In this test, I want to see whether the R1 model can read some C code and explain why the Clang autovectorizer algorithm decided not to compile SIMD instructions for the loop.
I have provided some simplified rules based on the Clang docs and the normal rules of autovectorizing algorithms.
The code is from the CPython code base.
Your job is to suggest optimizations to C code by finding loops which could be vectorized.
The clang compiler has decided that this loop cannot be vectorized, because it fails to meet one of the following rules. Your job is to work out which of these rules is not met in the code, then suggest an alternative block of code that would meet these requirements. The vectorizer will compile SIMD instructions into the program. This is efficient when the instructions in the code are the same for each iteration and they can be executed in parallel by multiple cores. If the loop contains complex branching logic, then it cannot be vectorized. If you decide that a loop cannot be vectorized without simple changes, explain why.
Here are the rules and constraints for the LLVM vectorizer algorithm:
The loop trip count (number of cycles) at entry to the loop at runtime.
The number of loop cycles can not change once entering the loop. e.g. if there is a conditional continue statement, it cannot be vectorized.
The loop must have a single entry and exit point. e.g. If there are break statements in the code, it has multiple exit points.
The Loop Vectorizer operates on loops and widens instructions to operate on multiple consecutive iterations.
The Loop Vectorizer supports loops with unknown trip count at compile time. The trip count does not need to be constant.
The Loop Vectorizer can vectorize loops with runtime checks of pointers.
The Loop Vectorizer can vectorize reductions (e.g., sum, product, XOR, AND, OR).
The Loop Vectorizer can vectorize inductions (e.g., saving the value of the induction variable into an array).
The Loop Vectorizer can vectorize if-conversions (e.g., flattening IF statements).
The Loop Vectorizer can vectorize pointer induction variables (e.g., using C++ iterators).
The Loop Vectorizer can vectorize reverse iterators (e.g., counting backwards).
The Loop Vectorizer can vectorize scatter/gather operations.
The Loop Vectorizer can vectorize mixed types (e.g., combining integers and floats).
The Loop Vectorizer can vectorize global structures alias analysis (e.g., accessing global structures).
The Loop Vectorizer can vectorize certain function calls (e.g., intrinsic math functions).
This is the C code to evaluate:
static digit v_iadd(digit *x, Py_ssize_t m, digit *y, Py_ssize_t n) { Py_ssize_t i; digit carry = 0;
assert(m >= n);
for (i = 0; i < n; ++i) {
carry += x[i] + y[i];
x[i] = carry & PyLong_MASK;
carry >>= PyLong_SHIFT;
assert((carry & 1) == carry);
}
for (; carry && i < m; ++i) {
carry += x[i];
x[i] = carry & PyLong_MASK;
carry >>= PyLong_SHIFT;
assert((carry & 1) == carry);
}
return carry;
}
Assistant: Okay, I'm trying to figure out why the clang compiler isn't vectorizing this loop. Let me look at the code step by step.
The function is called v_iadd and it takes four parameters: x, m, y, n. It returns a digit type. The goal seems to be adding two arrays of digits with possible carry-over.
Looking at the code, there are two loops here. The first loop runs from i=0 to i < n, which is the length of array y. Inside this loop, they add x[i] and y[i], then handle the carry. After that, there's a second loop that continues processing the carry for the remaining elements in x beyond n, up to m.
Now, I remember that vectorization requires the loop to have a predictable number of iterations without any changes once entered. Also, the control flow should be simple—no breaks or multiple exits unless handled properly.
In the first loop, i goes from 0 to n-1. That's straightforward. But then there's the second loop which starts at i (which is now n) and continues while carry is non-zero and i < m. This means the number of iterations in the second loop depends on the value of carry, which can change each time. So the trip count isn't fixed; it varies based on runtime conditions.
Clang probably can't vectorize this because the second loop's iteration count isn't known at compile time. Vectorization relies on knowing how many times the loop will run so that it can batch process multiple iterations into SIMD instructions. If the number of iterations is variable and unknown, especially dependent on data-dependent conditions like carry, the compiler can't safely vectorize.
Another point is the control flow. The first loop has a simple exit (i < n), but the second loop introduces a condition that depends on carry, which complicates things. This might make it hard for the vectorizer to predict how many iterations there will be, leading to potential issues with misaligned data or incorrect results if not handled properly.
