Average Error: 0.5 → 0.5
Time: 8.5s
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 r374107 = x;
        double r374108 = 1.0;
        double r374109 = r374107 - r374108;
        double r374110 = sqrt(r374109);
        double r374111 = sqrt(r374107);
        double r374112 = r374110 * r374111;
        return r374112;
}
↓
double f(double x) {
        double r374113 = x;
        double r374114 = 0.5;
        double r374115 = 0.125;
        double r374116 = r374115 / r374113;
        double r374117 = r374114 + r374116;
        double r374118 = r374113 - r374117;
        return r374118;
}

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)))