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floating-pointieee

How does 2.0 + 3.0 = 4.0 using a 5-bit floating point representation based on the IEEE format and round-to-even?

I am confused as to how 2.0 + 3.0 = 4.0 when rounding-to-even for this IEEE-based format of representing floating point numbers.

https://i.sstatic.net/P8LIR.jpg

Specifically the last part. The second to last one is also slightly confusing.

Could I please have an explanation? Thanks!

Solution

  • 2.0 is represented as 1.0 E 1 (1 * 2^1) (I'm writing mantissas in base 2.)

    3.0 is represented as 1.1 E 1 (3/2 * 2^1)

    Adding them together, you get an unnormalized mantissa of 10.1, which gets normalized to 1.01, and the exponent becomes 2^2. But because there is only room to represent 1 decimal place of the mantissa, we have a choice between using 1.0 and 1.1 for the mantissa, and round-to-even says we choose the "even" mantissa, which gets us to 1.0 E 2 (1 * 2^2, or 4.0).