Average Error: 0.5 → 0.4
Time: 19.1s
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 r29574 = x;
        double r29575 = 1.0;
        double r29576 = r29574 - r29575;
        double r29577 = sqrt(r29576);
        double r29578 = sqrt(r29574);
        double r29579 = r29577 * r29578;
        return r29579;
}
↓
double f(double x) {
        double r29580 = x;
        double r29581 = 0.5;
        double r29582 = r29580 - r29581;
        double r29583 = 0.125;
        double r29584 = r29583 / r29580;
        double r29585 = r29582 - r29584;
        return r29585;
}

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

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

  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 2019304 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))