I love these incomprehensible magic number optimizations. Every time I see one I wonder how many optimizations like this we missed back in the old days when we were writing all our inner loops in assembly?
Does anyone have a collection of these things?
Here is a short list:
https://graphics.stanford.edu/~seander/bithacks.html
It is not on the list, but #define CMP(X, Y) (((X) > (Y)) - ((X) < (Y))) is an efficient way to do generic comparisons for things that want UNIX-style comparators. If you compare the output against 0 to check for some form of greater than, less than or equality, the compiler should automatically simplify it. For example, CMP(X, Y) > 0 is simplified to (X > Y) by a compiler.
The signum(x) function that is equivalent to CMP(X, 0) can be done in 3 or 4 instructions depending on your architecture without any comparison operations:
https://www.cs.cornell.edu/courses/cs6120/2022sp/blog/supero...
It is such a famous example, that compilers probably optimize CMP(X, 0) to that, but I have not checked. Coincidentally, the expansion of CMP(X, 0) is on the bit hacks list.
There are a few more superoptimized mathematical operations listed here:
https://www2.cs.arizona.edu/~collberg/Teaching/553/2011/Reso...
Note that the assembly code appears to be for the Motorola 68000 processor and it makes use of flags that are set in edge cases to work.
Finally, there is a list of helpful macros for bit operations that originated in OpenSolaris (as far as I know) here:
https://github.com/freebsd/freebsd-src/blob/master/sys/cddl/...
There used to be an Open Solaris blog post on them, but Oracle has taken it down.
Enjoy!
For an entire book on this stuff, see Henry S. Warren Jr's Hackers Delight. The "three valued compare function" is in chapter 2, for example.
> It is not on the list, but #define CMP(X, Y) (((X) > (Y)) - ((X) < (Y))) is an efficient way to do generic comparisons for things that want UNIX-style comparators. If you compare the output against 0 to check for some form of greater than, less than or equality, the compiler should automatically simplify it. For example, CMP(X, Y) > 0 is simplified to (X > Y) by a compiler.
I guess this only applies when the compiler knows what version of > you are using?
Eg it might not work in C++ when < and > are overloaded for eg strings?
My comment had been meant for C, but it should apply to C++ too even when operator overloading is used, provided the comparisons are simple and inlined. If you add overloads for the > and < operators in your string example to a place where they would inline, and the overload compares .length(), this should simplify. For example, godbolt shows that CMP(X, Y) == 0 is optimized to one mov instruction and one cmp instruction despite operator overloads when I implement your string example:
https://godbolt.org/z/nGbPhz86q
If you did not inline the operator overloads and had them in another compilation unit, do not expect this to simplify (unless you use LTO).
If you have compound comparators in the operator overloads (such that on equality in one field, it considers a second for a tie breaker), I would not expect it to simplify, although the compiler could surprise me.
There's also this classic: https://en.wikipedia.org/wiki/Fast_inverse_square_root
That is an approximation. If approximations are acceptable, then here is a trick you might like. In loops that call cosf(i * C) and/or sinf(i * C), where i is incremented by 1 on each iteration and C is some constant expression, you can call cosf() and sinf() once (or twice if i starts at something other than 0 or 1) outside of the loop and use the angle addition formula to do accumulation via multiplication and addition inside the loop. The loop will run significantly faster.
Even if you only need one of cosf() or sinf(), many CPUs calculate both values at the same time, so taking the other is free. If you only need single precision values, you can do this in double precision to avoid much of the errors you would get by doing this in single precision.
This trick can be used to accelerate the RoPE relative positional encoding calculations used in inference for llama 3 and likely others. I have done this and seen a measurable speed up, although these calculations are such a small part of inference that it was a small improvement.
there is "Hacker's Delight" by Henry S. Warren, Jr.
You should look at supercompilation.
sometimes also known as superoptimization, which many of them also use SMT solvers like Z3 mentioned in the article
We didn't miss them. In those days they weren't optimizations. Multiplications were really expensive.
Multiplications of this word length, one should clarify. It's not that multiplication was an inherently more expensive or different operation back then (assuming from context here that the "old days" of coding inner loops in assembly language pre-date even the 32-bit ALU era). Binary multiplication has not changed in millennia. Ancient Egyptians were using the same binary integer multiplication logic 5 millennia ago as ALUs do today.
It was that generally the fast hardware multiplication operations in ALUs didn't have very many bits in the register word length, so multiplications of wider words had to be done with library functions that did long multiplication in (say) base 256.
