Average Error: 0.5 → 0.4
Time: 6.6s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\left(\frac{-1}{2} + x\right) + \frac{\frac{-1}{8}}{x}\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
\left(\frac{-1}{2} + x\right) + \frac{\frac{-1}{8}}{x}
double f(double x) {
        double r164560 = x;
        double r164561 = 1.0;
        double r164562 = r164560 - r164561;
        double r164563 = sqrt(r164562);
        double r164564 = sqrt(r164560);
        double r164565 = r164563 * r164564;
        return r164565;
}
↓
double f(double x) {
        double r164566 = -0.5;
        double r164567 = x;
        double r164568 = r164566 + r164567;
        double r164569 = -0.125;
        double r164570 = r164569 / r164567;
        double r164571 = r164568 + r164570;
        return r164571;
}

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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))