cs3330-chap3-arith-2

cs3330-chap3-arith-2 - 1 CS/ECE 3330 CS/ECE 3330 Computer...

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Unformatted text preview: 1 CS/ECE 3330 CS/ECE 3330 Computer Architecture Chapter 3 Floating Point ¡ Operations on integers ¡ Addition and subtraction ¡ Multiplication and division Last Time ¡ Dealing with overflow CS/ECE 3330 – Fall 2009 23 2 ¡ Representation for non-integral numbers ¡ Including very small and very large numbers ¡ Like scientific notation Floating Point ¡ –2.34 10 56 ¡ +0.002 10 –4 ¡ +987.02 10 9 ¡ In binary ¡ 1.xxxxxxx 2 2 yyyy normalized not normalized CS/ECE 3330 – Fall 2009 2 ¡ Types float and double in C 24 ¡ Defined by IEEE Std 754 - 1985 ¡ Developed in response to divergence of representations Floating Point Standard ¡ Portability issues for scientific code ¡ Now almost universally adopted ¡ Two representations ¡ Single precision (32-bit) ¡ Double precision (64-bit) CS/ECE 3330 – Fall 2009 25 3 IEEE Floating-Point Format S Exponent Fraction single: 8 bits double: 11 bits single: 23 bits double: 52 bits ¡ S: sign bit (0 Î non-negative, 1 Î negative) ¡ Normalize significand: 1.0 ≤ |significand| < 2.0 ¡ Always has a leading pre-binary-point 1 bit, so no need to represent it explicitly (hidden bit) Bias) (Exponent S 2 Fraction) (1 1) ( x − × + × − = CS/ECE 3330 – Fall 2009 ¡ Significand is Fraction with the “1.” restored ¡ Exponent: excess representation: actual exponent + Bias ¡ Ensures exponent is unsigned ¡ Single: Bias = 127; Double: Bias = 1023 26 ¡ Exponents 00000000 and 11111111 reserved ¡ Smallest value ¡ Exponent: 00000001 Single-Precision Range Î actual exponent = ______________ ¡ Fraction: 000…00 Î significand = ____ ¡ 1.0 2 –126 ≈ 1.2 10 –38 ¡ Largest value ¡ exponent: 11111110 1 - 127 = -126 1.0 CS/ECE 3330 – Fall 2009 Î actual exponent = ________________ ¡ Fraction: 111…11 Î significand ≈ ____ ¡ 2.0 2 +127 ≈ 3.4 10 +38 27 254 – 127 = +127 2.0 4 ¡ Exponents 0000…00 and 1111…11 reserved ¡ Smallest value ¡ Exponent: 00000000001 Double-Precision Range Î actual exponent = ______________ ¡ Fraction: 000…00 Î significand = ____ ¡ 1.0 2 –1022 ≈ 2.2 10 –308 ¡ Largest value ¡ Exponent: 11111111110 1 – 1023 = -1022 1.0 CS/ECE 3330 – Fall 2009 Î actual exponent = ___________________ ¡ Fraction: 111…11 Î significand ≈ ____ ¡ 2.0 2 +1023 ≈ 1.8 10 +308 28 2046 – 1023 = +1023 2.0 ¡ Relative precision ¡ all fraction bits are significant ¡ Single: approx 2 –23 Floating-Point Precision ¡ Equivalent to 23 log 10 2 ≈ 23 0.3 ≈ 6 decimal digits of precision ¡ Double: approx 2 –52 ¡ Equivalent to 52 log 10 2 ≈ 52 0.3 ≈ 16 decimal digits of precision CS/ECE 3330 – Fall 2009 29 5 ¡ Represent –0.75 ¡ –0.75 = (–1) 1 1.1 2 2 –1 ¡ S = 1 Floating-Point Example ¡...
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cs3330-chap3-arith-2 - 1 CS/ECE 3330 CS/ECE 3330 Computer...

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