Question

Floating point - gap

With integers, the "gap" between representations is always 1. This means that rounding a decimal to an integer can introduce an error of up to +0.5.
a) In a 32-bit IEEE floating point number, what is the smallest number greater than 0.5 that can be exactly represented? What is the gap between those two numbers? Briefly justify your answers.

Answer 1

In IEEE 32-bit floating-point format, the smallest number greater than 0.5 that can be exactly represented is 0.5 + 2^-24, and the gap between these numbers is 2^-24, approximately 5.96e-08.

Step-by-step explanation:

The question is about the representation of numbers in the IEEE 32-bit floating-point format, specifically regarding the precision and 'gap' between representable numbers immediately greater than 0.5. In a 32-bit IEEE floating point number, the smallest number greater than 0.5 that can be represented exactly is 0.5 + 2-24, because the precision of a floating-point number right after 0.5 is determined by the mantissa, which, in this case, allows for 23 bits of precision (plus one implicit bit) after the binary point. Therefore, the gap between 0.5 and the next representable number is 2-24, which is approximately 5.96e-08. This gap is determined by the least significant bit in the mantissa of the floating-point representation for numbers in this range.

Answer 2

The smallest number greater than 0.5 that can be exactly represented in a 32-bit IEEE floating point number is approximately 0.5000000596046448. The gap between 0.5 and this number is approximately 0.000000059604644775390625, which is 2 to the power of -24.

Step-by-step explanation:

The student has asked about the smallest number greater than 0.5 that can be exactly represented in a 32-bit IEEE floating point number and the gap between these two numbers. In a 32-bit IEEE floating point representation, the smallest number greater than 0.5 would be 0.5 plus the smallest increment that can be represented at this magnitude level. To compute this increment, we need to understand the structure of the floating point number: the sign bit, the exponent, and the significand (also known as mantissa).

For a 32-bit float in IEEE 754 standard, the exponent is 8 bits and the significand is 23 bits with an implicit leading bit (which is always 1 for normalized numbers). The value of 0.5 corresponds to an exponent that represents 1 (since 0.5 is 2-1) and all zeros in the significand, excluding the implicit leading bit. To find the smallest number larger than 0.5, we would increase the significand by 1 bit, which is the smallest possible increment for the significand at this exponent level.

Given the representation of 0.5 as 1.0 x 2-1, the next representable number would be 1.0...01 (with the trailing '1' being the least significant bit in the significand) x 2-1. This trailing '1' in a 23-bit significand represents 2-23. Thus we have
0.5 + 2-24 = 0.5 + 0.000000059604644775390625, which is approximately 0.5000000596046448. This means the gap or difference between the two representable numbers is 2-24, or about 0.000000059604644775390625.

Note that this is a simplified representation as representing floating-point numbers on actual hardware involves more nuance, including the potential for rounding errors.