Average Error: 0.5 → 0.4
Time: 4.0s
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 r64 = x;
        double r65 = 1.0;
        double r66 = r64 - r65;
        double r67 = sqrt(r66);
        double r68 = sqrt(r64);
        double r69 = r67 * r68;
        return r69;
}
↓
double f(double x) {
        double r70 = x;
        double r71 = 0.5;
        double r72 = 0.125;
        double r73 = 1.0;
        double r74 = r73 / r70;
        double r75 = r72 * r74;
        double r76 = r71 + r75;
        double r77 = r70 - r76;
        return r77;
}

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. Final simplification0.4

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

Reproduce

herbie shell --seed 2020025 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))