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+---
+tags: AoCO2025, Compiler, x86
+---
+
+## Study Notes: Induction variables and loops, Advent of Compiler Optimisations 2025
+
+These notes are based on the post [**Induction variables and loops**](https://xania.org/202512/09-induction-variables) and the YouTube video [**[AoCO 9/25] More Loops: Induction Variables**](https://www.youtube.com/watch?v=vZk7Br6Vh1U&list=PL2HVqYf7If8cY4wLk7JUQ2f0JXY_xMQm2&index=10) which are Day 9 of the [Advent of Compiler Optimisations 2025](https://xania.org/AoCO2025-archive) Series by [Matt Godbolt](https://xania.org/MattGodbolt).
+
+My notes focus on reproducing and verifying [Matt Godbolt](https://xania.org/MattGodbolt)'s teaching within a local development environment using `GNU toolchain` and `LLVM toolchain` on `Ubuntu`.
+
+Written by me and assisted by AI, proofread by me and assisted by AI. 
+
+#### Development Environment
+```bash
+$ lsb_release -d
+Description:	Ubuntu 24.04.3 LTS
+
+$ gcc -v
+gcc version 13.3.0 (Ubuntu 13.3.0-6ubuntu2~24.04.1)
+
+$ llvm-objdump -v
+Ubuntu LLVM version 18.1.8
+```
+
+## Introduction
+
+This note introduces a loop optimization known as `Induction Variable Elimination` **[1]**.
+
+The core concept is analogous to an `arithmetic sequence` **[2]**.
+
+The closed-form expression $a_n = a_1 + (n - 1) d$ is mathematically equivalent to the 
+recursive definition $a_{n + 1} = a_{n} + d$.
+
+To verify whether a compiler performs induction variable optimization, we compare two implementations:
+
+1. A loop using the formula $a_n = a_1 + (n - 1) d$
+2. A loop using the formula $a_{n + 1} = a_{n} + d$
+
+If the compiler successfully applies this optimization, 
+the resulting disassembly for both implementations should be identical.
+
+## Part 01: Closed-Form Expression
+
+$$
+a_n = a_1 + (n - 1) d
+$$
+
+```bash
+$ cat main.c
+```
+
+```c
+void sum(int* a, int n, int d) {
+  int cache = a[0];
+  for (int i = 0; i < n; ++i) {
+    a[i] = cache + i * d;
+  }
+}
+```
+
+```bash
+$ gcc -O2 -c main.c
+$ llvm-objdump -d --disassemble-symbols=sum --x86-asm-syntax=att main.o
+```
+
+```x86asm
+0000000000000000 <sum>:
+ 0: f3 0f 1e fa          endbr64 
+ 4: 8b 07                movl   (%rdi), %eax        ; Load a[0] into %eax outside the loop
+ 6: 85 f6                testl  %esi, %esi          ; Check if n <= 0
+ 8: 7e 1b                jle    0x25                ; If n <= 0, jump to return
+ a: 48 63 f6             movslq %esi, %rsi          ; Sign-extend n to 64-bit for address calculation
+ d: 48 8d 0c b7          leaq   (%rdi,%rsi,4), %rcx ; Calculate end address: a + (n * 4), store in %rcx
+11: 0f 1f 80 00 00 00 00 nopl   (%rax)              ; Instruction alignment (no-op) for performance
+; --- Loop Starts ---
+18: 89 07                movl   %eax, (%rdi) ; Write current value to a[i]
+1a: 48 83 c7 04          addq   $0x4, %rdi   ; Pointer Increment: Advance %rdi to the next int 
+1e: 01 d0                addl   %edx, %eax   ; [Induction Variable] cache = cache + d
+20: 48 39 cf             cmpq   %rcx, %rdi   ; Check if current pointer %rdi has reached end address %rcx
+23: 75 f3                jne    0x18         ; If not equal, jump back to 0x18
+; --- Loop Ends ---
+
+25: c3                   retq
+```
+
+## Part 02 : Recursive Definition
+
+$$
+a_{n + 1} = a_{n} + d
+$$
+
+```bash
+$ cat main.c
+```
+
+```c
+void sum(int* a, int n, int d) {
+  int cache = a[0];
+  for (int i = 0; i < n; ++i) {
+    a[i] = cache;
+    cache += d;
+  }
+}
+```
+
+```bash
+$ gcc -O2 -c main.c
+$ llvm-objdump -d --disassemble-symbols=sum --x86-asm-syntax=att main.o
+```
+
+```x86asm
+0000000000000000 <sum>:
+ 0: f3 0f 1e fa          endbr64
+ 4: 8b 07                movl    (%rdi), %eax        ; Load a[0] into %eax outside the loop
+ 6: 85 f6                testl   %esi, %esi          ; Check if n <= 0
+ 8: 7e 1b                jle     0x25 <sum+0x25>     ; If n <= 0, jump to return
+ a: 48 63 f6             movslq  %esi, %rsi          ; Sign-extend n to 64-bit for address calculation
+ d: 48 8d 0c b7          leaq    (%rdi,%rsi,4), %rcx ; Calculate end address: a + (n * 4), store in %rcx
+11: 0f 1f 80 00 00 00 00 nopl    (%rax)              ; Instruction alignment (no-op) for performance
+; --- Loop Starts ---
+18: 89 07                movl    %eax, (%rdi)    ; Write current value to a[i]
+1a: 48 83 c7 04          addq    $0x4, %rdi      ; Pointer Increment: Advance %rdi to the next int 
+1e: 01 d0                addl    %edx, %eax      ; [Induction Variable] cache = cache + d
+20: 48 39 cf             cmpq    %rcx, %rdi      ; Check if current pointer %rdi has reached end address %rcx
+23: 75 f3                jne     0x18 <sum+0x18> ; If not equal, jump back to 0x18
+; --- Loop Ends ---
+25: c3                   retq
+```
+
+## Conclusion
+The disassembly results for both implementations are identical.
+This confirms that the compiler successfully performs Induction Variable Elimination.
+It recognizes the linear relationship within the loop and optimizes 
+the multiplication into an incremental addition.
+
+## References
+1. https://en.wikipedia.org/wiki/Induction_variable
+2. https://en.wikipedia.org/wiki/Arithmetic_progression