Average Error: 0.5 → 0.4
Time: 13.3s
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 r27247540 = x;
double r27247541 = 1.0;
double r27247542 = r27247540 - r27247541;
double r27247543 = sqrt(r27247542);
double r27247544 = sqrt(r27247540);
double r27247545 = r27247543 * r27247544;
return r27247545;
}


double f(double x) {
double r27247546 = x;
double r27247547 = 0.5;
double r27247548 = r27247546 - r27247547;
double r27247549 = 0.125;
double r27247550 = r27247549 / r27247546;
double r27247551 = r27247548 - r27247550;
return r27247551;
}



Try it out

Results

 In Out
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 1
(FPCore (x)
:name "sqrt(x-1) * sqrt(x)"
(* (sqrt (- x 1.0)) (sqrt x)))