Average Error: 0.5 → 0.5
Time: 8.2s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\left(x - \frac{0.125}{x}\right) - 0.5\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
\left(x - \frac{0.125}{x}\right) - 0.5
double f(double x) {
        double r12060 = x;
        double r12061 = 1.0;
        double r12062 = r12060 - r12061;
        double r12063 = sqrt(r12062);
        double r12064 = sqrt(r12060);
        double r12065 = r12063 * r12064;
        return r12065;
}
↓
double f(double x) {
        double r12066 = x;
        double r12067 = 0.125;
        double r12068 = r12067 / r12066;
        double r12069 = r12066 - r12068;
        double r12070 = 0.5;
        double r12071 = r12069 - r12070;
        return r12071;
}

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 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1.0)) (sqrt x)))