# Is Code Faster Than Data? Switch Statements vs. Arrays

The last post explored compile time hashing a bit and at the end showed a way to use compile time code to assist a string to enum function, by having a minimal perfect hash that was guaranteed at compile time not to have any hash collisions.

That was pretty interesting to me, and made me wonder what else was possible with compile time (or compile time assisted) data structures. This would be mainly for static data structures that were read only. We could get creative and allow for deletes, inserts and data edits, but we’ll probably come back to that in a future post.

In pursuing other compile time data structures, there is one big question that would be easy to make assumptions about: Is custom code for compile time data structure lookup functionality faster than data?

In other words, could we hard code a hash table key to value lookup that was more performant than a general case hash table which was meant to be used by any data?

The answer to this is “usually it can be” IMO, since we could do lots of things specific to the data, like finding cheaper / better hash functions for the specific data input we were expecting. Since we aren’t doing inserts or deletes, or data modification, we could also allocate just the right amount of memory, in one shot, which would reduce memory waste and memory fragmentation.

But lets do some experiments and see. Let’s start off by comparing how a switch statement performs compared to some simple array types.

## Testing Details

I ran these tests in x86/x64 debug/release in visual studio 2015.

I made 5000 random uint32’s and stored them in these containers:

• std::vector
• std::array
• C style array
• dynamic memory
• Switch statement function
• Switch statement function with __assume(0) in default case

The __assume(0) in the switch default case is a hint to the optimizer to not do the usual bounds checking needed for handling a value not present in the switch statement. It’s microsoft specific, but there are equivelants in other compilers. Note that if you switch a value that would usually be handled by the default case, you would then have undefined behavior. It’s a speed up, but comes at a price. You can read about it here: msdn: __assume. This test code never passes values that would trigger the default case, so is arguably a good candidate for __assume, if you believe there are any good usages for __assume.

For a neat read on some details of undefined behavior check this out: Undefined behavior can result in time travel (among other things, but time travel is the funkiest)

I did the following tests.

• Sequential Sum: Add up the values at the indices 0-4999, in order. For the switch function, it was called with indices 0-4999 in order
• Shuffle Sum: Same as sequential sum, but in a shuffled order.
• Sparse Sum: Same as shuffle sum, but only doing the first 1000 indices. An alternate switch function was used which only contained values for those first 1000 indices, simulating the ability for us to strip out values never actually needed, to see if that has an impact on performance.

Timing was measured using std::chrono::high_resolution_clock, timing each test done 1,000 times in a row to make timing differences more apparent. I did this 50 times for each test to get an average and a standard deviation for those 50 samples.

The source code for the tests is at:
Github: Atrix256/RandomCode/ArrayAccessSpeeds

## Results

The results are below. Times are in milliseconds. The number in parentheses is the standard deviation, also in milliseconds.

Sequential Sum: Sum 5,000 numbers 1,000 times, in order.

 Debug Release Win32 x64 Win32 x64 std::vector 177.16 (4.68) 57.87 (6.48) 1.23 (0.25) 0.30 (0.46) std::array 94.53 (5.84) 53.61 (3.96) 1.25 (0.29) 0.34 (0.48) C array 8.67 (0.48) 9.90 (0.57) 1.22 (0.37) 0.30 (0.46) dynamic memory 8.96 (0.29) 10.01 (0.77) 1.27 (0.40) 0.30 (0.46) switch function 148.50 (3.48) 115.01 (4.83) 47.53 (3.86) 43.53 (1.75) switch function assume 0 135.96 (1.22) 95.54 (2.83) 46.33 (0.97) 38.91 (0.72)

Shuffle Sum: Sum 5,000 numbers 1,000 times, in random order.

 Debug Release Win32 x64 Win32 x64 std::vector 179.82 (8.13) 46.75 (1.05) 2.78 (0.42) 1.41 (0.19) std::array 89.94 (1.23) 39.47 (1.07) 2.58 (0.29) 1.42 (0.19) C array 8.59 (0.41) 8.63 (0.49) 2.55 (0.25) 1.39 (0.21) dynamic memory 8.23 (0.26) 8.41 (0.50) 2.73 (0.25) 1.40 (0.20) switch function 151.83 (0.95) 116.37 (16.91) 55.74 (1.97) 48.20 (0.56) switch function assume 0 142.07 (0.97) 103.01 (11.17) 55.45 (2.00) 42.09 (0.79)

Sparse Sum: Sum 1,000 of the 5,000 numbers 1,000 times, in random order.

 Debug Release Win32 x64 Win32 x64 std::vector 34.86 (0.46) 9.35 (0.48) 0.54 (0.14) 0.26 (0.25) std::array 17.88 (0.45) 7.93 (0.27) 0.52 (0.10) 0.25 (0.25) C array 1.72 (0.39) 1.70 (0.46) 0.52 (0.10) 0.24 (0.25) dynamic memory 1.67 (0.45) 1.70 (0.35) 0.55 (0.18) 0.26 (0.25) switch function 29.10 (0.49) 18.88 (0.92) 13.42 (0.43) 9.51 (0.43) switch function assume 0 26.74 (0.52) 18.19 (0.39) 12.15 (0.50) 9.23 (0.42)

## Observations

It’s interesting to see that std::array is 10 times slower than a c array or dynamic memory, in debug win32, and that std::vector is 20 times slower. In debug x64 std::array is only about 5 times slower, and std::vector is only a little bit slower than std::array. In release, std::vector and std::array are more in line with the cost of a c array, or dynamic memory.