So, the main issue is the variable trip count in the second loop due to the carry dependency. To fix this, maybe we can merge both loops into a single one that runs up to m, handling all cases without relying on carry for the iteration count beyond what's known at compile time.
The code cannot be vectorized because the second loop has a variable trip count dependent on the runtime value of carry, violating the requirement that the number of iterations must be fixed once the loop is entered.
Step-by-Step Explanation:
Loop Structure Analysis:
i = 0 to n-1, which is predictable.i = n and continues while carry is non-zero and i < m. This introduces a variable trip count.Violation of Vectorization Rules:
carry, which can change each iteration.Proposed Solution:
i = 0 to m-1.x, including those beyond n.Vectorized Code Suggestion:
static digit
v_iadd(digit *x, Py_ssize_t m, digit *y, Py_ssize_t n)
{
Py_ssize_t i;
digit carry = 0;
assert(m >= n);
for (i = 0; i < m; ++i) {
if (i < n) {
carry += x[i] + y[i];
} else {
carry += x[i];
}
x[i] = carry & PyLong_MASK;
carry >>= PyLong_SHIFT;
assert((carry & 1) == carry);
}
return carry;
}Explanation of Changes:
if statement to add y[i] only when i < n.m times, ensuring a fixed trip count suitable for vectorization.This modification allows the compiler to vectorize the loop since the iteration count is known and control flow is simplified.
Analysis of the Provided C Code The given C code is a function v_iadd that performs an addition operation on two arrays of digits (x and y) with lengths m and n, respectively. The function returns the carry after the addition.
Vectorization Analysis The code contains two loops:
The first loop iterates over the elements of y and adds corresponding elements from x and y. The carry is updated and propagated to the next iteration. The second loop continues the addition operation if there is a remaining carry after the first loop. It iterates over the remaining elements of x and updates the carry. Vectorization Constraints The Loop Vectorizer Rules and Constraints suggest that the vectorizer can handle loops with unknown trip count, runtime checks, reductions, inductions, and if-conversions. However, the code has some limitations that might prevent vectorization:
The loops have a complex control flow due to the carry propagation and the conditional continuation of the second loop. The loops contain assertions, which might be considered as undefined behavior if the conditions are not met. Non-Vectorizable Rule The rule that is not met in the code is:
The loop must have a single entry and exit point. The second loop has a conditional entry point (for (; carry && i < m; ++i)), which means that the loop might not be executed if the carry is zero. This conditional entry point creates multiple exit points, making the loop non-vectorizable.
Suggested Optimization To optimize the code for vectorization, we can consider the following alternatives:
Unroll the loops: If the lengths of the arrays are known at compile time, we can unroll the loops to reduce the number of iterations and simplify the control flow. Use SIMD-friendly data structures: If possible, we can use SIMD-friendly data structures, such as arrays of structs or structs of arrays, to improve the memory access patterns and reduce the number of loads and stores. Use intrinsic functions: We can use intrinsic functions, such as those provided by the compiler or libraries like SIMDPP, to perform the addition operation in a vectorized manner. Here's an example of how the code could be modified to use SIMD-friendly data structures and intrinsic functions:
#include <immintrin.h>
static digit
v_iadd(digit *x, Py_ssize_t m, digit *y, Py_ssize_t n)
{
Py_ssize_t i;
digit carry = 0;
assert(m >= n);
for (i = 0; i < n; i += 4) {
__m128i x_vec = _mm_loadu_si128((__m128i *)(x + i));
__m128i y_vec = _mm_loadu_si128((__m128i *)(y + i));
__m128i sum_vec = _mm_add_epi32(x_vec, y_vec);
__m128i carry_vec = _mm_srli_epi32(sum_vec, PyLong_SHIFT);
carry += _mm_extract_epi32(carry_vec, 0);
_mm_storeu_si128((__m128i *)(x + i), sum_vec);
}
for (; carry && i < m; ++i) {
carry += x[i];
x[i] = carry & PyLong_MASK;
carry >>= PyLong_SHIFT;
}
return carry;
}The function v_iadd has two loops. The first loop is the one starting at for (i = 0; i < n; ++i). The second loop continues with i from the previous loop up to m. The question is about the first loop, right? Because the problem states that the clang compiler decided this loop cannot be vectorized.
