Average Error: 0.5 → 0.5
Time: 3.6s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[x - \left(0.5 + 0.125 \cdot \frac{1}{x}\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
x - \left(0.5 + 0.125 \cdot \frac{1}{x}\right)
double f(double x) {
        double r19698 = x;
        double r19699 = 1.0;
        double r19700 = r19698 - r19699;
        double r19701 = sqrt(r19700);
        double r19702 = sqrt(r19698);
        double r19703 = r19701 * r19702;
        return r19703;
}
↓
double f(double x) {
        double r19704 = x;
        double r19705 = 0.5;
        double r19706 = 0.125;
        double r19707 = 1.0;
        double r19708 = r19707 / r19704;
        double r19709 = r19706 * r19708;
        double r19710 = r19705 + r19709;
        double r19711 = r19704 - r19710;
        return r19711;
}

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

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

  3. Final simplification0.5

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

Reproduce

herbie shell --seed 2020062 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))