It’s expected that std::array and std::vector are slower in debug because they do things like check indices for being out of bounds. They also have a lot more code associated with them which may not “optimize away” in debug, like it does in release (including inlining functions). It’s nice to see that they become basically free abstractions in release though. It gives us more confidence that they can be used in performance critical areas without too much extra worry, as long as we are ok with the debug slowdowns and want or need the debug functionality the types give us.

It’s also interesting to see that adding the __assume(0) to the default case in the switch statements actually improves performance a measurable amount in most tests.

The switch function IS NOT faster than the simple array access, and in fact is quite a bit slower!

Why is it so much slower?

One reason is because of the function call overhead associated with each lookup. When I marked the function as inline, the compiler refused to inline it. When i marked it as __forceinline to force it to inline, the compiler took A LONG TIME in the “generating code” section. I left it for 10 minutes and it still wasn’t done so killed it since it is unacceptably long. I tried a bit to get apples to apples comparisons by putting the test code into the function, but the optimizer ended up realizing it could run the function once and multiply it’s result by the number of test repeats I wanted to do. Trying to fool the optimizer meant doing things that weren’t quite an apples to apples comparison anymore, and made the results get slower. So, I’m leaving it at this: In the release assembly, you can see the function call to the switch function, where you don’t have one with eg. dynamic memory and function calls are not free, so no, the function call is not faster than not having a function call!

The second reason is how switch statements work compared to how arrays work. The switch statement first does a comparison to see if the value is within range of the maximum valued case (in other words, it does a range check, like what std::array and std::vector do in debug! It does this in debug and release both.), and then it does a jmp to a location of code where it then does a return with a constant value.

An array on the other hand just calculates the offset of a memory value based on the index passed in and the object size, and then reads that memory address.

Arrays have more direct access to the values than switch statements do.

It’s true that the switch statements could be optimized to behave more like arrays, and the whole function could go away in lieu of a memory lookup, but the optimizer doesn’t make that optimization here.

Here’s a good read on details of how switch statements are actually implemented in msvc under various conditions:
Something You May Not Know About the Switch Statement in C/C++

On the memory usage front, the simplest array (not worrying about things like geometric dynamic growth policies or memory padding) takes 20,000 bytes to store 5000 uint32’s.

In win32 (didn’t try x64 but I’m sure it’s similar), gutting the switch functions to only have a single case made a huge difference in executable size which shows us that switch statements are bad for memory efficiency too:
Debug went from 340 KB (348,160 bytes) to 244 KB (249,856 bytes).
Release went from 191 KB (196,096 bytes) to 133 KB (136,192 bytes).

## More Questions

I’d like to see how switch statements compare to hash tables. I think switch statements will still be slower, but it would be good to investigate the details and confirm.

This also makes me think that the “String To Enum” code in the last post, which makes use of a switch statement, is probably not the ideal code for doing that, as far as performance and memory usage go.

I think that instead of being switch statement based, it should probably be array based.

However, the idea that we can have compile time assurance that our hash has no collisions is really nice and we get that from doing case statements on run time hashes of our full set of valid inputs. It would be nice to find a way to leverage that while still getting the speed we want, as automatically as possible. At worst, code generation could be used, but it would be nicer to do something less rigid / heavy handed.

Lastly, I still believe that “ad hoc” custom fit code for specific data lookups have the best potential for performance. I want to keep looking into it to see about finding some good avenues for success.

Have any info to share? Please do!

Thanks for reading and I hope you find this useful for at least interesting.

# Exploring Compile Time Hashing

Never put off til run time what can be done at compile time.

C++11 gave us constexpr which lets us make C++ code that the compiler can run during compilation, instead of at runtime.

This is great because now we can use C++ to do some things that we previously had to use macros or templates for.

As with many of the newish C++ features, it feels like there are some rough edges to work out with constexpr, but it adds a lot of new exciting capabilities.

In this post we will explore some new possibilities, and get a better feel for this new world of compile time code execution. There are going to be some unexpected surprises along the way 😛

## Testing Details

The code in this post will be using some compile time crc code I found at Github Gist: oktal/compile-time-crc32.cc. I haven’t tested it for correctness or speed, but it serves the purpose of being a compile time hash implementation that allows us to explore things a bit.

I’ve compiled and analyzed the code in this post in visual studio 2015, in both debug/release and x86/x64. There are differences in behavior between debug and release of course, but x86 and x64 behaved the same. If you have different results with different compilers or different code, please share!

With that out of the way, onto the fun!

We are going to be looking at:

1. Simple Compile Time Hashing Behavior
2. Compile Time Hash Switching
3. Leveraging Jump Tables
4. Perfect Hashing
5. Minimally Perfect Hashing
6. Compile Time Assisted String To Enum

## Simple Compile Time Hashing Behavior

Let’s analyze some basic cases of trying to do some compile time hashing

    const char *hello1String = "Hello1";
unsigned int hashHello1 = crc32(hello1String);  // 1) Always Run Time.
unsigned int hashHello2 = crc32("Hello2");      // 2) Always Run Time.

