Average Error: 0.5 → 0.4
Time: 7.9s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\left(x - \frac{\frac{1}{8}}{x}\right) - \frac{1}{2}\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
\left(x - \frac{\frac{1}{8}}{x}\right) - \frac{1}{2}
double f(double x) {
        double r108661 = x;
        double r108662 = 1.0;
        double r108663 = r108661 - r108662;
        double r108664 = sqrt(r108663);
        double r108665 = sqrt(r108661);
        double r108666 = r108664 * r108665;
        return r108666;
}
↓
double f(double x) {
        double r108667 = x;
        double r108668 = 0.125;
        double r108669 = r108668 / r108667;
        double r108670 = r108667 - r108669;
        double r108671 = 0.5;
        double r108672 = r108670 - r108671;
        return r108672;
}

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

    \[\leadsto \color{blue}{x - \left(\frac{1}{8} \cdot \frac{1}{x} + \frac{1}{2}\right)}\]

  3. Simplified0.4

    \[\leadsto \color{blue}{\left(x - \frac{\frac{1}{8}}{x}\right) - \frac{1}{2}}\]

  4. Final simplification0.4

    \[\leadsto \left(x - \frac{\frac{1}{8}}{x}\right) - \frac{1}{2}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))