the quantity y=1-\sqrt{1+x^2} for x=10^{-6} is not so accurate due to numerical cancellation. indeed, the relative error associated to it is, approximately: in javascript

The numerical cancellation occurs when subtracting two very close numbers, resulting in a loss of significant digits. This can lead to inaccurate or unstable results. To avoid this, we can use an alternative method to calculate the quantity y = 1 - √(1 + x^2).

In JavaScript, here is one way to calculate the relative error associated with y for x = 10^(-6):

index.tsx
let x = 1e-6;
let y = 1 - Math.sqrt(1 + x ** 2);
let yApprox = (x ** 2) / (1 + Math.sqrt(1 + x ** 2)); // Approximation of y

let relativeError = Math.abs((y - yApprox) / yApprox);

console.log(relativeError);
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This code calculates the relative error between the exact value of y and the approximation of y using a formula that reduces numerical cancellation. By comparing the exact value and the approximation, we can measure the error as a relative difference between the two.