This ASM x86 function calculates the length of a null-terminated string. It iterates through the string character by character until it encounters the null terminator (ASCII 0). This is a fundamental operation used in ma...

The `calculate_string_length` function takes a pointer to a string in `rdi` and returns its length in `rax`. It first checks if the input pointer is null; if so, it returns -1. Otherwise, it initializes a counter `rax` to zero. The function then enters a loop, loading each byte of the string into `cl`. If `cl` is zero (the null terminator), the loop terminates. Otherwise, the counter `rax` is incremented, and the loop continues. This approach has a time complexity of O(n), where n is the length of the string, as each character is examined once. The space complexity is O(1) as it only uses a few registers. The invariant maintained is that `rax` always holds the number of characters processed so far that are not the null terminator. The loop correctly terminates upon finding the null byte, ensuring accurate length calculation.

FUNCTION calculate_string_length(string_pointer): IF string_pointer IS NULL THEN RETURN -1 END IF length = 0 LOOP: current_char = character at (string_pointer + length) IF current_char IS 0 THEN EXIT LOOP END IF length =...

This ASM x86 function computes the factorial of a non-negative integer recursively. The factorial of n (n!) is the product of all positive integers less than or equal to n. This function includes checks for negative inpu...

The `recursive_factorial` function calculates n! using recursion. It handles base cases: n=0 returns 1, and negative n returns -1. For n > 0, it recursively calls itself with n-1, stores n on the stack, and then multiplies the result of `factorial(n-1)` by n. The core challenge is detecting overflow. The `imul rdi, rcx` instruction is used, which sets the overflow flag (OF) if the result of the multiplication `rdi * rcx` cannot fit into a 64-bit register. If the OF is set, the function returns -1. The time complexity is O(n) due to n recursive calls, and the space complexity is O(n) due to the recursion depth on the stack. The invariant is that before the multiplication `imul rdi, rcx`, `rcx` holds `(n-1)!` and `rdi` holds `n`. The overflow check ensures that the intermediate and final results remain within the representable range of a 64-bit signed integer.

FUNCTION recursive_factorial(n): IF n < 0 THEN RETURN -1 // Error: Negative input END IF IF n == 0 THEN RETURN 1 // Base case END IF // Recursive step previous_factorial = recursive_factorial(n - 1) IF previous_factorial...

This ASM x86 code implements a matrix multiplication routine (C = A * B) optimized for speed using AVX2 SIMD instructions. It processes data in 256-bit (32-byte) chunks, performing 4 double-precision floating-point opera...

The `multiply_matrices_avx2` function computes C = A * B for N x N matrices, where N is a multiple of 8. It uses three nested loops: the outer loop iterates through rows of C (i), the middle loop iterates through columns of C (j), and the inner loop iterates through the dot product elements (k). The key optimization is the inner loop: instead of processing one element at a time, it loads 8 doubles from row i of A and 8 doubles from column j of B into YMM registers (`ymm1`, `ymm2`). These vectors are then multiplied (`vmulpd`) and accumulated into a sum vector (`ymm0`) using `vaddpd`. Finally, the accumulated sum vector is stored into C[i][j]. This vectorized approach leverages the parallelism of AVX2, performing 4 operations per clock cycle per YMM register. The time complexity is O(N^3), but the constant factor is significantly reduced compared to scalar multiplication. Space complexity is O(1) beyond input storage.

FUNCTION multiply_matrices_avx2(ptrA, ptrB, ptrC, N): SAVE registers (rbp, rbx, r12-r15) FOR i FROM 0 TO N-1: FOR j FROM 0 TO N-1 STEP 8: INITIALIZE sum_vector YMM0 to {0.0, ..., 0.0} ptrA_row_i = ptrA + i * N * sizeof(d...

This ASM x86 code implements a fast string scanning function to find the first occurrence of a target character. It attempts to optimize by processing the string in 8-byte chunks, though the provided implementation falls...

The `find_char_fast` function searches for the first occurrence of a `target_char` within a null-terminated `source` string. It initializes pointers and saves registers. An edge case for an empty string is handled. The core logic aims to process the string in 8-byte blocks. However, the current implementation within the `.scan_loop` falls back to a nested `.byte_check_loop` to compare each byte individually against the `target_char`. If a match is found, the pointer to that character is returned. If a null terminator is encountered before a match, it signifies the end of the string, and the function returns 0. If the entire 8-byte block is scanned without a match or null terminator, the main pointer advances by 8 bytes, and the process repeats. True optimization for 8-byte chunks would involve SIMD instructions or bitwise tricks to check all 8 bytes simultaneously. The time complexity is O(N) in the worst case (character not found or at the very end), where N is the string length. Space complexity is O(1).

