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In more complicated cases, compilers use a combination of LEA and ADD. For example, take a look at the following code, which is essentially a multiplication by 32:

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Basically, what you have here is y=x*2*2*2*2*2, which is equivalent to y=x*32. This code, generated by Intel s compiler, is actually quite surprising when you think about it. First of all, in terms of code size it is big one LEA and four ADDs are quite a bit longer than a single SHL. Second, it is surprising that this sequence is actually quicker than a simple SHL by 5, especially considering that SHL is considered to be a fairly high-performance instruction. The explanation is that LEA and ADD are both very low-latency, high-throughput instructions. In fact, this entire sequence could probably execute in less than three clock cycles (though this depends on the specific processor and on other environmental aspects). In contrast, SHL has a latency of four clocks cycles, which is why using it is just not as efficient. Let s examine another multiplication sequence:

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When variables are multiplied, the expected value of the result is E[XY ] = E[X]E[Y ] + Cov [X, Y ] (A.47)

This sequence, which was generated by GCC, uses LEA to multiply ESI by 3, and then uses SAL (SAL is the same instruction as SHL they share the same opcode) to further multiply by 4. These two operations multiply the operand by 12. The code then subtracts the operand from the result. This sequence essentially multiplies the operand by 11. Mathematically this can be viewed as: y=(x+x*2)*4-x.

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For computers, division is the most complex operation in integer arithmetic. The built-in instructions for division, DIV and IDIV are (relatively speaking) very slow and have a latency of over 50 clock cycles (on latest crops of NetBurst processors). This compares with a latency of less than one cycle for additions and subtractions (which can be executed in parallel). For unknown divisors, the compiler has no choice but to use DIV. This is usually bad for performance but is good for reversers because it makes for readable and straightforward code. With constant divisors, the situation becomes far more complicated. The compiler can employ some highly creative techniques for efficiently implementing division, depending on the divisor. The problem is that the resulting code is often highly unreadable. The following sections discuss reciprocal multiplication, which is an optimized division technique.

However, only when the variables are uncorrelated is there a simple expression for the variance: 2 2 2 2 Var[XY ] = 2 Y + 2 X + X Y (A.48) X Y and

The idea with reciprocal multiplication is to use multiplication instead of division in order to implement a division operation. Multiplication is 4 to 6 times

(A.49)

faster than division on IA-32 processors, and in some cases it is possible to avoid the use of division instructions by using multiplication instructions The idea is to multiply the dividend by a fraction that is the reciprocal of the divisor For example, if you wanted to divide 30 by 3, you would simply compute the reciprocal for 3, which is 1 3The result of such an operation is approximately 03333333, so if you multiply 30 by 03333333, you end up with the correct result, which is 10 Implementing reciprocal multiplication in integer arithmetic is slightly more complicated because the data type you re using can only represent integers To overcome this problem, the compiler uses fixed-point arithmetic Fixed-point arithmetic enables the representation of fractions and real numbers without using a floating movable decimal point.

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