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COM181 Computer Hardware Lecture 1b: Floating Point

COM181 Computer Hardware Lecture 1b: Floating Point. Ian McCrum (see pages 242-258/Chapter 3 in textbook). Fixed Point n umbers using Binary.

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COM181 Computer Hardware Lecture 1b: Floating Point

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  1. COM181 Computer HardwareLecture 1b: Floating Point Ian McCrum (see pages 242-258/Chapter 3 in textbook)

  2. Fixed Point numbers using Binary • Could just use weights of ½ , ¼ , 1/8, 1/16 – e.g in an 8 bit word have the 4 bits on the left carry weights of 8421 and the 4 bits on the right carry weights of 2-1, 2-2, 2-3 and 2-4 . • Better to use scientific notation; hence we need to store both a mantissa and an exponent. i.e 3.1428 x 1042 +/- 1.01101101 x 2+/-1010101

  3. IEEE 754 standard for 32 bit floating point. • All numbers must be normalised. A normalised binary number begins with a ‘1’ • You need not store the leading ‘1’ • To accommodate negative numbers you use signed arithmetic for the mantissa (not two’s complement – this makes comparisons easier) • and a “biased scheme” for the exponent. • This allows circuitry to compare IEEE754 floating point numbers easily (though it makes it complicated for humans!)

  4. IEEE754 format • Bit 31 is a sign bit, a ‘1’ means a negative number. • Bits 30-23 carry an 8 bit exponent • Bits 22-0 carry 23 bits of mantissa, which is actually 24 bits since we know the MSB is ‘1’ • There is a 64 bit version (see textbook) • There are a number of binary patterns that cannot be a number, these are useful, e.g to represent infinity or an invalid answer (NaN)

  5. Biased Exponent • By having the 8 bit exponent be 0x00 for the most negative value comparisons are easy • Hence for values of -/+127. You add 127 to give a range of 0x00 to 0xFF. • i.e -1 is stored as (-1+127 = 126 = 0111 1110) • i.e +1 is stored as 1+127 = 128 = 1000 0000 • i.e +128 is stored as 255 = 1111 1111.

  6. Examples (from textbook) -0.75ten = -3/4 =-3/22 = -3 x 2-2 -11 x 2-2 or -1.1 x 2-1 The sign bit is ‘1’ since the mantissa is negative. The exponent is -1 so we store (-1+127=126) The mantissa is just 1 since we discard the leading ‘1’ 1 0 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 31 30………………23 22………………………………………………………. 0 1 bit 8 bits 23 bits

  7. Examples 1 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Sign bit is ‘1’ Exponent is 1000 0001, this is 129, we subtract the 127 bias to get +2 Mantissa is 01000… we add the implied ‘1’ to get 101 = 1 + 0 x ½ + 1 x ¼ = 1.25 Hence - 1.25 x 2+2 = -5.010

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