// 3) error C2131: expression did not evaluate to a constant
//const char *hello3String = "Hello3";
//constexpr unsigned int hashHello3 = crc32(hello3String);
constexpr unsigned int hashHello4 = crc32("Hello4");  // 4) Debug: Run Time.  Release: Compile Time

printf("%X %X %X %Xn", hashHello1, hashHello2, hashHello4, crc32("hello5"));  // 5) Always Run Time. (!!!)


Let’s take a look at the assembly for the above code when compiled in debug. The assembly line calls to crc32 are highlighted for clarity.

    const char *hello1String = "Hello1";
00007FF717B71C3E  lea         rax,[string "Hello1" (07FF717B7B124h)]
00007FF717B71C45  mov         qword ptr [hello1String],rax
unsigned int hashHello1 = crc32(hello1String);  // 1) Always Run Time.
00007FF717B71C49  xor         edx,edx
00007FF717B71C4B  mov         rcx,qword ptr [hello1String]
00007FF717B71C4F  call        crc32 (07FF717B710C3h)
00007FF717B71C54  mov         dword ptr [hashHello1],eax
unsigned int hashHello2 = crc32("Hello2");      // 2) Always Run Time.
00007FF717B71C57  xor         edx,edx
00007FF717B71C59  lea         rcx,[string "Hello2" (07FF717B7B12Ch)]
00007FF717B71C60  call        crc32 (07FF717B710C3h)
00007FF717B71C65  mov         dword ptr [hashHello2],eax

// 3) error C2131: expression did not evaluate to a constant
//const char *hello3String = "Hello3";
//constexpr unsigned int hashHello3 = crc32(hello3String);
constexpr unsigned int hashHello4 = crc32("Hello4");  // 4) Debug: Run Time.  Release: Compile Time
00007FF717B71C68  xor         edx,edx
00007FF717B71C6A  lea         rcx,[string "Hello4" (07FF717B7B134h)]
00007FF717B71C71  call        crc32 (07FF717B710C3h)
00007FF717B71C76  mov         dword ptr [hashHello4],eax

printf("%X %X %X %Xn", hashHello1, hashHello2, hashHello4, crc32("hello5"));  // 5) Always Run Time. (!!!)
00007FF717B71C79  xor         edx,edx
00007FF717B71C7B  lea         rcx,[string "hello5" (07FF717B7B13Ch)]
00007FF717B71C82  call        crc32 (07FF717B710C3h)
00007FF717B71C87  mov         dword ptr [rsp+20h],eax
00007FF717B71C8B  mov         r9d,0BECA76E1h
00007FF717B71C91  mov         r8d,dword ptr [hashHello2]
00007FF717B71C95  mov         edx,dword ptr [hashHello1]
00007FF717B71C98  lea         rcx,[string "%X %X %X %Xn" (07FF717B7B150h)]
00007FF717B71C9F  call        printf (07FF717B711FEh)


As you can see, there is a “call crc32” in the assembly for every place that we call crc32 in the c++ code – 4 crc32 calls in the c++, and 4 crc32 calls in the asm. That means that all of these crc32 calls happen at run time while in debug mode.

I can sort of see the reasoning for always doing constexpr at runtime in debug mode, since you probably want to be able to step through constexpr functions to see how they operate. I’d bet that is the reasoning here.

Let’s see what it compiles to in release. Release is a little bit harder to understand since the optimizations make it difficult/impossible to pair the c++ lines with the asm.

00007FF68DC010BA  lea         rdx,[string "Hello1"+1h (07FF68DC02211h)]
00007FF68DC010C1  mov         ecx,7807C9A2h
00007FF68DC010C6  call        crc32_rec (07FF68DC01070h)
00007FF68DC010CB  lea         rdx,[string "Hello2"+1h (07FF68DC02219h)]
00007FF68DC010D2  mov         ecx,7807C9A2h
00007FF68DC010D7  mov         edi,eax
00007FF68DC010D9  call        crc32_rec (07FF68DC01070h)
00007FF68DC010DE  lea         rdx,[string "hello5"+1h (07FF68DC02221h)]
00007FF68DC010E5  mov         ecx,4369E96Ah
00007FF68DC010EA  mov         ebx,eax
00007FF68DC010EC  call        crc32_rec (07FF68DC01070h)
00007FF68DC010F1  mov         r9d,0BECA76E1h
00007FF68DC010F7  mov         dword ptr [rsp+20h],eax
00007FF68DC010FB  mov         r8d,ebx
00007FF68DC010FE  lea         rcx,[string "%X %X %X %Xn" (07FF68DC02228h)]
00007FF68DC01105  mov         edx,edi
00007FF68DC01107  call        printf (07FF68DC01010h)


We can see that in release, there are still 3 calls to crc32 which means that only one hash actually happens at compile time.

From the assembly we can easily see that “Hello1”, “Hello2” and “Hello5” are hashed at runtime. The assembly shows those strings as parameters to the function.

That leaves only “Hello4” remaining, which means that is the one that got hashed at compile time. You can actually see that on line 12, the value 0x0BECA76E1 is being moved into register r9d. If you step through the code in debug mode, you can see that the value of hashHello4 is actually 0x0BECA76E1, so that “move constant into register” on line 12 is the result of our hash happening at compile time. Pretty neat right?