So, looking at the first loop:
for (i = 0; i < n; ++i) {
carry += x[i] + y[i];
x[i] = carry & PyLong_MASK;
carry >>= PyLong_SHIFT;
assert((carry & 1) == carry);
}Now, according to the vectorizer rules, various conditions are to be met.
First, let's consider the loop structure. The loop is a for loop with a simple increment. The loop control variable is i, and the trip count is n, which is known in some sense (though maybe not compile-time constant, but the rule allows for runtime trip counts). So that's okay.
Next, the exit conditions—it's a single exit point. The loop doesn't have any breaks or multiple exits here. So that rule is satisfied.
Now, data dependencies. The key thing for vectorization is that the operations can be executed in parallel across iterations. Let's check for loop-carried dependencies.
Looking at the code:
carry += x[i] + y[i]; → here, 'carry' is accumulated across iterations. So each iteration depends on the value of 'carry' from the previous iteration. This is a loop-carried dependency. That would prevent vectorization because each iteration's carry value is dependent on the prior iteration. For vectorization, the operations need to be independent across the vectorized lanes (i.e., different iterations processed in parallel).
The assignment x[i] = ... writes to x[i], which is okay since each iteration writes to a different x[i], no cross-iteration dependencies here. So x and y are being read, and x is written, but each iteration has its own i. So that's fine. The problem is with 'carry' being a reduction variable with a data dependency across iterations.
Where else? The carry is modified in each iteration. The statement after adding to carry is carry >>= PyLong_SHIFT;. And this carry is reused in the next iteration. So there's a loop-carried dependency through 'carry', making it impossible to vectorize as is.
The rules mention that vectorizers can handle reductions. Oh, right, because certain reduction operations can be parallelized. When there's an associative and commutative operation, you can split the reduction into partial sums. For example, sum reduction can be vectorized by summing subparts and then combining them. But in this case, the situation is more complex. Let me check the specifics of the code.
Wait, the carry is computed as (x[i] + y[i] + previous carry) >> PyLong_SHIFT. But the carry from one iteration affects the next. So this is a serial dependency: each iteration's carry is calculated from the previous. This makes the loop-carried dependency a problem for vectorization because the value of 'carry' in iteration i+1 depends on iteration i.
Therefore, the loop could not be vectorized because of this loop-carried dependency on 'carry', which causes each iteration to depend on the outcome of the previous. The vectorizer cannot find a way to handle this reduction in a way that allows parallel execution of multiple iterations.
Another point: the loop has a reduction (carry) which accumulates in a way that's not trivially vectorizable. Reductions can sometimes be handled if the operation is associative. For example, a sum can be computed by summing parts of the vector and then combining the sums. However, the way 'carry' is modified here is part of the algorithm's dependency chain. Let's think about the code:
In each iteration, carry is updated as (carry + x[i] + y[i]) and then the lower bits are stored to x[i], then carry is right-shifted by SHIFT bits. Then the assert checks that carry is 0 or 1? Because the carry is computed by adding two digits (each with a max value of MASK, which is probably 2^SHIFT -1?) and the previous carry. For example, if the digit is stored as 30-bit numbers, then adding two digits gives up to 2*(2^30 -1) + 1 = 2^31 -1 + ... Wait, the maximum carry after adding x[i] and y[i] (each up to MASK, which is 2^k - 1) plus the previous carry which is 0 or 1. Then, the new carry after shifting would be (x[i] + y[i] + previous_carry) divided by 2^k. So carry would be up to ( (2*(2^k -1)) + 1 ) / 2^k ) = (2^{k+1} -1) / 2^k = 1.999... → so carry here can't be more than 1?
Wait, the assert checks "(carry & 1) == carry", which implies carry is either 0 or 1, but wait if carry is shifted right by SHIFT (say, 30), then carry after >> should be 0 or 1. Because the sum (x[i] + y[i] + previous carry) can be up to (2^30 -1) + (2^30 -1) +1 = 2^31 -1. When you shift right by 30, 2^31 is 2, but wait 2^31 -1 divided by 2^30 is just 1.999999..., so shifting right by 30 gives 1. So carry would be 0 or 1. Thus, the assert is checking that carry is indeed 0 or 1. So each iteration's carry contributes 0 or 1 to the next iteration.
But in terms of the loop's structure: the carry is carried forward. Each iteration depends on the carry from the previous one. Thus, this is a non-vectorizable loop-carried dependency.
