Average Error: 0.5 → 0.3
Time: 8.1s
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 r109236 = x;
        double r109237 = 1.0;
        double r109238 = r109236 - r109237;
        double r109239 = sqrt(r109238);
        double r109240 = sqrt(r109236);
        double r109241 = r109239 * r109240;
        return r109241;
}
↓
double f(double x) {
        double r109242 = -0.5;
        double r109243 = x;
        double r109244 = r109242 + r109243;
        double r109245 = 0.125;
        double r109246 = r109245 / r109243;
        double r109247 = r109244 - r109246;
        return r109247;
}

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

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019154 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))