I was actually surprised to see how many hashes remained happening at run time though, especially the one that is a parameter to printf. There really is no reason I can think of why they would need to remain happening at run time, versus happening at compile time, other than (this?) compiler not aggressively moving whatever it can to compile time. I really wish it worked more like that though, and IMO I think it should. Maybe in the future we’ll see compilers move more that direction.

## Compile Time Hash Switching

Something neat about being able to do hashing at compile time, is that you can use the result of a compile time hash as a case value in a switch statement!

Let’s explore that a bit:

    unsigned int hash = crc32("Hello1");  // 1) Run Time.
constexpr unsigned int hashTestHello2 = crc32("Hello2"); // 2) Debug: Run Time. Release: Not calculated at all.
switch (hash) { // 3) Uses variable on stack
case hashTestHello2: {  // 4) Compile Time Constant.
printf("An");
break;
}
case crc32("Hello3"): {  // 5) Compile Time Constant.
printf("Bn");
break;
}
// 6) error C2196: case value '1470747604' already used
/*
case crc32("Hello2"): {
printf("Cn");
break;
}
*/
default: {
printf("Cn");
break;
}
}


Something interesting to note is that if you have duplicate cases in your switch statement, due to things hashing to the same value (either duplicates, or actual hash collisions) that you will get a compile time error. This might come in handy, let’s come back to that later.

Let’s look at the assembly code in debug:

    unsigned int hash = crc32("Hello1");  // 1) Run Time.
00007FF77B4119FE  xor         edx,edx
00007FF77B411A00  lea         rcx,[string "Hello1" (07FF77B41B124h)]
00007FF77B411A07  call        crc32 (07FF77B4110C3h)
00007FF77B411A0C  mov         dword ptr [hash],eax
constexpr unsigned int hashTestHello2 = crc32("Hello2"); // 2) Debug: Run Time. Release: Not calculated at all.
00007FF77B411A0F  xor         edx,edx
00007FF77B411A11  lea         rcx,[string "Hello2" (07FF77B41B12Ch)]
00007FF77B411A18  call        crc32 (07FF77B4110C3h)
00007FF77B411A1D  mov         dword ptr [hashTestHello2],eax
switch (hash) { // 3) Uses variable on stack
00007FF77B411A20  mov         eax,dword ptr [hash]
00007FF77B411A23  mov         dword ptr [rbp+0F4h],eax
00007FF77B411A29  cmp         dword ptr [rbp+0F4h],20AEE342h
00007FF77B411A33  je          Snippet_CompileTimeHashSwitching1+71h (07FF77B411A51h)
00007FF77B411A35  cmp         dword ptr [rbp+0F4h],57A9D3D4h
00007FF77B411A3F  je          Snippet_CompileTimeHashSwitching1+63h (07FF77B411A43h)
00007FF77B411A41  jmp         Snippet_CompileTimeHashSwitching1+7Fh (07FF77B411A5Fh)
case hashTestHello2: {  // 4) Compile Time Constant.
printf("An");
00007FF77B411A43  lea         rcx,[string "An" (07FF77B41B144h)]
00007FF77B411A4A  call        printf (07FF77B4111FEh)
break;
00007FF77B411A4F  jmp         Snippet_CompileTimeHashSwitching1+8Bh (07FF77B411A6Bh)
}
case crc32("Hello3"): {  // 5) Compile Time Constant.
printf("Bn");
00007FF77B411A51  lea         rcx,[string "Bn" (07FF77B41B160h)]
00007FF77B411A58  call        printf (07FF77B4111FEh)
break;
00007FF77B411A5D  jmp         Snippet_CompileTimeHashSwitching1+8Bh (07FF77B411A6Bh)
}
// 6) error C2196: case value '1470747604' already used
/*
case crc32("Hello2"): {
printf("Cn");
break;
}
*/
default: {
printf("Cn");
00007FF77B411A5F  lea         rcx,[string "Cn" (07FF77B41B164h)]
00007FF77B411A66  call        printf (07FF77B4111FEh)
break;
}
}


We can see that the hash for “Hello1” and “Hello2” are both calculated at run time, and that the switch statement uses the stack variable [hash] to move the value into a register to do the switch statement.

Interestingly though, on lines 14 and 16 we can see it moving a constant value into registers to use in a cmp (compare) operation. 0x20AEE342 is the hash value of “Hello3” and 0x57A9D3D4 is the hash value of “Hello2” so it ended up doing those hashes at compile time, even though we are in debug mode. This is because case values must be known at compile time.

It’s interesting to see though that the compiler calculates hashTestHello2 at runtime, even though the only place we use it (in the case statement), it puts a compile time constant from a compile time hash. Odd.

Let’s see what happens in release:

00007FF7FA9B10B4  lea         rdx,[string "Hello1"+1h (07FF7FA9B2211h)]
00007FF7FA9B10BB  mov         ecx,7807C9A2h
00007FF7FA9B10C0  call        crc32_rec (07FF7FA9B1070h)
00007FF7FA9B10C5  cmp         eax,20AEE342h
00007FF7FA9B10CA  je          main+49h (07FF7FA9B10F9h)
00007FF7FA9B10CC  cmp         eax,57A9D3D4h
00007FF7FA9B10D1  je          main+36h (07FF7FA9B10E6h)
00007FF7FA9B10D3  lea         rcx,[string "Cn" (07FF7FA9B2220h)]
00007FF7FA9B10DA  call        printf (07FF7FA9B1010h)
// ... More asm code below, but not relevant


Release is a little more lean which is nice. On line 3 we calculate the hash of “Hello1” at runtime, then on lines 4 and 6, we compare it against the constant values of our compile time hashes for “Hello2” and “Hello3”. That is all that is done at runtime, which is more in line with what we’d like to see the compiler do. It’s still a little bit lame that it didn’t see that “Hello1” was a compile time constant that was being hashed, and did it at runtime, but at least it got most of the hashing to happen at compile time.