So this code in the headlined article would not be "three instructions" but three calls to internal helper library functions used by the compiler for long-word multiplication, comparison, and bitwise AND; not markedly more optimal than three internal helper function calls for the three original modulo operations, and in fact less optimal than the bit-twiddled modulo-powers-of-2 version found halfway down the headlined article, which would only need check the least significant byte and not call library functions for two of the 32-bit modulo operations.
Bonus points to anyone who remembers the helper function names in Microsoft BASIC's runtime library straight off the top of xyr head. It is probably a good thing that I finally seem to have forgotten them. (-: They all began with "B$" as I recall.
Most 8-bit CPUs didn't even have a hardware multiply instruction. To multiply on a 6502, for example, or a Z80, you have to add repeatedly. You can multiply by a power of 2 by shifting left, so you can get a bigger result by switching between shifting and adding or subtracting. Although, again, on these earlier CPUs you can only shift by one bit at a time, rather than by a variable number of bits.
There's also the difference between multiplying by a hard-coded value, which can be implemented with shifts and adds, and multiplying two variables, which has to be done with an algorithm.
The 8086 did have multiply instructions, but they were implemented as a loop in the microcode, adding the multiplicand, or not, once for each bit in the multiplier. More at https://www.righto.com/2023/03/8086-multiplication-microcode.... Multiplying by a fixed value using shifts and adds could be faster.
The prototype ARM1 did not have a multiply instruction. The architecture does have a barrel shifter which can shift one of the operands by any number of bits. For a fixed multiplication, it's possible to compute multiplying by a power of two, by (power of two plus 1), or by (power of two minus 1) in a single instruction. The latter is why ARM has both a SUB (subtract) instruction, computing rd := rs1 - Operand2, and a RSB (Reverse SuBtract) instruction, computing rd := Operand2 - rs1. The second operand goes through the barrel shifter, allowing you to write an instruction like 'RSB R0, R1, R1, #4' meaning 'R0 := (R1 << 4) - R1', or in other words '(R1 * 16) - R1', or R1 * 15.
ARMv2 added in MUL and MLA (MuLtiply and Accumulate) instructions. The hardware ARM2 implementation uses a Booth's encoder to multiply 2 bits at a time, taking up to 16 cycles for 32 bits. It can exit early if the remaining bits are all 0s.
Later ARM cores implemented an optional wider multiplier (that's the 'M' in 'ARM7TDMI', for example) that could multiply more bits at a time, therefore executing in fewer cycles. I believe ARM7TDMI was 8-bit, completing in up to 4 cycles (again, offering early exit). Modern ARM cores can do 64-bit multiplies in a single cycle.
The base RISC-V instruction set does not include hardware multiply instructions. Most implementations do include the M (or related) extensions that provide them, but if you are building a processor that doesn't need it, you don't need to include it.
> Multiplications of this word length, one should clarify. It's not that multiplication was an inherently more expensive or different operation back then (assuming from context here that the "old days" of coding inner loops in assembly language pre-date even the 32-bit ALU era). Binary multiplication has not changed in millennia. Ancient Egyptians were using the same binary integer multiplication logic 5 millennia ago as ALUs do today.
Well, we can actually multiply long binary numbers asymptotically faster than Ancient Egyptians.
> Binary multiplication has not changed in millennia. Ancient Egyptians were using the same binary integer multiplication logic 5 millennia ago as ALUs do today.
It turns out that multiplication in modern ALUs is very different. The Pentium, for instance, does multiplication using base-8, not base-2, cutting the number of additions by a factor of 3. It also uses Booth's algorithm, so much of the time it is subtracting, not adding.
Related, Computerphile had a video a few months ago where they try to put compute time relative to human time, similar to the way one might visualize an atom by making the proton the size of a golfball. I think it can help put some costs into perspective and really show why branching maters as well as the great engineering done to hide some of the slowdowns. But definitely some things are being marked simply by the sheer speed of the clock (like how the small size of a proton hides how empty an atom is)
https://youtube.com/watch?v=PpaQrzoDW2Iand divides were worse. (1 cycle add, 10 cycle mult, 60 cycle div)
That's fair but mod is division, or no? So realistically the new magic number version would be faster. Assuming there is 32 bit int support. Sorry, this is above my paygrade.
Many compiles will compute div-by-a-constant using the invert, multiply, and shift off the remainder trick. Once you have that, you can do mod-by-a-constant as a derivative and usually still beat 1-bit or 2-bit division.
Yeah, I'm thinking more of ones that remove all the divs from some crazy math functions for graphics rendering and replace them all with bit shifts or boolean ops.