Average Error: 0.5 → 0.4
Time: 12.1s
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 r229955 = x;
        double r229956 = 1.0;
        double r229957 = r229955 - r229956;
        double r229958 = sqrt(r229957);
        double r229959 = sqrt(r229955);
        double r229960 = r229958 * r229959;
        return r229960;
}
↓
double f(double x) {
        double r229961 = x;
        double r229962 = -0.5;
        double r229963 = 0.125;
        double r229964 = r229963 / r229961;
        double r229965 = r229962 - r229964;
        double r229966 = r229961 + r229965;
        return r229966;
}

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 2019149 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))