Average Error: 0.5 → 0.5
Time: 8.0s
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 r11999 = x;
        double r12000 = 1.0;
        double r12001 = r11999 - r12000;
        double r12002 = sqrt(r12001);
        double r12003 = sqrt(r11999);
        double r12004 = r12002 * r12003;
        return r12004;
}
↓
double f(double x) {
        double r12005 = x;
        double r12006 = 0.5;
        double r12007 = 0.125;
        double r12008 = r12007 / r12005;
        double r12009 = r12006 + r12008;
        double r12010 = r12005 - r12009;
        return r12010;
}

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 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1.0)) (sqrt x)))