Average Error: 0.5 → 0.4
Time: 12.6s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[x + \left(\frac{-1}{2} - \frac{\frac{1}{8}}{x}\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
x + \left(\frac{-1}{2} - \frac{\frac{1}{8}}{x}\right)
double f(double x) {
        double r229188 = x;
        double r229189 = 1.0;
        double r229190 = r229188 - r229189;
        double r229191 = sqrt(r229190);
        double r229192 = sqrt(r229188);
        double r229193 = r229191 * r229192;
        return r229193;
}
↓
double f(double x) {
        double r229194 = x;
        double r229195 = -0.5;
        double r229196 = 0.125;
        double r229197 = r229196 / r229194;
        double r229198 = r229195 - r229197;
        double r229199 = r229194 + r229198;
        return r229199;
}

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

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

  4. Final simplification0.4

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

Reproduce

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