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1 ( ) Session 9 Floating Point Systems Group Department of Computer Science ETH Zürich 1
2 Floating Point Recap for the Assignment 2
3 Floating Point Representation Numerical Form Scientific Notation Sign s 0, 1 Significant M [1.0, 2.0) Exponent E Encoding Bit Pattern MSB is sign bit s exp E, exp E frac M, exp E F = 1 s M 2 E s exp frac /64 float double extended 3
4 Casting Integer Types What happens here? 1. unsigned int foo; 2. long bar = (long) foo; Floats What happens here? 1. int i; 2. long long l; 3. float f; 4. double d; 5. i = (int) f; 6. i = (int) d; 7. f = (float) d; 8. d = (double) i; 9. f = (float) f; 4
5 Floats <-> Integers Casting between floats, doubles and integers generally changes the bit representation! 1. int i = 0xABCDABCD; 2. float f = (float)i; int *i2 = (int *)&f; printf( %x, %x, I, *i2); // Prints // abcdabcd, cea864a8 5
6 Floats <-> Integers From To Descrption double/float float f= ; float f2= ; int long long l=0x7fffffffffffffff; long long l2=0xffffffff; int (int)f =? (int)f2 =? double double d = (double)l; double d2 = (double)l2; Truncates the fractional part, Out of range, NaN -> TMin In general exact conversion iff int < 54 bits l == (long long)d; l2 == (long long)d2; int Float Will round according to rounding mode float f2=1.50f; float f3=1.50f; printf("%f, %i, %i\n", f2+f3, (int)(f2+f3), (int)f2 + (int)f3); //
7 Normalized / Denormalized Normalized: exp!= {000 0, 111 1} Good for bigger values Not equi-spaced Denormalized: exp == Good for very small values Equi-spaced [-1 + eps, 1 - eps] And zero 7
8 NORMALIZED! Exponent There must be a way to express negative exponents -> Encode as biased value E = Exp Bias Bias = 2 e-1-1: For Single precision? For Double precision? Exponent in general never all zeros and all ones! 8
9 NORMALIZED! Significant We know that M [1.0, 2.0) We always have one leading 1 Remove that leading 1 to stave one bit! What are the max and min values for the significant? 9
10 DENORMALIZED Exponent There must be a way to express values very close to 0: exponent must be as negative as possible. Exp is all zero and the exponent is evaluated as E = - Bias
11 DENORMALIZED Significant We are close to zero: M [0.0, 1.0) We always have one leading 0 11
12 Special Values Fraction Exponent Description Infinity (+ / -) If an operation overflows!= Not-a-Number (NaN) No numeric value can be determined sqrt(-1) Zero is in fact all zero like integer zero (there is also a -0 in float) 12
13 -0? In IEEE arithmetic, it is natural to define log 0 = - and log x to be a NaN when x < 0. Suppose that x represents a small negative number that has underflowed to zero. Thanks to signed zero, x will be negative, so log can return a NaN. However, if there were no signed zero, the log function could not distinguish an underflowed negative number from 0, and would therefore have to return -. 13
14 Tiny floating point example s exp frac Typical exam question 8-bit floating point representation the sign bit is in the most significant bit. the next four bits are the exponent, with a bias of 7. the last three bits are the frac Same general form as IEEE Format normalized, denormalized representation of 0, NaN, infinity 14
15 Conversion Step 1: Normalize the Numbers Step 2: Round to fit in fraction Step 3: Post-normalize to deal with rounding effects Value Binary Define 15 to be 13, i.e. 15 := 13 15
16 Conversion Step 1: Normalize the Numbers Set binary point s.t. has leading 1 Start with bias exponent = 7, decrement if need to left shift Value Binary Fraction Exponent (no shift) (4 shift) 16
17 Conversion Step 2: Round to fit in fraction We have 3 bit fractions Value Fraction GRS Rounded
18 Conversion Step 3: Post-normalize to deal with rounding effects Overflow in fraction due to rounding? (Not here) Shift right and increment exponent Value Binary
19 A possible Exam Question? You have a 8 bit floating point representation with 3 fraction bits. Give the floating point representations of
20 Multiplication Exact result: F new = 1 s 1 s 2 M 1 M 2 2 E 1+E 2 while( M 1 M 2 2 ) {M=M>>1; E++} Round M to fit fraction bits Check if exponent still in range 20
21 Addition Signed align and add (Assume E1 > E2) Shift the first operand by the difference of their exponents Add the M and s bits Apply shift and exponent adjustments till M is in Round 21
22 winter99/ pdf 22
23 What Every Computer Scientist Should Know About Floating-Point Arithmetic: 23
24 Assignment 08 Floating Point 24
25 Now its your turn! Implement your floating point handler in C! No use of floats/doubles Use the given skeleton 25
26 Your float_t You will represent the float as a struct 1. typedef struct float_t { 2. uint8_t sign; 3. uint8_t exponent; 4. uint32_t mantissa; 5. }; Challenge: Can you use bit fields for this and simply cast the pointer? 26
27 Conversion The only time you are allowed to use floats is when conversion it to your float_t 1. float_t fp_encode(float x); 2. float fp_decode(float_t x); 27
28 Approach Create some float numbers and convert them into your float_t. Choose good representatives Do some add, multiply, negations with your implemented functions and with the floats Compare at the end. 28
29 Approach Example 1. void main() { 2. float f1 = 1.123; 3. float f2 = 550; 4. float f3; 5. float_t ft1 = fp_encode(f1); 6. float_t ft2 = fp_encode(f2); 7. float_t ft2 8. float_t ft3; f3 = f1+f2; 11. ft3 = fp_add(ft1, ft2); assert(f3 == fp_decode(ft3)); 14. } 29
30 Submission Once you committed your final solution, write an to me! Subect: [CASP] Submission Content: Briefly describe what is working / what is not working Make sure your solution compiles! (with Wall) You can also submit your last homework 31
31 Have a nice weekend 32
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