So the vectorizer can't handle this because the loop has a dependency that prevents multiple iterations from being executed in parallel. Specifically, the value of 'carry' in iteration i+1 is dependent on iteration i. So even though it's a reduction, it's a serial reduction where each iteration's carry is input to the next. Therefore, the loop cannot be vectorized as is.
The rule that's being violated here is probably the presence of a loop-carried dependency that isn't a vectorizable reduction. In the list of rules, it's mentioned that the vectorizer can handle reductions, but perhaps only certain types, like sum where partial sums can be calculated independently.
So, to vectorize this loop, the dependency on carry must be resolved. But how?
Wait, perhaps if the algorithm can be restructured to minimize the loop-carried dependency. Alternatively, can we find a way to compute multiple iterations at a time?
Alternatively, maybe the carry propagation is the issue, and breaking the dependency might be possible.
But let's think: this is a multi-digit addition with carry propagation. Since the carry is carried over digit to digit, this is inherently a sequential process. Therefore, vectorizing it might not be straightforward.
In traditional scenarios, additions can be vectorized if the carry is not involved. For example, in big integer addition where you can split into chunks, but each chunk can be added with carry-in and carry-out, which allows some vectorization, but the carries would require some form of carry propagation.
But in this loop, carry is affected in each iteration, and the same carry variable is used, leading to a dependency chain.
So in this case, perhaps the loop cannot be vectorized without significant changes to the algorithm, because the carry is serial in nature. However, if we could calculate how the carry propagates through multiple digits, perhaps through a technique like "carry-lookahead", but that might complicate things beyond what simple loop transformations can do.
Alternatively, unrolling the loop and trying to manage multiple carries could work, but for automatic vectorization by the compiler, that's probably too complex.
In this code, the loop cannot be vectorized because each iteration depends on the carry from the previous iteration, which creates a loop-carried dependency that the vectorizer cannot handle. Since the carry is computed from the previous iteration and then used in the next, this creates a chain that can't be parallelized.
So the reason the loop can't be vectorized is because of a loop-carried dependency that's not a vectorizable reduction. The rule here is that vectorizable loops can't have loop-carried dependencies except for reductions that can be parallelized. But in this case, the reduction is part of a carry chain that's serial in nature, so it can't be vectorized under the current rules.
Therefore, the code would need to be modified to eliminate the loop-carried dependency. But how?
Perhaps reorganizing the loop to compute the sum without carrying over in each step. For example, compute the sum and a mask, then compute the carry for multiple steps at once, but this seems complicated. Another approach is to make carry a vector and handle multiple elements with their own carries, but that's not straightforward.
Alternatively, since the carry after each step is only 0 or 1 (as per the assert), maybe the code can be transformed into two separate loops. The first loop adds x[i] + y[i] without considering the carry, storing the sum and the carry generated. Then, a second loop propagates the carry. This would break the single dependency chain.
Wait, that's an interesting idea. Let's think: what if we separate the addition and the carry propagation into two separate processes? For example:
But then propagating the carry from sum[i] to sum[i+1] would again create a dependency. However, in the original code, the carry is cumulative. The code is effectively performing a big integer addition, propagating the carry as it goes.
But in the original code, the code updates x[i], then shifts the carry for the next iteration. This is serial because the carry from i to i+1 depends on the cumulative result.
An alternative approach to make it vectorizable could involve processing the chunks in a way that allows each iteration to generate both a digit and a carry, then having the carry from the first iteration affect the next.
However, the fundamental issue is that the carry is passed from one iteration to the next, making the iterations dependent. So unless the vectorizer can handle such a dependency (which requires the values to be computed serially), it can't vectorize this.
Another possible approach is to block the loop into chunks where each block can compute a carry, and then handle the propagation of the carry after each block. For example, unroll the loop in blocks of 4 and compute the possible carries for each block, but this would complicate the logic and might require manual work.
But since the compiler's automatic vectorizer can't handle this, perhaps the only way is to manually split the loop into a part that can be vectorized and a part that handles the carry.
Alternatively, there might be a way to model this as a reduction. Wait, here, the carry is being accumulated in a way, but it's not a simple sum. Let me think again:
The variable 'carry' is modified as:
carry = (prev_carry + x[i] + y[i]) >> PyLong_SHIFT.
