Average Error: 0.5 → 0.4
Time: 3.5s
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 r1275 = x;
        double r1276 = 1.0;
        double r1277 = r1275 - r1276;
        double r1278 = sqrt(r1277);
        double r1279 = sqrt(r1275);
        double r1280 = r1278 * r1279;
        return r1280;
}
↓
double f(double x) {
        double r1281 = x;
        double r1282 = 0.5;
        double r1283 = 0.125;
        double r1284 = r1283 / r1281;
        double r1285 = r1282 + r1284;
        double r1286 = r1281 - r1285;
        return r1286;
}

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)))