Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2022/digital-systems-linear-algebra-introduction-to-formal-proof/2/c/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Digital systems, linear algebra, introduction to formal proof / 2 / c / Solution by Codex 0 2026-10-25
Keep total in the callee-saved register r4, and the first returned sum in the callee-saved register r5. Suppose the surrounding function has made an aligned stack frame with the local seven-halfword array at the current sp, occupying fourteen bytes in a sixteen-byte slot. One possible frame setup is push {r4,r5,r6,lr} followed by sub sp,sp,#16; the earlier code initializes total, regular, and special before this fragment.
    ldr  r0, =regular       @ assembler literal load of the global array address
    movs r1, #7
    bl   sum16
    movs r5, r0             @ first sum must survive the next call

    add  r0, sp, #0         @ current frame's local special array
    movs r1, #7
    bl   sum16
    muls r0, r5, r0         @ second sum times first sum
    adds r4, r4, r0         @ update total
The AAPCS32 permits each call to overwrite r0–r3, r12, and condition flags, but requires preservation of r4–r8, r10, r11, and sp. Thus both r4 and r5 remain valid even if sum16 is replaced by another conforming implementation. bl changes the link processor register lr, so the enclosing function saves its own return address in its prologue. Its epilogue can use add sp,sp,#16 and pop {r4,r5,r6,pc}. The frame uses thirty-two bytes in total, preserving the required eight-byte stack pointer alignment at both calls.[ref]
The two calls may be evaluated in this order. Assume the product and final addition fit the signed 32-bit result expected by the source program; unlike each individual seven-element sum, their product need not fit. The assembly language operations retain their low 32 bits if this assumption fails, whereas signed integer overflow in the C programming language has different semantics.
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