Average Error: 0.5 → 0.3
Time: 12.1s
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 r275973 = x;
        double r275974 = 1.0;
        double r275975 = r275973 - r275974;
        double r275976 = sqrt(r275975);
        double r275977 = sqrt(r275973);
        double r275978 = r275976 * r275977;
        return r275978;
}
↓
double f(double x) {
        double r275979 = x;
        double r275980 = 0.5;
        double r275981 = 0.125;
        double r275982 = r275981 / r275979;
        double r275983 = r275980 + r275982;
        double r275984 = r275979 - r275983;
        return r275984;
}

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

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1.0)) (sqrt x)))