This program demonstrates how to use integers instead of floating point operations using the example of fractal images.
Using integers in that way is specifically interesting if there is no FPU available. The background is discussed in my article »Calculating Fractals With Integer Operations«.
Have a look at the file config.h to modify the following options.
The code by default is configured to be multithreaded, making use of parallization on multiple CPUs.
This package contains several implementation variants of the inner loop in the following files:
iterated.cis a strait forward implementation of the iteration loop usingdoubles.iteratel.cis an implementation of the same algorithm using integers of typelonginstead.iterate.Sis an implementation done in Intel x86_64 assembler. It contains a traditional implementation using stack variables (#define CONSERVATIVE) and a high performance implementation.
Read my article »Fractals And Intel x86_64 Assembler« for implementation details of the assembler variant.
This package also contains the original variants in C and 68000 assembler written on Amiga 500 around 1990.
It was in the late ’80s when I started to become interested in fractals. At the time then computers were unbelievable slow compared to what we have today thus I tried to speed up the algorithms in several ways.
This article is about using integer operations to calculate fractal images, more specifically I’ll explain how to replace floating point operations by integer operations (It is not about the theory behind fractals. There’s a lot of literature about that.).
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I used an Amiga 500 computer which was an incredible system in the late ’80s. The system design was far ahead of its time and also the performance was pretty high. Intel PCs at the time then looked like an old rusty bicycle compared to the Amiga computer. The Amiga’s core was a Motorola 68000 CPU which internally was a full 32 bit microprocessor using the CISC design. Thus, it was a pleasure to write assembler code compared to other CPUs. But still, most systems had no floating point unit as it is common for modern computers of today. Thus floating point operations had to be implemented in software.
CPUs internally have a set of registers (which could be considered as local variables) of a fixed bit width. In case of the 68000 these are 32 bit wide registers.1 Almost every CPU can execute the four basic arithmetical integer operations add, subtract, multiply, and divide2 where the arguments to these ops are the registers. Additionally there are logical operations such as bit shifting.
Not all operations are executed at the same speed. Add/subtract and bit shifting typically is faster than multiplications and divisions.3 All other mathematical operations and functions have to be constructed using a sequence of these basic arithmetic instructions and thus are obviously slower. Such functions can be e.g. operations on integers which are longer than the native bit size of the registers, floating point arithmetics, or even more complex algorithms such as trigonometric functions or similar.
The goal is to avoid expensive operations particularly in the inner loops of algorithms.
Replacing floating point operations by integer operations technically is a fixed point arithmetic. That is that we move the decimal point to another fixed position during the algorithm and finally move it back to get the result. This sounds weird but is pretty easy explained with an example. Consider the two random decimal numbers 3.14 and 0.69. If we add or muliply these two the results are 3.83 = 3.14 + 0.69, and 2.16 = 3.14 * 0.69. Unfortunately our CPU cannot deal with decimal numbers thus we loose the fractional part and in turn we will loose a lot of precision: 3 = 3 + 0, and 0 = 3 * 0.
To solve this we move the decimal point 2 digits to the right. Mathematically this is multiplying by 100 in this example. Thus 314 = 3.14 * 100 and 69 = 0.69 * 100. Now we can add them since they are integer numbers: 383 = 314 + 69. To get the result just divide by 100 again which is 3.83 = 383 / 100.
This works with any multiplier f. Every variable is considered to be multiplied by f, e.g. a → a · f. We then have to make sure that the equations stay in balance if the operands are multiplied by f.
If a + b = c then also a · f + b · f = c · f. At multiplications (a · b = c) the result has to be divided by f again, because a · f · b · f = c · f², thus c ·f = (a · f · b · f) / f.
As already mentioned, multiplications and divisions are expensive, thus there is another optimization. If the factor f is chosen to be a power of 2, multiplications and divisions (of integer numbers) can be replaced by bit shift operations. Shifting a number to the left by one bit is equal to multiplying with 2.
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The following code shows the inner loop which is executed for each pixel of a fractal image.
int iterate(double real0, double imag0)
{
double realq, imagq, real, imag;
int i;
real = real0;
imag = imag0;
for (i = 0; i < MAXITERATE; i++)
{
realq = real * real;
imagq = imag * imag;
if ((realq + imagq) > 4)
break;
imag = real * imag * 2 + imag0;
real = realq - imagq + real0;
}
return i;
}There are several operations in it. To rewrite this to integer operations, additions can be directly done as discussed before. The result of the multiplications has to be shifted to the right by the number of bits which expresses the multiplication factor. The functions is called with the parameters real0 and imag0 already multiplied by f.
int iterate(nint_t real0, nint_t imag0)
{
nint_t realq, imagq, real, imag;
int i;
real = real0;
imag = imag0;
for (i = 0; i < MAXITERATE; i++)
{
realq = (real * real) >> NORM_BITS;
imagq = (imag * imag) >> NORM_BITS;
if ((realq + imagq) > (nint_t) 4 * NORM_FACT)
break;
imag = ((real * imag) >> (NORM_BITS - 1)) + imag0;
real = realq - imagq + real0;
}
return i;
}In this code snippet the cpp macro NORM_FACT represents the factor f and NORM_BITS the number of bits. Thus NORM_FACT = 2NORM_BITS. Line number 16 shows a bit shift to the right by just NORM_BITS – 1. This is because the result is multiplied by two again which is a bit shift to the left by one.
