====== 2. Data representation in memory and real numbers  ======

  * for lecturer: [[..:..:..:internal:tutorials:02:start|cvičení 2]]

===== Exercise outline =====

  - basic data types (integer number) storage in computer memory, program in C
  - integer representation, addition, subtraction, multiplication, division
  - real number represetation (in IEEE 754)

===== What should I repeat before the first exercise =====

  - binary representation and hexadecimal numbers
  - C language syntax
  - terms little and big endian
  - the second complement code
  - logic operations (and, or, invert, rotation, ...)
  - two's complement code and IEEE 754.
  - to understand the program from the last class.

===== What shall we do on the first exercise =====

The exercise will be based on following C code. It prints number representation in computer memory. We shall modify the code during the class.

<code>
#include <stdio.h>

#define PRINT_MEM(a) print_mem((unsigned char*)&(a), sizeof(a))

void print_mem(unsigned char *ptr, int size) {
  int i;
  printf("address = 0x%08lx\n", (long unsigned int)ptr);

  for (i = 0; i < size; i++) {
    printf("0x%02x ", *(ptr+i));
  }

  printf("\n");
}

int main() {
  unsigned int unsig = 5;
  int sig = -5;

  printf("value = %d\n", unsig);
  PRINT_MEM(unsig);

  printf("\nvalue = %d\n", sig);
  PRINT_MEM(sig);

  return 0;
}
</code>

<note>
To compile the program: gcc -Wall -ansi -pedantic ./program.c
</note>

Python language alternative

<code>
#!/usr/bin/python3

import struct

a = 1234.56789
b = -11.1111111111111111111111
c = a + b

buf = struct.pack('<f', c)

print ('C float LE:' + ' '.join(["{0:02x}".format(b) for b in buf]))
print ('C float LE:' + str(struct.unpack('<f', buf)[0]))

buf = struct.pack('>f', c)

print ('C float BE:' + ' '.join(["{0:02x}".format(b) for b in buf]))
print ('C float BE:' + str(struct.unpack('>f', buf)[0]))

buf = struct.pack('<d', c)

print ('C double LE:' + ' '.join(["{0:02x}".format(b) for b in buf]))
print ('C double LE:' + str(struct.unpack('<d', buf)[0]))

buf = struct.pack('>d', c)

print ('C double BE:' + ' '.join(["{0:02x}".format(b) for b in buf]))
print ('C double BE:' + str(struct.unpack('>d', buf)[0]))
</code>

See Python struct module for more [[https://docs.python.org/3/library/struct.html|struct — Interpret bytes as packed binary data]].


===== Tasks =====

  - Compile and execute program above
    * Explain output of the program
    * Alter the code to print representation of other data types (float, char, double, long, etc...)
    * Alter the code to print numbers and their representation between -16...15.
    * Modify the program to add and subtract integer numbers and print arguments and results (in "normal" form and computer representation).
    * Try to add and subtract negative and positive numbers. Try to cause integer overflow and underflow.
  - Integer addition and subtraction in two's complement representation
    * add and subtract two integer numbers. For example 5+(-6) and 5-(-6).
    * repeat the operations with different numbers and check your results with the computer program from the first exercise.
    * When the underflow and overflow can happen? How can we detect, that it had occured?
  - Integer multiplication
    * multiply two integers, For example 7*6.
    * is there any difference when multiplying negative numbers? (e.g. -7*6, (-7)*(-6), etc...) 
    * show how to speed-up the multiplier? (use many adders instead repetitively using one).
  - Integer division
    * divide integers 42/7, 43/7
    * does the algorithm change when we use negative numbers?
  - Real number representation (IEEE 754)
    * binary representation of real numbers (float - 32bit, double - 64bit)
    * show binary representation of -0.75. Check your result with program from the last exercise.
    * find decimal number for float binary number 0xC0A00000 in IEEE754.
    * explain who to add numbers 9.999*10^1 and 1.1610*10^(-1) in decimal representation. Assume that it is possible to store only 4 digits in mantisa and 2 digits in exponent.
      * Hint: 1) align numbers 2) addition 3) normalization 4) round numbers.
    * in binary representation add 0.5 and -0.4375
    * in decimal representation show multiplication of 1.110*10^10 * 9.200*10^(-5)
    * in binary representation multiply 0.5 and -0.4375

===== Usefull links =====

  * [ [[http://www2.its.strath.ac.uk/courses/c/|http://www2.its.strath.ac.uk/courses/c/]] ] - (Short) Introduction to C language
  * [ [[http://www.gnu.org/software/libc/manual/html_node/Formatted-Output.html|http://www.gnu.org/software/libc/manual/html_node/Formatted-Output.html]] ] - **printf** function documentation. The printf is used for formated output.
  * [[https://bootlin.com/doc/legacy/command-line/command_memento.pdf|Basic command-line operations for Unix and GNU/Linux systems]] on [[https://bootlin.com/|Bootlin]]
  * Description of IEEE 754 numbers representation on [[https://en.wikipedia.org/wiki/IEEE_754|Wikipedii]]