What if we change the “hash” variable to be constexpr?

    // Release: this function just prints "Cn" and exits.  All code melted away at compile time!
constexpr unsigned int hash = crc32("Hello1");  // 1) Debug: Run Time
constexpr unsigned int hashTestHello2 = crc32("Hello2"); // 2) Debug: Run Time
switch (hash) { // 3) Debug: Compile Time Constant.
case hashTestHello2: {   // 4) Debug: Compile Time Constant.
printf("An");
break;
}
case crc32("Hello3"): {   // 5) Debug: Compile Time Constant.
printf("Bn");
break;
}
default: {
printf("Cn");
break;
}
}


Let’s check it out in debug first:

constexpr unsigned int hash = crc32("Hello1");  // 1) Debug: Run Time
00007FF71F7A1ABE  xor         edx,edx
00007FF71F7A1AC0  lea         rcx,[string "Hello1" (07FF71F7AB124h)]
00007FF71F7A1AC7  call        crc32 (07FF71F7A10C3h)
00007FF71F7A1ACC  mov         dword ptr [hash],eax
constexpr unsigned int hashTestHello2 = crc32("Hello2"); // 2) Debug: Run Time
00007FF71F7A1ACF  xor         edx,edx
switch (hash) { // 3) Debug: Compile Time Constant.
00007FF71F7A1AE0  mov         dword ptr [rbp+0F4h],0CEA0826Eh
00007FF71F7A1AEA  cmp         dword ptr [rbp+0F4h],20AEE342h
00007FF71F7A1AF4  je          Snippet_CompileTimeHashSwitching2+72h (07FF71F7A1B12h)
00007FF71F7A1AF6  cmp         dword ptr [rbp+0F4h],57A9D3D4h
00007FF71F7A1B00  je          Snippet_CompileTimeHashSwitching2+64h (07FF71F7A1B04h)
00007FF71F7A1B02  jmp         Snippet_CompileTimeHashSwitching2+80h (07FF71F7A1B20h)
case hashTestHello2: {   // 4) Debug: Compile Time Constant.
printf("An");
00007FF71F7A1B04  lea         rcx,[string "An" (07FF71F7AB144h)]
case hashTestHello2: {   // 4) Debug: Compile Time Constant.
printf("An");
00007FF71F7A1B0B  call        printf (07FF71F7A11FEh)
break;
00007FF71F7A1B10  jmp         Snippet_CompileTimeHashSwitching2+8Ch (07FF71F7A1B2Ch)
}
case crc32("Hello3"): {   // 5) Debug: Compile Time Constant.
printf("Bn");
00007FF71F7A1B12  lea         rcx,[string "Bn" (07FF71F7AB160h)]
00007FF71F7A1B19  call        printf (07FF71F7A11FEh)
break;
00007FF71F7A1B1E  jmp         Snippet_CompileTimeHashSwitching2+8Ch (07FF71F7A1B2Ch)
}
default: {
printf("Cn");
00007FF71F7A1B20  lea         rcx,[string "Cn" (07FF71F7AB164h)]
00007FF71F7A1B27  call        printf (07FF71F7A11FEh)
break;
}
}


The code does a runtime hash of “Hello1” and “Hello2” on lines 4 and 9. Then, on line 12, it moves the compile time hash value of “Hello1” into memory. On line 13 it compares that against the compile time hash value of “Hello3”. On line 15 it compares it against the compile time hash value of “Hello2”.

Now let’s check release:

00007FF7B7B91074  lea         rcx,[string "Cn" (07FF7B7B92210h)]
00007FF7B7B9107B  call        printf (07FF7B7B91010h)


Awesome! It was able to do ALL calculation at compile time, and only do the printf at run time. Neat!

As one final test, let’s see what happens when we put a crc32 call straight into the switch statement.

    constexpr unsigned int hashTestHello2 = crc32("Hello2"); // 1) Debug: Run Time. Release: Not calculated at all.
switch (crc32("Hello1")) {  // 2) Always Run Time (!!!)
case hashTestHello2: {  // 3) Compile Time Constant.
printf("An");
break;
}
case crc32("Hello3"): {  // 4) Compile Time Constant.
printf("Bn");
break;
}
default: {
printf("Cn");
break;
}
}


