What would be the IEEE 754 double precision floating point representation of 1.3

Question

What would be the IEEE 754 double precision floating point representation of 1.3248735989328012498123 3898124124 × 10-17. For explanation, I want you to document the steps you perform, in this order: (1) What is n in decimal fixed point form (ddd.ddddd); (2) What is n in binary fixed point form (bbb.bbbb), storing the first 110 bits following the binary point); (3) What is the normalized binary number, written in the form 1.bbbbb...bbb × 2e , storing 54 bits following the binary point) (4) What are the 52 mantissa bits, after the bits in bit positions -53, -54, ... are eliminated using the round to nearest, ties to even mode; exclude the 1. part; (5) What is the biased exponent in decimal and in binary? (6) Write the 64-bits of the number in the order: s e m; and (7) Write the final answer as a 16- hexdigit number

Explanation / Answer

There are several ways to represent real numbers on computers. Fixed point places a radix point somewhere in the middle of the digits, and is equivalent to using integers that represent portions of some unit. For example, one might represent 1/100ths of a unit; if you have four decimal digits, you could represent 10.82, or 00.01. Another approach is to use rationals, and represent every number as the ratio of two integers.

Floating-point representation – the most common solution – uses scientific notation to encode numbers, with a base number and an exponent. For example, 123.456 could be represented as 1.23456 × 102. In hexadecimal, the number 123.abc might be represented as 1.23abc × 162. In binary, the number 10100.110 could be represented as 1.0100110 × 24.

Floating-point solves a number of representation problems. Fixed-point has a fixed window of representation, which limits it from representing very large or very small numbers. Also, fixed-point is prone to a loss of precision when two large numbers are divided.

Floating-point, on the other hand, employs a sort of "sliding window" of precision appropriate to the scale of the number. This allows it to represent numbers from 1,000,000,000,000 to 0.0000000000000001 with ease, and while maximizing precision (the number of digits) at both ends of the scale.

Storage Layout

IEEE floating point numbers have three basic components: the sign, the exponent, and the mantissa. The mantissa is composed of the fraction and an implicit leading digit (explained below). The exponent base (2) is implicit and need not be stored.

The following figure shows the layout for single (32-bit) and double (64-bit) precision floating-point values. The number of bits for each field are shown (bit ranges are in square brackets, 00 = least-significant bit):

Laid out as bits, floating point numbers look like this:

The Sign Bit

The sign bit is as simple as it gets. 0 denotes a positive number, and 1 denotes a negative number. Flipping the value of this bit flips the sign of the number.

The Exponent

The exponent field needs to represent both positive and negative exponents. To do this, a bias is added to the actual exponent in order to get the stored exponent. For IEEE single-precision floats, this value is 127. Thus, an exponent of zero means that 127 is stored in the exponent field. A stored value of 200 indicates an exponent of (200–127), or 73. For reasons discussed later, exponents of 127 (all 0s) and +128 (all 1s) are reserved for special numbers.

For double precision, the exponent field is 11 bits, and has a bias of 1023.

The Mantissa

The mantissa, also known as the significand, represents the precision bits of the number. It is composed of an implicit leading bit (left of the radix point) and the fraction bits (to the right of the radix point).

To find out the value of the implicit leading bit, consider that any number can be expressed in scientific notation in many different ways

Floating Point Components Sign Exponent Fraction Single Precision 1 [31] 8 [30–23] 23 [22–00] Double Precision 1 [63] 11 [62–52] 52 [51–00]

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