I recently came across a blog post by Brett Berry on Why 5 x 3 = 5 + 5 + 5 Was Marked Wrong. Discussion on how children should be taught aside, the post imho is an interesting read; the tl;dr of which is that while 5 x 3 = 15 and therefore is equal to 5 + 5 + 5, it is equilvalent only to 3 + 3 + 3 + 3 + 3.
This is due to the fact that the first factor of the multiplation equation represents the number of times that the second factor is to be accumulated. Therefore 5 x 3 is equivalent to an operation wherein the number 3 is accumulated5 times, i.e. 3 + 3 + 3 + 3 + 3.
This may seem like a frivilous distinction; after all, the function f() = 5 x 3 will always yield 15. However whether the function implements the operation internally as 3 + 3 + 3 + 3 + 3 or 5 + 5 + 5 will materially impact the efficiency of the function. The two respective implementations fail the equivalence test when their execution time is considered.
Take for example the implementation of an 8-bit multiplication function
CUPC/8 assembly
mul_n: resb 1 math_mul: st [mul_n], r0 xor r0, r0 .loop: push r0 ld r0, [mul_n] eq r0, #0 bzf .done sub r0, #1 st [mul_n], r0 pop r0 add r0, r1 b .loop .done: pop r0 pop pcl pop pch
The mul_n variable, received as a function parameter in register r0 represents the number of times the value from register r1 should be accumulated. As the Arithmetic Logic Unit takes the same amount of time to add two 8-bit numbers regardless of their values, it makes sense to limit mul_n, thereby limiting the number of loop iterations executed. Calling math_mul(3, 5) would perform 5 + 5 + 5 and is clearly preferable efficiency-wise to math_mul(5, 3) (3 + 3 + 3 + 3 + 3>).
Not all that is equal is equivalent.
