Average Error: 0.5 → 0.4
Time: 12.3s
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 r12195 = x;
        double r12196 = 1.0;
        double r12197 = r12195 - r12196;
        double r12198 = sqrt(r12197);
        double r12199 = sqrt(r12195);
        double r12200 = r12198 * r12199;
        return r12200;
}
↓
double f(double x) {
        double r12201 = x;
        double r12202 = 0.5;
        double r12203 = 0.125;
        double r12204 = r12203 / r12201;
        double r12205 = r12202 + r12204;
        double r12206 = r12201 - r12205;
        return r12206;
}

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

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

  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 2019199 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1.0)) (sqrt x)))