Average Error: 0.5 → 0.3
Time: 7.9s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\frac{\frac{-1}{8}}{x} + \left(x + \frac{-1}{2}\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
\frac{\frac{-1}{8}}{x} + \left(x + \frac{-1}{2}\right)
double f(double x) {
        double r107066 = x;
        double r107067 = 1.0;
        double r107068 = r107066 - r107067;
        double r107069 = sqrt(r107068);
        double r107070 = sqrt(r107066);
        double r107071 = r107069 * r107070;
        return r107071;
}
↓
double f(double x) {
        double r107072 = -0.125;
        double r107073 = x;
        double r107074 = r107072 / r107073;
        double r107075 = -0.5;
        double r107076 = r107073 + r107075;
        double r107077 = r107074 + r107076;
        return r107077;
}

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

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

  4. Final simplification0.3

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

Reproduce

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