Average Error: 0.5 → 0.4
Time: 9.1s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\left(x - 0.5\right) - \frac{0.125}{x}\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
\left(x - 0.5\right) - \frac{0.125}{x}
double f(double x) {
        double r223398 = x;
        double r223399 = 1.0;
        double r223400 = r223398 - r223399;
        double r223401 = sqrt(r223400);
        double r223402 = sqrt(r223398);
        double r223403 = r223401 * r223402;
        return r223403;
}
↓
double f(double x) {
        double r223404 = x;
        double r223405 = 0.5;
        double r223406 = r223404 - r223405;
        double r223407 = 0.125;
        double r223408 = r223407 / r223404;
        double r223409 = r223406 - r223408;
        return r223409;
}

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

    \[\leadsto \color{blue}{x - \left(0.125 \cdot \frac{1}{x} + 0.5\right)}\]

  3. Simplified0.4

    \[\leadsto \color{blue}{\left(x - 0.5\right) - \frac{0.125}{x}}\]

  4. Final simplification0.4

    \[\leadsto \left(x - 0.5\right) - \frac{0.125}{x}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1.0)) (sqrt x)))