Applying C - Fixed Point Arithmetic
Written by Harry Fairhead
Monday, 11 May 2020
Article Index
Applying C - Fixed Point Arithmetic
Rescaling
Printing Fixed Point

## Printing Fixed Point

Finally, how do we print a fixed point value as a decimal?

This is more difficult than you might imagine and is often implemented incorrectly. You can get the integer part of the result simply by shifting:

`temp>>8`

and this can be printed as if it was just an integer, which is exactly what it is. You can also isolate the bits of the fractional part using a mask:

`temp&0xFF`

and at this point the temptation is to print this as if it was an int:

`printf("%d.%d \n",temp>>8,temp&0xFF);`

but it isn’t and this doesn’t work.

The reason is that the bit pattern is weighted by inverse powers of 2 and the bit pattern doesn’t correspond to the usual decimal values.

A simple example will illustrate the problem and give us the solution. Consider a fixed point value with the fractional part bit pattern:

`10000000`

which, if just treated as a decimal value, 128, gives a decimal fraction of:

`.128`

which is obviously wrong.

With the binary point inserted, the fractional part corresponds to:

`0.10000000`

and this is 0.5 in decimal, not 0.128.

The fractional part of the fixed point represents a value in the range 0 to less than 1, but as an unsigned binary integer it represents a range from 0 to 2s. We can use this to obtain a decimal representation. For example, for the binary fraction:

`0.10000000`

the integer representation is:

`10000000`

which is 128, and this corresponds to the decimal fraction:

`128/28 = 128/256 = 1/2 = 0.5`

Of course, we can’t do the division to get 0.5 because we only have integer arithmetic, but if we multiply by 1000, a power of ten sufficient to give an integer, then we can do the entire calculation using integer arithmetic:

`1000*128/28 = 1000*128/256 = 500`

which when printed with a decimal point in front is:

`0.500`

You can see that the power of ten gives you the corresponding number of decimal places.

Thus, if f is the fractional part converted to an integer, then 1000*f/2s is the decimal representation of the fraction.

To print the temperature we use:

`printf("%d.%d \n",temp>>8,(temp&0xFF)*1000/(1<<8));`

This works with obvious adjustments for any scale factor and any number of decimal places.

It is important to note that the multiplication has to be performed before the division and this introduces the possibility of overflow. The rule is that the number of decimal places, d, has to satisfy:

d*2s<232 or d<232-s

for a 32-bit int.

For example, if s is 16:

d<232-16<216<65536

the largest value of d that avoids an overflow is 10,000. If s is 8 then d can be as large as 10,000,000.

#### Omitted from this extract but in complete chapter:

• Overflow During a Computation
• Avoiding Overflow with 64-bits
• Simulating 64-bit Multiply
• GCC Fixed Point Extensions
• Using a Fixed Point Library
• Decimal Fixed Point
• Some Speed Comparisons

## Summary

• Fixed-point arithmetic is just scaled integer arithmetic and is therefore simple and fast. However, modern floating point hardware can often match its speed.

• You have to select the total number of bits and the number of bits to use as the binary fraction.

• Fixed-point multiplication and division require rescaling to bring the binary point back to the original location.

• Printing a fixed-point value is easy for the integer part, but the fractional part needs more care.

• Because of the way the scaling works, fixed-point calculations can overflow, even though the final result can be represented in the given format. This can be avoided by working with more bits or by applying the rescaling during the calculation with the subsequent loss of precision.

• There is no standard library for fixed point computation. GCC has some built-in functions, but these only work with specific hardware. An alternative that works with all hardware is libfixmath.

• As well as using binary scaling in fixed point, you can also use decimal scaling. This is slower, but sometimes it is more direct.

• It is worth remembering the fixed point doesn’t always have a speed advantage over hardware-implemented floating point. Always benchmark before making a choice.

#### Now available as a paperback or ebook fromAmazon.

Applying C For The IoT With Linux

1. C,IoT, POSIX & LINUX
2. Kernel Mode, User Mode & Syscall
3. Execution, Permissions & Systemd
Extract Running Programs With Systemd
4. Signals & Exceptions
Extract  Signals
5. Integer Arithmetic
Extract: Basic Arithmetic As Bit Operations
6. Fixed Point
Extract: Simple Fixed Point Arithmetic
7. Floating Point ***NEW
8. File Descriptors
Extract: Simple File Descriptors
Extract: Pipes
9. The Pseudo-File System
Extract: The Pseudo File System
Extract: Memory Mapped Files
10. Graphics
Extract: framebuffer
11. Sockets
Extract: Sockets The Client
Extract: Socket Server
Extract:  Condition Variables
13. Cores Atomics & Memory Management
Extract: Applying C - Cores
14. Interupts & Polling
Extract: Interrupts & Polling
15. Assembler
Extract: Assembler

Also see the companion book: Fundamental C

<ASIN:1871962609>

<ASIN:1871962617>

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