Average Error: 0.5 → 0.5
Time: 8.9s
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 r11888 = x;
        double r11889 = 1.0;
        double r11890 = r11888 - r11889;
        double r11891 = sqrt(r11890);
        double r11892 = sqrt(r11888);
        double r11893 = r11891 * r11892;
        return r11893;
}
↓
double f(double x) {
        double r11894 = x;
        double r11895 = 0.5;
        double r11896 = 0.125;
        double r11897 = r11896 / r11894;
        double r11898 = r11895 + r11897;
        double r11899 = r11894 - r11898;
        return r11899;
}

Error

Bits error versus x

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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 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019179 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1.0)) (sqrt x)))