Average Error: 0.5 → 0.4
Time: 8.8s
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 r12005 = x;
        double r12006 = 1.0;
        double r12007 = r12005 - r12006;
        double r12008 = sqrt(r12007);
        double r12009 = sqrt(r12005);
        double r12010 = r12008 * r12009;
        return r12010;
}
↓
double f(double x) {
        double r12011 = x;
        double r12012 = 0.5;
        double r12013 = 0.125;
        double r12014 = r12013 / r12011;
        double r12015 = r12012 + r12014;
        double r12016 = r12011 - r12015;
        return r12016;
}

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 0.4

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

  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 2019194 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1.0)) (sqrt x)))