Although this probably sounds great, there is a limitation and this is the range of numbers. Integer numbers have a fixed width of bits and thus a maximum number of different values.. If we use a multiplication factor as show above, we can do arithmetics on fractional numbers and thus also on smaller numbers between 0 and 1 but the maximum number of different values expressible by an integer does still not change. A 32 bit integer has a range from -2147483648 to 2147483647. If we choose the factor f to be 213 = 8192, then the tiniest fractional number is 1 / 8192 which is approx. 0.000122.
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Some time ago I wrote this article about using integer operations to calculate fractal images. Currently, I’m giving a course which prepares for malware analysis. Among other things we deal a lot with Intel assembler and how compilers create and optimize code.
The full code of everything discussed below (and also of the article referred to above) is found here on Github.
One of the exercises was to write the integer version of the iteration function manually in assembler. Here is the original version in C:
int iterate(nint_t real0, nint_t imag0)
{
nint_t realq, imagq, real, imag;
int i;
real = real0;
imag = imag0;
for (i = 0; i < MAXITERATE; i++)
{
realq = (real * real) >> NORM_BITS;
imagq = (imag * imag) >> NORM_BITS;
if ((realq + imagq) > (nint_t) 4 * NORM_FACT)
break;
imag = ((real * imag) >> (NORM_BITS - 1)) + imag0;
real = realq - imagq + real0;
}
return i;
}NORM_BITS, NORM_FACT, and MAXITERATE are preprozessor macros defining integer constants, and nint_t is typedef’ed to long.
I did some benchmarks experimenting with several variants that the code could be written in assembler. For the benchmark I used gettimeofday(2) directly before and after calculating the image. The iteration loop is set to MAXITERATE = 50 000 and I ran every attempt 5 times and then calculated the average time value. At an image resolution of 640×400 the iteration loop would be executed 12.8×10⁹ (12.8 billion) times in the worst case. At the given coordinates and zoom level the iteration loop is actually executed 2.8×10⁹ times.
I ran the programs directly on the console without X11 to avoid interference with other tasks and to reduce the influence of multi-tasking as much as possible.
As a time reference I used the original C program compiled with gcc option -O2. I consider this time as 100%4 I used gcc version 4.9.2 on a 64 bit Intel Core 2 Duo 1.2Ghz (family 6, model 15, stepping 13, ucode 0x3a). I observed that the iteration loop is compiled in the same way, independently if optimized with -O2 or -O3.
The benchmark showed that there are several different optimizations which influence the performance, some of which are not really obvious. The following listing shows the solution which performed most. It needs about 91,7% of the time compared to the compiler’s optimized code. This is a huge improvement.
.section .text
.align 16
.global iterate
iterate:
mov $(4 * NORM_FACT),%rdx
mov %rdi,%r8 // real = real0
mov %rsi,%r9 // imag = imag0
mov $MAXITERATE,%ecx // i = 64
jmp .Litloop
.align 16
.Litloop:
mov %r8,%r10
imul %r10,%r10 // realq = real * real
sar $NORM_BITS,%r10 // realq >>= 13
mov %r9,%r11
imul %r11,%r11 // imagq = imag * imag
sar $NORM_BITS,%r11 // imagq >>= 13
lea (%r10,%r11),%rax // realq + imagq
cmp %rdx,%rax // > 4 * NORM_FACT ?
jg .Litbrk
imul %r8,%r9 // imag *= real
sar $(NORM_BITS - 1),%r9 // imag >>= NORM_BITS - 1
add %rsi,%r9 // imag += imag0
sub %r11,%r10 // realq - imagq
lea (%rdi,%r10),%r8
dec %ecx // i--
jne .Litloop
.Litbrk:
mov $MAXITERATE,%eax // 64 - i
sub %ecx,%eax
ret
The inner loop is between line numbers 12 and 33. The first important improvement is that the loop, i.e. the target of the branch in line #33 shall be aligned to 16 bytes. This is according to the »Intel 32/64 Optimization Reference Manual« and this is also done by the compilier’s optimizer. The benchmark showed that if it is not properly aligned that the runtime is at 93,9% which is more than 2% slower.
Line #32 and #33 shows the loop counter decrement and the conditional branch. Although the Intel core provides a LOOP instruction the benchmark shows that using LOOP is significantly slower than doing a DEC/JNE combination, independently if the target is properly aligned or not. The execution time with LOOP is at about 96,6%. If we use SUB $1,%ecx it slightly slower, about 0,2%.
Another trick which gains performance is to use the LEA instruction instead of a MOV/ADD combination. LEA actually is a three-operand instruction. Line #21 is equivalent to
mov %r10,%rax
add %r11,%rax
There are many reasons to not write any code in assembler today. But although modern code optimizers are highly efficient, manual written code with thoroughly chosen instructions may still perform better.
Footnotes
-
The Intel i286 had 16 bit registers, the i386 up to the Pentium generation had 32 bit registers, and modern Intel cores have 64 bit registers. In the embedded area a lot of 8 bit CPU are in uses such as e.g. the AVR cores, or the 8051-based cores. ↩
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E.g. the AVR core has no divide instruction. ↩
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On the M68000 a multiplication used up to 70 CPU cycles in contrast to an addition which took just 4 to 8 cycles. ↩
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The absolute time on my computer was slightly more than 30 seconds for one run. ↩