Here’s the debug assembly:

    constexpr unsigned int hashTestHello2 = crc32("Hello2"); // 1) Debug: Run Time. Release: Not calculated at all.
00007FF72ED51B7E  xor         edx,edx
00007FF72ED51B80  lea         rcx,[string "Hello2" (07FF72ED5B12Ch)]
00007FF72ED51B87  call        crc32 (07FF72ED510C3h)
00007FF72ED51B8C  mov         dword ptr [hashTestHello2],eax
switch (crc32("Hello1")) {  // 2) Always Run Time (!!!)
00007FF72ED51B8F  xor         edx,edx
00007FF72ED51B91  lea         rcx,[string "Hello1" (07FF72ED5B124h)]
00007FF72ED51B98  call        crc32 (07FF72ED510C3h)
00007FF72ED51B9D  mov         dword ptr [rbp+0D4h],eax
00007FF72ED51BA3  cmp         dword ptr [rbp+0D4h],20AEE342h
00007FF72ED51BAF  cmp         dword ptr [rbp+0D4h],57A9D3D4h
00007FF72ED51BB9  je          Snippet_CompileTimeHashSwitching3+5Dh (07FF72ED51BBDh)
00007FF72ED51BBB  jmp         Snippet_CompileTimeHashSwitching3+79h (07FF72ED51BD9h)
case hashTestHello2: {  // 3) Compile Time Constant.
printf("An");
00007FF72ED51BBD  lea         rcx,[string "An" (07FF72ED5B144h)]
00007FF72ED51BC4  call        printf (07FF72ED511FEh)
break;
00007FF72ED51BC9  jmp         Snippet_CompileTimeHashSwitching3+85h (07FF72ED51BE5h)
}
case crc32("Hello3"): {  // 4) Compile Time Constant.
printf("Bn");
00007FF72ED51BCB  lea         rcx,[string "Bn" (07FF72ED5B160h)]
00007FF72ED51BD2  call        printf (07FF72ED511FEh)
break;
00007FF72ED51BD7  jmp         Snippet_CompileTimeHashSwitching3+85h (07FF72ED51BE5h)
}
default: {
printf("Cn");
00007FF72ED51BD9  lea         rcx,[string "Cn" (07FF72ED5B164h)]
00007FF72ED51BE0  call        printf (07FF72ED511FEh)
break;
}
}


It was able to do the case value crc’s at compile time, but the other two it did at runtime. Not surprising for debug. Let’s check release:

00007FF6A46D10B4  lea         rdx,[string "Hello1"+1h (07FF6A46D2211h)]
00007FF6A46D10BB  mov         ecx,7807C9A2h
00007FF6A46D10C0  call        crc32_rec (07FF6A46D1070h)
00007FF6A46D10C5  cmp         eax,20AEE342h
00007FF6A46D10CA  je          main+49h (07FF6A46D10F9h)
00007FF6A46D10CC  cmp         eax,57A9D3D4h
00007FF6A46D10D1  je          main+36h (07FF6A46D10E6h)
00007FF6A46D10D3  lea         rcx,[string "Cn" (07FF6A46D2220h)]
00007FF6A46D10DA  call        printf (07FF6A46D1010h)
// ... More asm code below, but not relevant


It did the hash for “Hello1” at run time (why??), but it did the others at release. A bit disappointing that it couldn’t make the “Hello1” hash be compile time, but we saw this behavior before so nothing new there.

## Leveraging Jump Tables

In the above switch statements, the tests for hashes were very much “If hash is a, do this, else if hash is b, do that”. It was an if/else if/else if style chain.

Switch statements actually have the ability to become jump tables though, which let them get to the right case value with fewer comparisons.

Check out this code:

    // Debug: Jump Table
// Release: Just does the constant case, everything else goes away
unsigned int i = 3;
switch (i) {
case 0: printf("An"); break;
case 1: printf("Bn"); break;
case 2: printf("Cn"); break;
case 3: printf("Dn"); break;
case 4: printf("En"); break;
case 5: printf("Fn"); break;
case 6: printf("Gn"); break;
case 7: printf("Hn"); break;
default: printf("Nonen"); break;
}


Here’s the debug assembly:

