Average Error: 0.5 → 0.4
Time: 10.9s
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 r417981 = x;
        double r417982 = 1.0;
        double r417983 = r417981 - r417982;
        double r417984 = sqrt(r417983);
        double r417985 = sqrt(r417981);
        double r417986 = r417984 * r417985;
        return r417986;
}
↓
double f(double x) {
        double r417987 = x;
        double r417988 = 0.5;
        double r417989 = 0.125;
        double r417990 = r417989 / r417987;
        double r417991 = r417988 + r417990;
        double r417992 = r417987 - r417991;
        return r417992;
}

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

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

  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 2019174 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1.0)) (sqrt x)))