While looking at the ARM instruction set, you notice that the ADD
instruction has the so-called "Flexible second operand", which can be abused for quick multiplication. For example, the following instruction multiplies register r1
by 17 (shifting it left by 4 bits and adding to itself):
ADD r1, r1, r1, LSL #4
; LSL means "logical shift left"
The subtraction instruction has that feature too. So multiplication by 15 is also easy to implement:
RSB r1, r1, r1, LSL #4
; RSB means "reverse subtract", i.e. subtract r1 from (r1 << 4)
So this is an excellent optimization opportunity! Certainly, when you write in a C program x *= 15
, the compiler will translate that to some inefficient code, while you could replace it by x = (x << 4) - x
, and then the compiler will generate better code!
Quickly, you come up with the following cunning plan:
Write a program or a subroutine, in any language, that receives one parameter
m
, and outputs a C optimizing macro, like the following (form = 15
):#define XYZZY15(x) (((x) << 4) - (x))
or (which is the same)
#define XYZZY15(x) (-(x) + ((x) << 4))
???
- Profit!
Notes:
- The macro's name is essential: magic requires it.
- Parentheses around
x
and the whole expression are required: ancient C arcana cannot be taken lightly. - You can assume
m
is between 2 and 65535. - The input and output can be passed through function parameters,
stdin/stdout
or in any other customary way. - In C, addition and subtraction have tighter precedence than bit shift
<<
; use parentheses to clarify order of calculations. More parentheses than necessary is OK. - It is absolutely vital that the C expression be optimal, so e.g.
x+x+x
is not acceptable, because(x<<1)+x
is better. - Expressions
x + x
andx << 1
are equivalent, because both require one ARM instruction to calculate. Also,(x << 1) + (x << 2)
(multiplication by 6) requires two instructions, and cannot be improved. - You are amazed to discover that the compiler detects that
(x << 0)
is the same asx
, sox + (x << 0)
is an acceptable expression. However, there is no chance the compiler does any other optimizations on your expression. Even though multiplication by 17 can be implemented in one instruction, multiplication by 17*17 cannot be implemented in two instructions, because there is no C expression to reflect that:
#define XYZZY289(x) x + (x << 4) + (x + (x << 4) << 4) // 3 operations; optimal
In other words, the compiler doesn't do common sub-expression elimination. Maybe you will fix that bug in version 2...
This is code-golf: the shortest code wins (but it must be correct, i.e. produce correct and optimal expressions for all m
)
Please also give example outputs for m = 2, 10, 100, 14043 and 65535
(there is no requirement on the length of the expression - it can contain extra parentheses, whitespace and shift left by 0 bits, if that makes your code simpler).
2 <= m <= 65535
, there will be a worst case value where the number of bitshifts is a maximum. Is it guaranteed that even for this worst case that it will be more efficient to do this many shifts instead of one multiplication? \$\endgroup\$x+x+x
and(x<<1)+x
. \$\endgroup\$m = 24
where that would make a difference. \$\endgroup\$((x<<8)+1)+(((x<<8)+1)<<5)
, is that 3 ops, or 2? \$\endgroup\$