13.3 Floating-point numbers, representation and manipulation

Objective

Show understanding that binary representations can give rise to rounding errors.

Introduction

Computers represent real numbers using floating-point notation. This is a way of expressing numbers with a finite number of digits, even if the number is very large or very small. However, because binary numbers have a limited number of digits, not all real numbers can be represented exactly. This leads to rounding errors.

Floating-Point Representation

Floating-point numbers are typically represented in a format similar to scientific notation. The representation consists of three parts:

• Sign bit: Indicates whether the number is positive or negative (0 for positive, 1 for negative).
• Exponent: Represents the power of 2 by which the significand is multiplied.
• Significand (Mantissa): Represents the significant digits of the number.

The standard format for floating-point numbers is IEEE 754.

IEEE 754 Single Precision (32-bit) Format

The IEEE 754 single-precision format uses 32 bits, divided as follows:

Bit Position	Sign	Exponent	Significand
31-23	Sign bit	Exponent (8 bits)	Significand (23 bits)
22-0	-	-	-

The exponent is biased to allow for both positive and negative exponents to be represented using a single set of bits. The bias for single precision is 127.

The significand is normalized, meaning it is in the form 1.xxxxx, where xxxxx is a sequence of digits.

Rounding Errors

Due to the finite number of bits available for the significand, many real numbers cannot be represented exactly in floating-point format. This results in rounding errors. These errors occur when a number cannot be represented with sufficient precision and is therefore rounded to the nearest representable value.

Consider the following example:

The decimal number 0.1 cannot be represented exactly in binary. When it is converted to a floating-point number, it is rounded to the nearest representable value, which is approximately 0.100000000000000005551115123125 in binary.

This demonstrates that even seemingly simple decimal numbers can have subtle errors when represented in binary.

Examples of Rounding Errors

1. Addition of numbers that cannot be represented exactly:

   Adding two numbers that have slightly different binary representations can result in a value that is not exactly representable. For example, adding two very close numbers like 0.1 and 0.2 might not yield exactly 0.3 due to rounding errors.

2. Multiplication of numbers that cannot be represented exactly:

   Similarly, multiplying numbers that have non-exact binary representations can also lead to rounding errors.

Mitigation Strategies

While rounding errors cannot be completely eliminated, several strategies can help to mitigate their impact:

• Using higher precision data types: Using double-precision floating-point numbers (64 bits) provides more bits for the significand, reducing the likelihood of rounding errors.
• Careful algorithm design: Some algorithms are more susceptible to rounding errors than others. Careful algorithm design can minimize the accumulation of these errors.
• Error analysis: In critical applications, it is important to analyze the potential for rounding errors and take appropriate measures.

Conclusion

Floating-point numbers are a powerful tool for representing real numbers in computers, but they are subject to rounding errors due to the finite precision of binary representation. Understanding these errors and employing mitigation strategies is crucial for ensuring the accuracy of numerical computations.