Average Error: 0.5 → 0.4
Time: 2.8s
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 r1337 = x;
        double r1338 = 1.0;
        double r1339 = r1337 - r1338;
        double r1340 = sqrt(r1339);
        double r1341 = sqrt(r1337);
        double r1342 = r1340 * r1341;
        return r1342;
}
↓
double f(double x) {
        double r1343 = x;
        double r1344 = 0.125;
        double r1345 = r1344 / r1343;
        double r1346 = 0.5;
        double r1347 = r1345 + r1346;
        double r1348 = r1343 - r1347;
        return r1348;
}

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 2020047 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))