Average Error: 0.5 → 0.3
Time: 3.1s
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 r1350 = x;
        double r1351 = 1.0;
        double r1352 = r1350 - r1351;
        double r1353 = sqrt(r1352);
        double r1354 = sqrt(r1350);
        double r1355 = r1353 * r1354;
        return r1355;
}
↓
double f(double x) {
        double r1356 = x;
        double r1357 = 0.125;
        double r1358 = r1357 / r1356;
        double r1359 = 0.5;
        double r1360 = r1358 + r1359;
        double r1361 = r1356 - r1360;
        return r1361;
}

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 0.3

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))