Average Error: 0.5 → 0.4
Time: 4.6s
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 r2740 = x;
        double r2741 = 1.0;
        double r2742 = r2740 - r2741;
        double r2743 = sqrt(r2742);
        double r2744 = sqrt(r2740);
        double r2745 = r2743 * r2744;
        return r2745;
}
↓
double f(double x) {
        double r2746 = x;
        double r2747 = 0.125;
        double r2748 = r2747 / r2746;
        double r2749 = 0.5;
        double r2750 = r2748 + r2749;
        double r2751 = r2746 - r2750;
        return r2751;
}

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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))