Need Floating Point Precision Using Unsigned Int

Achieving Floating-Point Precision with Unsigned Integers in C

Explore techniques for representing and manipulating fractional numbers using unsigned integers, offering a robust alternative to native floating-point types in resource-constrained environments or when specific precision is required.

Floating-point numbers are essential for representing fractional values in programming. However, in certain scenarios, such as embedded systems, high-performance computing, or when precise control over numerical representation is needed, using native floating-point types (like float or double in C) might not be ideal. They can introduce non-deterministic behavior, require significant processing power, or consume more memory than desired. This article delves into methods for achieving floating-point precision using unsigned integers, a technique often referred to as fixed-point arithmetic.

Understanding Fixed-Point Representation

Fixed-point arithmetic represents fractional numbers by implicitly assuming a fixed position for the decimal (or binary) point within an integer. Instead of storing the exponent and mantissa separately, as floating-point numbers do, fixed-point numbers store a scaled integer value. For example, to represent numbers with two decimal places, you could store the value multiplied by 100. So, 1.23 would be stored as 123. The key is to consistently apply the scaling factor during all operations.

flowchart TD
    A[Real Number] --> B{Choose Scaling Factor (e.g., 100)};
    B --> C[Multiply Real Number by Scaling Factor];
    C --> D[Round to Nearest Integer];
    D --> E[Store as Integer (Fixed-Point)];
    E --> F{Perform Integer Arithmetic};
    F --> G[Divide Result by Scaling Factor];
    G --> H[Obtain Real Number Result];

Workflow for Fixed-Point Number Representation and Arithmetic

Implementing Fixed-Point Arithmetic with Unsigned Integers

Let's consider representing numbers with a fixed number of fractional bits. A common approach is to use a Q format, such as Qm.n, where m is the number of integer bits and n is the number of fractional bits. For an unsigned integer, the total number of bits is m + n. The value represented by an integer I in Qn format (meaning n fractional bits) is I / 2^n. This allows for efficient bitwise operations for multiplication and division by powers of two.

#include <stdio.h>
#include <stdint.h>

// Define the number of fractional bits (Q-format: Q0.16)
#define FRAC_BITS 16
#define SCALE_FACTOR (1U << FRAC_BITS)

// Convert a float to fixed-point unsigned integer
uint32_t float_to_fixed(float val) {
    return (uint32_t)(val * SCALE_FACTOR);
}

// Convert a fixed-point unsigned integer to float
float fixed_to_float(uint32_t val) {
    return (float)val / SCALE_FACTOR;
}

// Fixed-point addition
uint32_t fixed_add(uint32_t a, uint32_t b) {
    return a + b;
}

// Fixed-point subtraction
uint32_t fixed_sub(uint32_t a, uint32_t b) {
    return a - b;
}

// Fixed-point multiplication
uint32_t fixed_mul(uint32_t a, uint32_t b) {
    // Multiply as 64-bit to prevent overflow before shifting
    return (uint32_t)(((uint64_t)a * b) >> FRAC_BITS);
}

// Fixed-point division
uint32_t fixed_div(uint32_t a, uint32_t b) {
    // Shift 'a' left before division to maintain precision
    return (uint32_t)(((uint64_t)a << FRAC_BITS) / b);
}

int main() {
    float f1 = 1.5f;
    float f2 = 0.75f;

    uint32_t fixed1 = float_to_fixed(f1);
    uint32_t fixed2 = float_to_fixed(f2);

    printf("Float 1: %.4f -> Fixed 1: %u\n", f1, fixed1);
    printf("Float 2: %.4f -> Fixed 2: %u\n", f2, fixed2);

    uint32_t fixed_sum = fixed_add(fixed1, fixed2);
    printf("Fixed Sum: %u -> Float Sum: %.4f\n", fixed_sum, fixed_to_float(fixed_sum));

    uint32_t fixed_product = fixed_mul(fixed1, fixed2);
    printf("Fixed Product: %u -> Float Product: %.4f\n", fixed_product, fixed_to_float(fixed_product));

    uint32_t fixed_quotient = fixed_div(fixed1, fixed2);
    printf("Fixed Quotient: %u -> Float Quotient: %.4f\n", fixed_quotient, fixed_to_float(fixed_quotient));

    return 0;
}

C code demonstrating basic fixed-point arithmetic operations using uint32_t.

Considerations and Trade-offs

While fixed-point arithmetic offers benefits like deterministic behavior and potentially higher performance on certain architectures, it comes with its own set of challenges:

• Overflow/Underflow: You must carefully manage the range of your numbers to prevent overflow during intermediate calculations, especially in multiplication and division. Using wider integer types (e.g., uint64_t for intermediate products) is a common strategy.
• Precision Loss: Fixed-point numbers have a constant absolute precision, unlike floating-point numbers which have a constant relative precision. This means that for very small numbers, fixed-point can offer higher relative precision, but for very large numbers, it might lose significant relative precision.
• Complexity: Implementing fixed-point arithmetic requires more manual management of scaling factors and bit shifts, increasing code complexity compared to using native floating-point types.
• Square Root and Transcendental Functions: Implementing functions like sqrt, sin, cos, log, etc., for fixed-point numbers is significantly more complex and often requires lookup tables or iterative algorithms.

Choosing the Right Approach

The decision to use fixed-point arithmetic over floating-point depends heavily on the application's requirements. Fixed-point is often preferred in:

• Embedded Systems: Where hardware floating-point units are absent or slow, and memory is limited.
• Digital Signal Processing (DSP): For audio processing, image filtering, and control systems where predictable behavior and specific precision are crucial.
• Financial Calculations: Where exact decimal representation is paramount, though often decimal fixed-point (base 10 scaling) is used rather than binary fixed-point.

For general-purpose applications on modern processors with FPU support, native floating-point types are usually more convenient and performant.