    // Debug: Jump Table
// Release: Just does the constant case, everything else goes away
unsigned int i = 3;
00007FF69CF9238E  mov         dword ptr [i],3
switch (i) {
00007FF69CF92395  mov         eax,dword ptr [i]
00007FF69CF92398  mov         dword ptr [rbp+0D4h],eax
00007FF69CF9239E  cmp         dword ptr [rbp+0D4h],7
00007FF69CF923A5  ja          $LN11+0Eh (07FF69CF92434h) 00007FF69CF923AB mov eax,dword ptr [rbp+0D4h] 00007FF69CF923B1 lea rcx,[__ImageBase (07FF69CF80000h)] 00007FF69CF923B8 mov eax,dword ptr [rcx+rax*4+1244Ch] 00007FF69CF923BF add rax,rcx 00007FF69CF923C2 jmp rax case 0: printf("An"); break; 00007FF69CF923C4 lea rcx,[string "An" (07FF69CF9B144h)] 00007FF69CF923CB call printf (07FF69CF911FEh) 00007FF69CF923D0 jmp$LN11+1Ah (07FF69CF92440h)
case 1: printf("Bn"); break;
00007FF69CF923D2  lea         rcx,[string "Bn" (07FF69CF9B160h)]
00007FF69CF923D9  call        printf (07FF69CF911FEh)
00007FF69CF923DE  jmp         $LN11+1Ah (07FF69CF92440h) case 2: printf("Cn"); break; 00007FF69CF923E0 lea rcx,[string "Cn" (07FF69CF9B164h)] 00007FF69CF923E7 call printf (07FF69CF911FEh) 00007FF69CF923EC jmp$LN11+1Ah (07FF69CF92440h)
case 3: printf("Dn"); break;
00007FF69CF923EE  lea         rcx,[string "Dn" (07FF69CF9B168h)]
00007FF69CF923F5  call        printf (07FF69CF911FEh)
00007FF69CF923FA  jmp         $LN11+1Ah (07FF69CF92440h) case 4: printf("En"); break; 00007FF69CF923FC lea rcx,[string "En" (07FF69CF9B16Ch)] 00007FF69CF92403 call printf (07FF69CF911FEh) 00007FF69CF92408 jmp$LN11+1Ah (07FF69CF92440h)
case 5: printf("Fn"); break;
00007FF69CF9240A  lea         rcx,[string "Fn" (07FF69CF9B170h)]
00007FF69CF92411  call        printf (07FF69CF911FEh)
00007FF69CF92416  jmp         $LN11+1Ah (07FF69CF92440h) case 6: printf("Gn"); break; 00007FF69CF92418 lea rcx,[string "Gn" (07FF69CF9B174h)] 00007FF69CF9241F call printf (07FF69CF911FEh) 00007FF69CF92424 jmp$LN11+1Ah (07FF69CF92440h)
case 7: printf("Hn"); break;
00007FF69CF92426  lea         rcx,[string "Hn" (07FF69CF9B178h)]
00007FF69CF9242D  call        printf (07FF69CF911FEh)
00007FF69CF92432  jmp         $LN11+1Ah (07FF69CF92440h) default: printf("Nonen"); break; 00007FF69CF92434 lea rcx,[string "Nonen" (07FF69CF9AFDCh)] 00007FF69CF9243B call printf (07FF69CF911FEh) }  On line 8 and 9 it checks to see if the value we are switching on is greater than 7, and if so, jumps to 07FF69CF92434h, which is the “default” case where it prints “None”. If the number is not greater than 7, it calculates an address based on the value we are switching on, and then jumps to it, on line 14. Instead of testing for every possible value in an if/else if/else if/else if type setup, it can go IMMEDIATELY to the code associated with the particular case statement. This is a jump table and can be a big speed improvement if you have a lot of cases, or if the code is called a lot. I’ll spare you the release assembly. The compiler can tell that this is a compile time known result and only has the printf for the “D” without any of the other logic. Let’s go back to our crc32 code and try and leverage a jump table:  // Debug / Release: Does hash and jump table. // Note: It AND's against 7 and then tests to see if i is greater than 7 (!!!) unsigned int i = crc32("Hello") & 7; switch (i) { case 0: printf("An"); break; case 1: printf("Bn"); break; case 2: printf("Cn"); break; case 3: printf("Dn"); break; case 4: printf("En"); break; case 5: printf("Fn"); break; case 6: printf("Gn"); break; case 7: printf("Hn"); break; default: printf("Nonen"); break; }  I’ll show you just the debug assembly for brevity. It does the hash and jump table at runtime for both debug and release. Interestingly, even though we do &7 on the hash value, the switch statement STILL makes sure that the value being switched on is not greater than 7. It can never be greater than 7, and that can be known at compile time, but it still checks. This is true even of the release assembly!  // Debug / Release: Does hash and jump table. // Note: It AND's against 7 and then tests to see if i is greater than 7 (!!!) unsigned int i = crc32("Hello") & 7; 00007FF7439C24CE xor edx,edx 00007FF7439C24D0 lea rcx,[string "Hello" (07FF7439CB180h)] 00007FF7439C24D7 call crc32 (07FF7439C10C3h) 00007FF7439C24DC and eax,7 00007FF7439C24DF mov dword ptr [i],eax switch (i) { 00007FF7439C24E2 mov eax,dword ptr [i] 00007FF7439C24E5 mov dword ptr [rbp+0D4h],eax 00007FF7439C24EB cmp dword ptr [rbp+0D4h],7 00007FF7439C24F2 ja$LN11+0Eh (07FF7439C2581h)
00007FF7439C24F8  mov         eax,dword ptr [rbp+0D4h]
00007FF7439C24FE  lea         rcx,[__ImageBase (07FF7439B0000h)]
00007FF7439C2505  mov         eax,dword ptr [rcx+rax*4+12598h]
00007FF7439C250F  jmp         rax
case 0: printf("An"); break;
00007FF7439C2511  lea         rcx,[string "An" (07FF7439CB144h)]
case 0: printf("An"); break;
00007FF7439C2518  call        printf (07FF7439C11FEh)
00007FF7439C251D  jmp         $LN11+1Ah (07FF7439C258Dh) case 1: printf("Bn"); break; 00007FF7439C251F lea rcx,[string "Bn" (07FF7439CB160h)] 00007FF7439C2526 call printf (07FF7439C11FEh) 00007FF7439C252B jmp$LN11+1Ah (07FF7439C258Dh)
case 2: printf("Cn"); break;
00007FF7439C252D  lea         rcx,[string "Cn" (07FF7439CB164h)]
00007FF7439C2534  call        printf (07FF7439C11FEh)
00007FF7439C2539  jmp         $LN11+1Ah (07FF7439C258Dh) case 3: printf("Dn"); break; 00007FF7439C253B lea rcx,[string "Dn" (07FF7439CB168h)] 00007FF7439C2542 call printf (07FF7439C11FEh) 00007FF7439C2547 jmp$LN11+1Ah (07FF7439C258Dh)
case 4: printf("En"); break;
00007FF7439C2549  lea         rcx,[string "En" (07FF7439CB16Ch)]
00007FF7439C2550  call        printf (07FF7439C11FEh)
00007FF7439C2555  jmp         $LN11+1Ah (07FF7439C258Dh) case 5: printf("Fn"); break; 00007FF7439C2557 lea rcx,[string "Fn" (07FF7439CB170h)] 00007FF7439C255E call printf (07FF7439C11FEh) 00007FF7439C2563 jmp$LN11+1Ah (07FF7439C258Dh)
case 6: printf("Gn"); break;
00007FF7439C2565  lea         rcx,[string "Gn" (07FF7439CB174h)]
00007FF7439C256C  call        printf (07FF7439C11FEh)
00007FF7439C2571  jmp         $LN11+1Ah (07FF7439C258Dh) case 7: printf("Hn"); break; 00007FF7439C2573 lea rcx,[string "Hn" (07FF7439CB178h)] 00007FF7439C257A call printf (07FF7439C11FEh) 00007FF7439C257F jmp$LN11+1Ah (07FF7439C258Dh)
default: printf("Nonen"); break;
00007FF7439C2581  lea         rcx,[string "Nonen" (07FF7439CAFDCh)]
00007FF7439C2588  call        printf (07FF7439C11FEh)
}


