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ECT 358

ECT 358. Lecture 16 Floating Point. Whoever gossips to you will be a gossip of you. A gossip betrays a confidence,        but a trustworthy man keeps a secret. Proverbs 11:13. Fixed and Floating Point. Fixed Point Unsigned Signed Floating Point. 1101 (13) 10 Multiplicand M

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ECT 358

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  1. ECT 358 Lecture 16 Floating Point

  2. Whoever gossips to you will be a gossip of you. A gossip betrays a confidence,        but a trustworthy man keeps a secret. Proverbs 11:13

  3. Fixed and Floating Point • Fixed Point • Unsigned • Signed • Floating Point

  4. 1101 (13)10 Multiplicand M 1011 (11)10 Multiplier Q 1101 1101 Partial Products 0000 1101 10001111 (143)10 Product Binary Multiplication • Similar to algorithm used to multiply by hand • Multiplying two n-bit numbers can result in a 2n-bit product

  5. Binary Multiplication B[1] B[0] A[1] A[0] A[0]B[1] A[0]B[0] A[1]B[1] A[1]B[0] _____________________________ S[3] S[2] S[1] S[0] NOTE: S[0] = A[0]&&B[0] S[1] = A[0]&&B[1]^ A[1]&&B[0] S[2] = A[1]&&B[1]^A[0]&&B[1]&& A[1]&&B[0] S[3] = A[1]&&B[1]&&A[0]&&B[1]&& A[1]&&B[0]

  6. Floating Point Numbers • IEEE Format • Single precision • The sign is for the fraction • Exponent uses excess 127 so all exponents are positive Exponent (8 bits) Fraction (23 bits) Sign (1 bits)

  7. S Exponent Fraction 0 0 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 3 E A A A A A B Floating Point Numbers Example • Convert from decimal to floating point in hex • Convert decimal into binary fixed point • .333333 = .010101010101010101… • Normalize (shift until in the form of 1.nnnn) • Shift 2 places left • 1.010101010101010… x2-2 • Remove the leading 1 • .01010101010101… x2-2 • Fraction is 01010101010101… • Exponent is -2 + 127 = 12510 and in binary 0111 11012 • Sign is 0 (positive fraction)

  8. Floating Point Numbers Example • Convert from Hex floating point to Decimal • Convert Hex into binary fixed point • 4234800016 = 0100 0010 0011 0100 1000 0000 0000 0000 • Exponent = 132 – 127 = 5 • Fraction = .01101001000000 • Return the leading 1 • Fraction = 1.011010010000000 • Locate binary point (shift right 5 places) • Fixed point Binary 0010 1101 . 0010 0000 • Convert from fixed point binary to decimal • Decimal = 45.125 Exponent Fraction

  9. Floating Point Numbers Examples • Convert 1.3333333 to Hex floating point • Answer 3FAA AAAB • Convert 17.0 to Hex floating point • Answer 4188 0000 • Convert 4234 8000 to Decimal floating point • Answer 45.125 • Convert 42E4 8000 to Decimal floating point • Answer 114.25

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