Average Error: 0.5 → 0.4
Time: 13.4s
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 r10186 = x;
        double r10187 = 1.0;
        double r10188 = r10186 - r10187;
        double r10189 = sqrt(r10188);
        double r10190 = sqrt(r10186);
        double r10191 = r10189 * r10190;
        return r10191;
}
↓
double f(double x) {
        double r10192 = x;
        double r10193 = 0.5;
        double r10194 = r10192 - r10193;
        double r10195 = 0.125;
        double r10196 = r10195 / r10192;
        double r10197 = r10194 - r10196;
        return r10197;
}

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 2019322 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))