FUNCTION find_char_fast(source_string, target_char): current_pos = source_string IF source_string is empty THEN RETURN 0 WHILE TRUE: // Attempt to process 8 bytes at a time (simplified here) temp_pos = current_pos FOR i...

This ASM x86 code implements an optimized unsigned integer division by a constant (7) using the 'magic number' method. Instead of a slow division instruction, it performs a multiplication by a pre-calculated magic number...

The function `divide_by_7_unsigned` aims to compute `dividend / 7` without using the `DIV` instruction. It leverages the property that for a constant divisor `D`, there exist a multiplier `M` and a shift amount `S` such that `N / D ≈ (N * M) >> S`. The magic number `M = 0x2A05F200674A4500` and shift `S = 3` are pre-calculated for `D = 7`. The algorithm multiplies the dividend (`RAX`) by `M`, resulting in a 128-bit product in `RDX:RAX`. The higher 64 bits (`RDX`) are then right-shifted by `S` bits. This shifted value is a good approximation of the quotient. For unsigned division, this approximation is often sufficient or can be adjusted with a few extra checks. The time complexity is O(1) as it involves a fixed number of arithmetic operations, significantly faster than the O(log N) of a typical division instruction.

FUNCTION divide_by_7_unsigned(dividend): magic_number = 0x2A05F200674A4500 shift_amount = 3 product_high, product_low = dividend * magic_number quotient = product_high >> shift_amount RETURN quotient

This ASM x86 code implements a function to find the maximum of two 64-bit integers using conditional move (`cmov`) instructions. Instead of using a conditional jump (like `jl` or `jg`), it compares the two numbers and th...

The `find_max` function determines the larger of two 64-bit integers, `a` (in `RDI`) and `b` (in `RSI`). It initializes the result register `RBX` with `a`. Then, it compares `a` and `b` using `cmp rdi, rsi`. The `cmovg rbx, rsi` instruction conditionally moves the value of `RSI` (which is `b`) into `RBX` *only if* `RDI` was greater than `RSI` (i.e., `a > b`). If `a` was not greater than `b` (meaning `a <= b`), `RBX` remains unchanged, holding the value of `a`. This effectively selects the maximum value. Finally, the maximum value stored in `RBX` is moved to `RAX` for the return value. This avoids a conditional jump, which can be beneficial if branch prediction is unreliable. The time complexity is O(1) as it involves a fixed number of instructions.

FUNCTION find_max(a, b): result = a IF a > b THEN result = a // This is implicitly handled by initialization ELSE result = b END IF RETURN result // Using CMOV logic: FUNCTION find_max_cmov(a, b): result = a IF a > b THE...

This ASM x86 code implements a function to calculate the population count (number of set bits) of a 64-bit integer using the `POPCNT` instruction. This instruction is a dedicated CPU feature that efficiently counts the n...

The `count_set_bits` function takes a 64-bit integer in `RAX` and returns the count of its set bits. It leverages the `POPCNT` instruction, which is a specialized CPU instruction designed for this exact purpose. The instruction `popcnt rbx, rax` reads the 64-bit value from `RAX`, counts the number of bits that are set to '1', and stores this count (which will be between 0 and 64) into `RBX`. Finally, the count from `RBX` is moved to `RAX` to be returned to the caller. The `POPCNT` instruction is highly optimized and typically performs this operation in a very small number of clock cycles, making it much faster than software-based bit counting algorithms. The time complexity is effectively O(1) as it's a single hardware instruction.

FUNCTION count_set_bits(number): IF POPCNT instruction is available THEN count = POPCNT(number) ELSE // Fallback to software implementation (e.g., bit shifting) count = 0 WHILE number > 0: IF number AND 1 == 1 THEN count...

This ASM x86 code demonstrates summing elements of an array using indirect indexed memory addressing. It calculates the effective memory address by combining a base pointer, an index register, and a scale factor. This ad...

The `sum_array_indirect_indexed` function calculates the sum of elements in an array. It iterates through the array using an index (`R12`). For each element, it computes the effective memory address using the indirect indexed addressing mode: `[base + index * scale]`. Here, `RDI` is the base address, `R12` is the index, and `8` is the scale factor because the array elements are QWORDs (8 bytes). The value at the calculated address is loaded into `RAX`, and then added to the running sum stored in `RBX`. The loop continues until all `RSI` elements have been processed. The final sum is then moved to `RAX` for the return value. This method is efficient for accessing array elements sequentially. Time complexity is O(N) where N is the number of elements, and space complexity is O(1).

FUNCTION sum_array_indirect_indexed(base_address, num_elements): sum = 0 index = 0 WHILE index < num_elements: effective_address = base_address + index * sizeof(element) element_value = MEMORY[effective_address] sum = su...