Each 'carry' is added to the next iteration. But due to the shifting, each iteration's contribution to 'carry' is scaled.
This seems like it's not a standard reduction. Compiler may not recognize this as a pattern it can vectorize.
So, the code as written can't be vectorized. The user has to restructure the code to handle carry in a way that's not dependent on previous iterations. But how?
An alternative approach could be to split the addition into two parts: first, compute the digits (x[i] + y[i]) without adding the carry from previous steps, then perform the carry propagation in a separate pass. But this might not work because the carry from each digit affects the next. However, maybe in this specific scenario, the carry can only be 0 or 1.
Wait, the assert says (carry & 1) == carry, implying that carry is 0 or 1 after each shift. Let me see why. Let's suppose PyLong_SHIFT is 30. Then, the sum x[i] + y[i] (each <= 2^30-1) would be up to 2*(2^30-1) = 2^31-2. Adding the previous carry (0 or 1) gives a maximum of 2^31-1. Dividing by 2^30 (shifting by 30), that's 2^31-1 >> 30 which is 1 (as 2^30 is 1073741824, so 2^30 * 2 = 2147483648, and 2^31-1 is 2147483647, which shifted right by 30 is 1 (so the new carry is 1), and the remainder is 2^30-1. But then the new x[i] is (carry + x[i] + y[i]) & PyLong_MASK, which masks the lower 30 bits.
Wait, the carry is updated as the sum divided by 2^30. So the sum of x[i], y[i], and the old carry is stored in the digit, but with carry >>= SHIFT (divided by 2^30). So carry can only be 0 or 1 hereU
| from random import randint | |
| # Constants | |
| PyLong_SHIFT = 30 | |
| PyLong_BASE = 1 << PyLong_SHIFT | |
| PyLong_MASK = (PyLong_BASE - 1) | |
| def v_iadd(x, m, y, n): | |
| CYCLES = 0 | |
| carry = 0 | |
| assert m >= n | |
| for i in range(m): | |
| if i < n: | |
| carry += x[i] + y[i] | |
| else: | |
| carry += x[i] | |
| x[i] = carry & PyLong_MASK | |
| carry >>= PyLong_SHIFT | |
| assert (carry & 1) == carry | |
| CYCLES += 1 | |
| return carry, CYCLES | |
| def v_iadd_original(x, m, y, n): | |
| carry = 0 | |
| CYCLES = 0 | |
| assert m >= n | |
| for i in range(n): | |
| carry += x[i] + y[i] | |
| x[i] = carry & PyLong_MASK | |
| carry >>= PyLong_SHIFT | |
| assert (carry & 1) == carry | |
| CYCLES += 1 | |
| for i in range(n, m): | |
| carry += x[i] | |
| x[i] = carry & PyLong_MASK | |
| carry >>= PyLong_SHIFT | |
| assert (carry & 1) == carry | |
| CYCLES += 1 | |
| return carry, CYCLES | |
| # test | |
| for i in range(1000): | |
| x = [randint(0, 2**30 - 1) for _ in range(i)] | |
| y = [randint(0, 2**30 - 1) for i in range(i - randint(0, i))] | |
| m = len(x) | |
| n = len(y) | |
| x1 = x.copy() | |
| y1 = y.copy() | |
| carry1, _cycles1 = v_iadd(x1, m, y1, n) | |
| x2 = x.copy() | |
| y2 = y.copy() | |
| carry2, _cycles2 = v_iadd_original(x2, m, y2, n) | |
| assert x1 == x2, (x1, x2) | |
| assert carry1 == carry2 | |
| assert _cycles1 == _cycles2 | |
| print("Test passed (m = %d, n = %d) %d cycles vs %d cycles" % (m, n, _cycles1, _cycles2)) |
Given code:
static digit
v_lshift(digit *z, digit *a, Py_ssize_t m, int d)
{
Py_ssize_t i;
digit carry = 0;
assert(0 <= d && d < PyLong_SHIFT);
for (i=0; i < m; i++) {
twodigits acc = (twodigits)a[i] << d | carry;
z[i] = (digit)acc & PyLong_MASK;
carry = (digit)(acc >> PyLong_SHIFT);
}
return carry;
}Alright, so I'm trying to figure out how to optimize this C code using vectorization. The function in question is v_lshift, which takes an array of digits and performs a left shift operation along with handling the carry-over between each digit.
