Average Error: 0.5 → 0.3
Time: 9.4s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[x - \left(\frac{0.125}{x} + 0.5\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
x - \left(\frac{0.125}{x} + 0.5\right)
double f(double x) {
        double r12296 = x;
        double r12297 = 1.0;
        double r12298 = r12296 - r12297;
        double r12299 = sqrt(r12298);
        double r12300 = sqrt(r12296);
        double r12301 = r12299 * r12300;
        return r12301;
}
↓
double f(double x) {
        double r12302 = x;
        double r12303 = 0.125;
        double r12304 = r12303 / r12302;
        double r12305 = 0.5;
        double r12306 = r12304 + r12305;
        double r12307 = r12302 - r12306;
        return r12307;
}

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

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))