)) (sqrt x)))