First off, looking at the loop structure: it's a simple for-loop iterating over m elements from 0 up to m-1. Inside this loop, there are some bitwise operations on a[i] shifted by d bits combined with any existing carry (carry). The result is stored in z[i], and the new value of carry is computed.
Now, vectorization typically requires that each iteration within a loop performs scalar operations that can be replicated across multiple iterations. This way, all these scalar instructions can be unrolled into single instruction threads for each register or processing unit (like SIMD units). However, in this case, the inner operation seems to involve some dependencies between consecutive elements due to how carry is updated and used.
Let me break down what's happening inside the loop:
For each iteration i:
Now, considering vectorization:
Vector instructions operate on multiple data elements simultaneously (e.g., 4 or more). To do this effectively, the operations inside must be independent across these vectors. However:
The carry value is used in each iteration and updated before storing into z[i]. This creates a dependency where each element depends on its predecessor because carry holds information that affects subsequent iterations. Since it's using an array shift (using (twodigits)a[i] << d), the operation within might be vectorizable if a[i] is treated as a vector. But here, we're dealing with individual elements and their shifts.
Carry Dependency: The carry from one iteration affects the next because each z[i+1] depends on carry computed after processing z[i]. This creates an inherent sequential dependency that makes it difficult to parallelize across multiple iterations using vector instructions, as each step relies on previous results.
Non-Associative Operations: Vectorization requires operations within a vector to be independent and associative so they can reassociate without changing the result when unrolled into separate instruction threads. However, with carry being updated in place after each iteration affects next steps, this isn't possible because it creates an order-dependent dependency.
Unroll the Loop Manually: If you know that certain loop counts are fixed or can be predetermined (like if m is a multiple of 4), perhaps unrolling the loop to process several elements at once could help bypass some dependencies, allowing vectorization within each unrolled segment. But this would require manual code changes and may not fully utilize vector units.
Redesign with Vector Operations: Introducing vectors for a[i] or considering that multiple carries can be handled in parallel might allow more efficient processing of consecutive elements without relying on sequential carry updates, but I'm unsure how to do that exactly given the current structure.
Alternative Data Structure: If the data is restructured (like using a different indexing pattern), perhaps dependencies could be minimized or removed entirely.
Before attempting any manual optimization, it would make sense to test whether vectorization even makes sense for this function in its current form due to potential performance gains being limited by these dependency issues. Additionally, measuring the impact of unrolling versus not might show negligible benefits if dependencies prevent effective parallel processing.
The loop cannot be effectively vectorized because each iteration depends on the result from the previous one (due to carry updates), creating a sequential dependency that prevents independent and simultaneous execution of multiple iterations using vector instructions. To address this, consider manual unrolling or restructuring your data layout.
If vectorizing is not feasible due to dependencies:
static digit v_lshift(digit *z, digit *a, Py_ssize_t m, int d) {
if (m == 0)
return carry;
// Unroll the loop by a factor of N for manual processing.
const size_t N = 4; // Adjust based on vector width and efficiency goals.
for (Py_ssize_t i = 0; i < m;) {
digit new_carry = carry;
twodigits acc;
if ((N * d) > PyLLONG_MAX || ... ) { /* Check data types to prevent overflow */}
// Handle potential overflows or errors
}
for (size_t j = 0; j < N && i + 1 <= m;) {
digit a_i = a[i];
twodigits shifted_a = __lz "__shl"__(a_i, d); /* Example shift operation */
// Compute z's corresponding element
if ((shifted_a.hi & mask) != (digit)(acc >> PyLong_SHIFT))
new_carry ^= ...; /* Update carry based on higher bits shifted out */
digit a_next = i < m -1 ? a[i+1] : 0;
// Store the result
z[i++] = (digit)shifted_a.lo;
if (++i == m)
break;
acc.hi = ...; /* Process next element's shift and carry */
}
new_carry ^= ...; /* Handle any remaining carries after unrolling */
carry = new_carry;
}
}This code structure attempts to process multiple elements per iteration, reducing the impact of sequential dependencies. However, it requires careful handling of data types (like digit and twodigits) as well as potential overflow checks depending on your specific implementation details.
By implementing such unrolling or restructuring approaches, you can mitigate dependency issues while maintaining functionality.