Consider the expression $(a - 1) * (((b + c) / 3) + d)$. Let X be the minimum number of registers required by an optimal code generation (without any register spill) algorithm for a load/store architecture, in which (i) only load and store instructions can have memory operands and (ii) arithmetic instructions can have only register or immediate operands. The value of X is _________.

We are evaluating the expression:

Let's determine the minimum number of registers required using optimal evaluation order (evaluating the heavier subtree first):

1. Evaluate the right operand $(((b+c)/3)+d)$ first:

   • LOAD R1, b (R1 = b)

   • LOAD R2, c (R2 = c)

   • ADD R1, R1, R2 (R1 = b + c, R2 becomes free)

   • DIV R1, R1, 3 (R1 = (b + c) / 3, immediate operand)

   • LOAD R2, d (R2 = d)

   • ADD R1, R1, R2 (R1 = ((b + c) / 3) + d, R2 becomes free)

2. Evaluate the left operand $(a - 1)$:

   • LOAD R2, a (R2 = a)

   • SUB R2, R2, 1 (R2 = a - 1, immediate operand)

3. Perform the final multiplication:

   • MUL R1, R2, R1 (R1 = (a - 1) * (((b + c) / 3) + d))

Throughout the computation, only $2$ registers ($R_1$ and $R_2$) are used simultaneously with no register spilling. Since at least $2$ registers are required to hold the operands for the binary multiplication operator, the minimum number of registers is 2.