Average Error: 0.5 → 0.4
Time: 10.2s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\frac{-1}{2} - \left(\frac{\frac{1}{8}}{x} - x\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
\frac{-1}{2} - \left(\frac{\frac{1}{8}}{x} - x\right)
double f(double x) {
        double r292805 = x;
        double r292806 = 1.0;
        double r292807 = r292805 - r292806;
        double r292808 = sqrt(r292807);
        double r292809 = sqrt(r292805);
        double r292810 = r292808 * r292809;
        return r292810;
}
↓
double f(double x) {
        double r292811 = -0.5;
        double r292812 = 0.125;
        double r292813 = x;
        double r292814 = r292812 / r292813;
        double r292815 = r292814 - r292813;
        double r292816 = r292811 - r292815;
        return r292816;
}

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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019143 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))