Things to know about Numeric Computation

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Phebe Todd
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1 Things to know about Numeric Computation

2 Classes of Numbers Countable Sets of Numbers: N: Natural Numbers {1, 2, 3, 4...}. Z: Integers (contains N) {..., -3, -2, -1, 0, 1, 2, 3,...} Q: Rational Numbers (contains Z) {0, 1, 1/2, 1/3,..., -1, -1/2, -1/3,... 2, 2/1, 2/2, 2/3... }

3 Classes of Numbers Uncountable Sets of Numbers R: Real Numbers {1, 2, , } includes all rational numbers (Q) and irrational numbers (everything in between members of Q) C: Complex Numbers {1, 2, , i } includes all of R and the imaginary number, i = sqrt(-1)

4 Integers in C short int x; long int x; int x; unsigned long int x; long long int x;

5 Short Ints short int x; Numeric Range: Storage Size: 2 bytes (16 bits) When to use? Almost never.

6 Long Ints long int x; Numeric Range: Storage Size: 4 bytes (32 bits) When to use? Most cases. Note, is same as: int x;

7 Unsigned Long unsigned long int x; Numeric Range: Storage Size: 4 bytes (32 bits) When to use? Need to count really high, but don t want to use 64-bits. Note, is same as: unsigned int x;

8 Long Long long long int x; Numeric Range: Storage Size: 8 bytes (64 bits) When to use? Counting to really big numbers (like seconds since the beginning of the universe), addressing >4GB of RAM

9 Integer Overflow long int x; x = ; x = x + 1; x is now !! When ints exceed their range, they wrap around to the other end. There is no warning.

10 Float is not a class of number Float is a numeric representation that approximates real numbers. Double is a type of float that does a better job at the approximation.

11 C Types float x; double x;

12 What is floating? Floating point refers to how the decimal-point can be positioned in the representation. So it works for big numbers... and small numbers

13 Float float x; Biggest Number: 3.4 * Smallest Nonzero Number: 1.2 * Storage Size: 4 bytes (32 bits) When to use? Want fast math and willing to sacrifice precision

14 Double float x; Biggest Number: 1.8 * Smallest Nonzero Number: 2.2 * Storage Size: When to use? 8 bytes (64 bits) Accuracy.

15 Float for Audio 32-bit floats can store 24-bit integer samples, range ( ), without ANY loss of information. Smallest audible number on a 24-bit audio system is approximately 1.0e-7 Corresponds to 120 db dynamic range on a 24-bit audio system

16 Numbers in MaxMSP [ int ] long int x; [ float ] float x; [ ~ ] float x;

17 Special Values NAN Not a number, e.g. result of division by zero. INFINITY, Result when overflow occurs -INFINITY Use the MSP object bitsafe~ to remove these potentially bad numbers from a signal stream.

18 Float Denormal Mode The float 32-bit format has poor accuracy for math operations resulting in small numbers. To address this deficiency the IEEE standards committee created a special denormal mode. Computation on floats switches to this mode when very small numbers are encountered.

19 The Denormal Problem Unfortunately, every math operation in denormal mode is about ten to one hundred times slower than in normal mode To get good performance in signal processing objects it is necessary to prevent denormal mode.

20 Denormals on Intel On the PPC architecture, denormal mode could be disabled system-wide. Computations resulting in very small numbers would return zero instead of going into denormal mode. On the Intel x86 architecture, denormal mode CANNOT be turned off*. It sucks, big time. * It can be turned off for SSE instructions, but that is effectively useless for most applications.

21 Example float x; x = 1.0; while(1) { } x = x * 0.5; x will underflow to denormal mode after about 120 iterations of this loop. This is a common problem in the implementation of IIR filters.

22 Denormal Symptoms There is no audio. (Signals are very very quiet) CPU suddenly jumps up to 100% and everything stops working.

23 Example Fixed float x; x = 1.0; while(1) { x = x * 0.5; } if(x < 1.0e-20) { } x = 0.; Test for small numbers and squash them.

24 Roundoff Error We are all familiar with the decimal approximation: 1/3 = The missing digits (3s we don t have space to write) are the roundoff error

25 Roundoff Error It turns out that this ness is a property of the base-ten number system. Floats on computers are base-two (binary). They have roundoff error in different cases, e.g.: 1/10 = 0.1 This number has an infinite repeating-decimal representation in base-two.

26 Roundoff Error Problems float x; x = 0.0; while(x!= 1000.) { } x = x + 0.1; 0.1 doesn t have a perfect representation in floats, so x never reaches the target exactly.

27 Fixing Roundoff Error Problems float x; x = 0.0; while(x < 999.9) { } x = x + 0.1; x = We know the final value we want so we can force it to be that at the end. This is a common problem in linear interpolation.

CS 33. Data Representation (Part 3) CS33 Intro to Computer Systems VIII 1 Copyright 2018 Thomas W. Doeppner. All rights reserved.

CS 33. Data Representation (Part 3) CS33 Intro to Computer Systems VIII 1 Copyright 2018 Thomas W. Doeppner. All rights reserved. CS 33 Data Representation (Part 3) CS33 Intro to Computer Systems VIII 1 Copyright 2018 Thomas W. Doeppner. All rights reserved. Byte-Oriented Memory Organization 00 0 FF F Programs refer to data by address