Average Error: 0.5 → 0.4
Time: 13.1s
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 r18279 = x;
        double r18280 = 1.0;
        double r18281 = r18279 - r18280;
        double r18282 = sqrt(r18281);
        double r18283 = sqrt(r18279);
        double r18284 = r18282 * r18283;
        return r18284;
}
↓
double f(double x) {
        double r18285 = x;
        double r18286 = 0.125;
        double r18287 = r18286 / r18285;
        double r18288 = 0.5;
        double r18289 = r18287 + r18288;
        double r18290 = r18285 - r18289;
        return r18290;
}

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 2019305 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))