Average Error: 0.5 → 0.3
Time: 13.4s
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 r275943 = x;
        double r275944 = 1.0;
        double r275945 = r275943 - r275944;
        double r275946 = sqrt(r275945);
        double r275947 = sqrt(r275943);
        double r275948 = r275946 * r275947;
        return r275948;
}
↓
double f(double x) {
        double r275949 = x;
        double r275950 = 0.5;
        double r275951 = 0.125;
        double r275952 = r275951 / r275949;
        double r275953 = r275950 + r275952;
        double r275954 = r275949 - r275953;
        return r275954;
}

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 2019171 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1.0)) (sqrt x)))