Average Error: 0.5 → 0.4
Time: 2.8s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[x - \left(0.5 + \frac{0.125}{x}\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
x - \left(0.5 + \frac{0.125}{x}\right)
double f(double x) {
        double r1318 = x;
        double r1319 = 1.0;
        double r1320 = r1318 - r1319;
        double r1321 = sqrt(r1320);
        double r1322 = sqrt(r1318);
        double r1323 = r1321 * r1322;
        return r1323;
}
↓
double f(double x) {
        double r1324 = x;
        double r1325 = 0.5;
        double r1326 = 0.125;
        double r1327 = r1326 / r1324;
        double r1328 = r1325 + r1327;
        double r1329 = r1324 - r1328;
        return r1329;
}

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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2020047 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))