Average Error: 0.5 → 0.5
Time: 9.2s
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 r11261 = x;
        double r11262 = 1.0;
        double r11263 = r11261 - r11262;
        double r11264 = sqrt(r11263);
        double r11265 = sqrt(r11261);
        double r11266 = r11264 * r11265;
        return r11266;
}
↓
double f(double x) {
        double r11267 = x;
        double r11268 = 0.5;
        double r11269 = 0.125;
        double r11270 = r11269 / r11267;
        double r11271 = r11268 + r11270;
        double r11272 = r11267 - r11271;
        return r11272;
}

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 2019195 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1.0)) (sqrt x)))