17 - Floating Point 1 Topics IEEE754Standard RoundingMode...

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1 Floating Point
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2 Topics Fractional Binary Numbers IEEE 754 Standard Rounding Mode FP Operations Floating Point in C Suggested Reading: 2.4
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3 Encoding Rational Numbers Form V = Very useful when        >> 0 or       <<1 An Approximation to real arithmetic From programmer’s perspective Uninteresting Arcane and incomprehensive y x 2 × V V
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4 Fractional Binary Numbers b m b m –1 b 2 b 1 b 0 b –1 b –2 b –3 b n • • • • • • . 1 2 4 2 m –1 2 m • • • • • • 1/2 1/4 1/8 2 n
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5 Fractional Binary Numbers Bits to right of “binary point” represent fractional powers of 2 Represents rational number:        2 i - = m n i i b
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6 Fractional Numbers to Binary Bits unsigned result_bits=0, current_bit=0x80000000 for (i=0;i<32;i++) { x *= 2 if ( x>= 1 ) { result_bits |= current_bit ; if ( x == 1) break ; x -= 1 ; } current_bit >> 1 ; }
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7 Fraction Binary Number Examples Value Binary Fraction 0.2 0.00110011[0011] Observations: The form 0.11111…11 represent numbers just below 1.0 which is  noted as 1.0- ε Binary Fractions can only exactly represent x/2 k Others have repeated bit patterns
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8 Encoding Rational Numbers Until 1980s Many idiosyncratic formats, fast speed, easy  implementation, less accuracy IEEE 754 Designed by W. Kahan for Intel processors (Turing  Award 1989) Based on a small and consistent set of principles,  elegant, understandable, hard to make go fast
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9 IEEE Floating-Point Representation Numeric form V=(-1) s × ×  2 E Sign bit  s  determines whether number is negative or positive Significand  M    normally a fractional value in range [1.0,2.0). Exponent  E  weights value by power of two
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10 IEEE Floating-Point Representation Encoding   s is sign bit exp  field encodes  E frac  field encodes  M Sizes Single precision (32 bits): 8 exp bits, 23 frac bits Double precision (64 bits): 11 exp bits, 52 frac bits s exp frac
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11 Normalize Values Condition   exp     000 0  and  exp     111 1 Exponent coded as biased value E  =  Exp – Bias Exp : unsigned value denoted by  exp Bias : Bias value Single precision: 127 (Exp: 1…254, E : -126…127)
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This note was uploaded on 10/19/2011 for the course CS 000 taught by Professor Jing during the Fall '10 term at Fudan University.

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17 - Floating Point 1 Topics IEEE754Standard RoundingMode...

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