Average Error: 0.5 → 0.4
Time: 14.1s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\left(x - 0.5\right) - \frac{0.125}{x}\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
\left(x - 0.5\right) - \frac{0.125}{x}
double f(double x) {
        double r218561 = x;
        double r218562 = 1.0;
        double r218563 = r218561 - r218562;
        double r218564 = sqrt(r218563);
        double r218565 = sqrt(r218561);
        double r218566 = r218564 * r218565;
        return r218566;
}
↓
double f(double x) {
        double r218567 = x;
        double r218568 = 0.5;
        double r218569 = r218567 - r218568;
        double r218570 = 0.125;
        double r218571 = r218570 / r218567;
        double r218572 = r218569 - r218571;
        return r218572;
}

Error

Bits error versus x

Try it out

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

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

  4. Final simplification0.4

    \[\leadsto \left(x - 0.5\right) - \frac{0.125}{x}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1.0)) (sqrt x)))