Calculator Inputs
Results
| # | x | Standard y | Fast y | |Error| | Rel. Error | Iterations |
|---|
Formula Used
We compute the inverse square root of a positive number x as y = 1 / √x. The reference value uses high-precision Math.sqrt.
The fast method is based on the well-known technique popularized in computer graphics. It starts from a remarkable initial estimate using a bit-level transform and then applies Newton–Raphson refinement:
Given x > 0, initial guess y₀ via magic constant 0x5f3759df.
Refine k times using: y_{n+1} = y_n * (1.5 - 0.5 * x * y_n²)After each iteration, the estimate approaches 1 / √x. You control the number of refinements with Fast method iterations.
How to Use This Calculator
- Enter a positive value for x. For batch mode, paste multiple values.
- Choose decimal places and whether to compute standard, fast, or both.
- Optionally enable iteration steps to inspect Newton updates in detail.
- Click Add Row for a single value or Batch Calculate for many.
- Export your table as CSV or generate a PDF report.
Tip: Use 0 iterations to view the raw magic-constant estimate.
Frequently Asked Questions
It is the reciprocal of the square root: given x > 0, the value is 1 / √x. It appears in normalization, physics, and graphics workloads.
With one or two Newton refinements, the fast method typically matches the standard calculation to several decimal places. The table shows absolute and relative errors for transparency.
Increase iterations if you need tighter agreement with the reference value. Each Newton update roughly doubles correct digits for typical ranges of x.
x must be positive. Very large or tiny magnitudes may show floating-point limits. The calculator guards against non-positive inputs.
Comparison highlights numerical behavior, error trends, and the effect of Newton refinements. It teaches how approximation improves toward the exact result.
Reference Values (Common Inputs)
| x | 1/√x |
|---|---|
| 0.25 | 2.00000000 |
| 0.50 | 1.41421356 |
| 1.00 | 1.00000000 |
| 2.00 | 0.70710678 |
| 3.00 | 0.57735027 |
| 10.00 | 0.31622777 |
Values rounded to eight decimals for quick reference.
Fast Method Convergence (Sample Inputs)
Auto-generated comparison of the initial guess and Newton refinements against the reference value.
| x | y0 | y1 | y2 | Ref | |y2 - Ref| |
|---|
Example Data
We pre-populate the table with five values to demonstrate usage.
- 0.25 → expected y = 2
- 1 → expected y = 1
- 2 → expected y ≈ 0.70710678
- 3.5 → expected y ≈ 0.53452248
- 10 → expected y ≈ 0.31622777