Average Error: 0.5 → 0.3
Time: 9.7s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[x - \left(\frac{0.125}{x} + 0.5\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
x - \left(\frac{0.125}{x} + 0.5\right)
double f(double x) {
        double r17638 = x;
        double r17639 = 1.0;
        double r17640 = r17638 - r17639;
        double r17641 = sqrt(r17640);
        double r17642 = sqrt(r17638);
        double r17643 = r17641 * r17642;
        return r17643;
}
↓
double f(double x) {
        double r17644 = x;
        double r17645 = 0.125;
        double r17646 = r17645 / r17644;
        double r17647 = 0.5;
        double r17648 = r17646 + r17647;
        double r17649 = r17644 - r17648;
        return r17649;
}

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 0.3

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))