Average Error: 0.5 → 0.4
Time: 10.9s
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 r197133 = x;
        double r197134 = 1.0;
        double r197135 = r197133 - r197134;
        double r197136 = sqrt(r197135);
        double r197137 = sqrt(r197133);
        double r197138 = r197136 * r197137;
        return r197138;
}
↓
double f(double x) {
        double r197139 = x;
        double r197140 = 0.5;
        double r197141 = -0.125;
        double r197142 = r197141 / r197139;
        double r197143 = r197140 - r197142;
        double r197144 = r197139 - r197143;
        return r197144;
}

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 2019125 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))