Average Error: 0.5 → 0.3
Time: 8.4s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\left(x - \frac{\frac{1}{8}}{x}\right) - \frac{1}{2}\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
\left(x - \frac{\frac{1}{8}}{x}\right) - \frac{1}{2}
double f(double x) {
        double r167047 = x;
        double r167048 = 1.0;
        double r167049 = r167047 - r167048;
        double r167050 = sqrt(r167049);
        double r167051 = sqrt(r167047);
        double r167052 = r167050 * r167051;
        return r167052;
}
↓
double f(double x) {
        double r167053 = x;
        double r167054 = 0.125;
        double r167055 = r167054 / r167053;
        double r167056 = r167053 - r167055;
        double r167057 = 0.5;
        double r167058 = r167056 - r167057;
        return r167058;
}

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

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))