CORDIC Algorithm Calculator

Interactive CORDIC Calculator

Enter the angle for sine/cosine or rotation.

Please enter a valid angle.

The starting (x, y) coordinates of the vector to be rotated.

What is a CORDIC Calculator?

CORDIC, an acronym for **COordinate Rotation DIgital Computer**, is a simple and hardware-efficient algorithm used to calculate a wide range of transcendental functions, including trigonometric and hyperbolic functions. A **calculator design using CORDIC** leverages this algorithm to perform complex math without needing hardware multipliers. Instead, it relies on iterative additions, subtractions, and bit-shift operations, which are much faster and cheaper to implement in digital circuits like FPGAs and microcontrollers. This method was first described by Jack Volder in 1959 and was fundamental to the design of early digital calculators.

This type of calculator is primarily used by engineers, computer scientists, and students working with low-power embedded systems or hardware design. A common misunderstanding is that modern computers still rely heavily on CORDIC for all trig functions; while many CPUs have dedicated floating-point instructions, CORDIC remains essential in applications where resource constraints are critical. You can learn more about its applications in our guide to Digital Signal Processing.

The CORDIC Formula and Explanation

The CORDIC algorithm operates in one of two modes: rotation or vectoring. This calculator primarily uses the **rotation mode**. The goal is to rotate a starting vector (x₀, y₀) by a target angle θ to find its new coordinates (xₙ, yₙ). The core idea is to decompose the target angle θ into a series of smaller, predefined angles, and apply a micro-rotation for each.

The iterative formulas for the circular rotation mode are:

x_i+1 = x_i – d_i * y_i * 2⁻ⁱ
y_i+1 = y_i + d_i * x_i * 2⁻ⁱ
z_i+1 = z_i – d_i * arctan(2⁻ⁱ)

The term d_i determines the direction of rotation at each step. The multiplication by 2⁻ⁱ is a simple binary bit-shift, avoiding a costly multiplication operation. The process continues until the angle accumulator z approaches zero.

Formula Variables

Variable	Meaning	Unit	Typical Range
x_i, y_i	Vector coordinates at iteration i	Unitless (or same as input)	Depends on initial vector
z_i	Remaining angle to rotate	Radians	-π to +π
d_i	Direction of micro-rotation (+1 or -1)	N/A	-1, 1
2⁻ⁱ	The “shift” amount, replaces tan(θ)	Unitless	Approaches 0 as i increases
arctan(2⁻ⁱ)	Pre-computed angle for each micro-rotation	Radians	Pre-calculated lookup table

One artifact of this process is that the vector’s magnitude grows at each step. The total gain approaches a constant value of approximately 1.647 after many iterations. The final vector must be scaled down by this gain factor to get the correct result. More details on this can be found in our article on Fixed-Point Arithmetic.

Practical Examples

Example 1: Calculate sin(45°) and cos(45°)

To calculate the sine and cosine, we start with a unit vector on the X-axis and rotate it.

• Inputs: Angle = 45°, Initial Vector = (1, 0), Iterations = 16
• Process: The calculator will rotate the vector (1, 0) by 45° using the iterative CORDIC method.
• Results (approximate):
  • cos(45°) ≈ 0.7071
  • sin(45°) ≈ 0.7071
  • Final Vector: (0.7071, 0.7071)

Example 2: Rotate a Vector by 90°

Here, we apply the CORDIC algorithm to an arbitrary vector.

• Inputs: Angle = 90°, Initial Vector = (2, 3), Iterations = 16
• Process: The algorithm rotates the vector (2, 3) counter-clockwise by 90°.
• Results (approximate):
  • Final Vector: (-3, 2)

How to Use This calculator design using cordic

1. Select Function: Choose ‘Sine & Cosine’ to find trig values or ‘Vector Rotation’ to rotate a specific point.
2. Enter Angle: Input the desired angle and select whether it’s in ‘Degrees’ or ‘Radians’.
3. Provide Vector (if applicable): If you chose ‘Vector Rotation’, enter the starting X and Y coordinates.
4. Set Iterations: Choose the number of iterations. 16 is a good balance of speed and precision for most applications.
5. Calculate: Click the ‘Calculate’ button to see the results. The primary result shows the final values, while the table and chart provide a detailed breakdown of the CORDIC process. For more on interpreting hardware-level calculations, see our Embedded Systems Design Guide.

Key Factors That Affect calculator design using cordic

• Number of Iterations: More iterations lead to higher precision but require more computation time. The precision increases by about one bit per iteration.
• Numerical Representation: CORDIC is often implemented using fixed-point arithmetic to save hardware resources, which can introduce quantization errors. Our Verilog HDL Introduction covers this topic in more detail.
• Gain Factor (K): The algorithm inherently scales the vector’s magnitude. This gain must be corrected for, either by pre-scaling the input or post-scaling the output.
• Angle Range: The basic CORDIC algorithm is limited to a specific range (e.g., -90° to +90°). Angles outside this range require initial quadrant correction.
• Lookup Table (LUT) Size: The pre-computed arctangent values are stored in a LUT. The size of this table impacts hardware resource usage.
• Coordinate System: CORDIC can be adapted for circular, linear, and hyperbolic coordinate systems to calculate different sets of functions.

Frequently Asked Questions (FAQ)

Why use CORDIC instead of Taylor series?

CORDIC avoids complex multiplications, making it much more efficient for hardware with limited resources, like early calculators or FPGAs.

What does the gain ‘K’ mean?

The K-factor is a constant scaling artifact of the CORDIC process. Because each micro-rotation slightly increases the vector’s length, the final vector is longer than the original. K is the total scaling factor, which is divided out at the end to get the correct magnitude.

Are the units for x and y important?

The units for the vector coordinates (x, y) are relative. If you input a vector in meters, the output will also be in meters. The algorithm is unit-agnostic.

How do I change the angle from degrees to radians?

Use the ‘Unit’ dropdown next to the Angle input field. The calculator handles the conversion automatically before starting the calculation.

Can this calculator handle hyperbolic functions?

This specific **calculator design using cordic** is configured for the circular system (sine, cosine). A different set of iterative equations is needed for hyperbolic functions (sinh, cosh).

What is an “iteration”?

An iteration is a single step in the CORDIC algorithm, consisting of a micro-rotation, an angle update, and shift/add operations. Explore more iterative methods in our article on Numerical Analysis Techniques.

Is the chart always a circle?

The chart shows the path on a 2D plane. If the initial vector lies on the unit circle, the final vector will also lie on the unit circle (after gain correction), but the intermediate steps will be slightly outside of it.

Why is the final result sometimes slightly inaccurate?

The accuracy is limited by the number of iterations and the use of standard floating-point numbers in JavaScript. CORDIC is an approximation algorithm, but its precision can be increased with more iterations.