Average Error: 0.5 → 0.4
Time: 12.8s
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 r15205 = x;
        double r15206 = 1.0;
        double r15207 = r15205 - r15206;
        double r15208 = sqrt(r15207);
        double r15209 = sqrt(r15205);
        double r15210 = r15208 * r15209;
        return r15210;
}
↓
double f(double x) {
        double r15211 = x;
        double r15212 = 0.5;
        double r15213 = r15211 - r15212;
        double r15214 = 0.125;
        double r15215 = r15214 / r15211;
        double r15216 = r15213 - r15215;
        return r15216;
}

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 2019303 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))