If we make “i” be a constexpr variable, it doesn’t affect debug, but in release, it is able to melt away all the code and just prints the correct case.

00007FF6093F1074  lea         rcx,[string "An" (07FF6093F2210h)]
00007FF6093F107B  call        printf (07FF6093F1010h)


## Perfect Hashing

Perfect hashing is when you have a known set of inputs, and a hash function such that there is no collision between any of the inputs.

Perfect hashing can be great for being able to turn complex objects into IDs for faster lookup times, but the IDs are not usually going to be contiguous. One ID might be 3, and another might be 45361. Perfect hashing can still be useful though.

The code below shows show some compile time perfect hashing could be achieved.

    // Debug: Does have some sort of jump table setup, despite the cases not being continuous.
// Release: prints "An".  All other code melts away at compile time.
static const unsigned int c_numBuckets = 16;
static const unsigned int c_salt = 1337;

constexpr unsigned int i = crc32("Identifier_A", c_salt) % c_numBuckets;
switch (i) {
case (crc32("Identifier_A", c_salt) % c_numBuckets): printf("An"); break;
case (crc32("Identifier_B", c_salt) % c_numBuckets): printf("Bn"); break;
case (crc32("Identifier_C", c_salt) % c_numBuckets): printf("Cn"); break;
case (crc32("Identifier_D", c_salt) % c_numBuckets): printf("Dn"); break;
case (crc32("Identifier_E", c_salt) % c_numBuckets): printf("En"); break;
case (crc32("Identifier_F", c_salt) % c_numBuckets): printf("Fn"); break;
case (crc32("Identifier_G", c_salt) % c_numBuckets): printf("Gn"); break;
case (crc32("Identifier_H", c_salt) % c_numBuckets): printf("Hn"); break;
default: printf("Nonen"); break;
}


One nice thing about doing perfect hashing at compile time like this is that if you ever have a hash collision, you’ll have a duplicate case in your switch statement, and will get a compile error. This means that you are guaranteed that your perfect hash is valid at runtime. With non compile time perfect hashes, you could easily get into a situation where you added some more valid inputs and may now have a hash collision, which would make hard to track subtle bugs as two input values would be sharing the same index for whatever read and/or write data they wanted to interact with.

You might notice that i am doing %16 on the hash instead of %8, even though there are only 8 items I want to test against.

The reason for that is hash collisions.

When you hash a string you are effectively getting a pseudo random number back that will always be the same for when that string is given as input.

For the above to work correctly and let me modulus against 8, i would have to roll an 8 sided dice 8 times and get no repeats.

There are 8! (8 factorial) different ways to roll that 8 sided dice 8 times and not get any collisions.

There are 8^8 different ways to roll the 8 sided dice 8 times total.

To get the chance that we roll an 8 sided dice 8 times and get no repeats (no hash collisions), the probability is 8! / 8^8 or 0.24%.

The salt in the crc32 function allows us to effectively re-roll our dice. Each salt value is a different roll of the dice.

How that fits in is that 0.24% of all salt values should let us be able to do %8 on our hashes and not have a hash collision.

Those odds are pretty bad, but they aren’t SUPER bad. We could just brute force search to find a good salt value to use and then hard code the salt in, like i did in the example above.

Unfortunately, the probability I calculated above only works if the hash function gives uniformly distributed output and is “truly random”.

In practice no hash function or pseudo random number generator is, and in fact this crc32 function has NO salt values which make for no collisions! I brute force tested all 2^32 (4.2 billion) possible salt values and came up with no salt that worked!

To get around that problem, instead of trying to get a perfect fit of 8 hashes with values 0-7 without collisions, i opted to go for 8 hashes with values 0-15 with no collisions. That changes my odds for the better, and there are in fact many salts that satisfy that.

It’s the equivalent of rolling a 16 sided dice 8 times without repeats.

Thinking about things a bit differently than before, the first roll has a 100% chance of not being a duplicate. The next roll has a 15/16 chance of not being duplicate. The next has a 14/16 chance and so on until the 8th roll which has a 9/16 chance.

Multiplying those all together, we end up with a 12.08% chance of rolling a 16 sided dice 8 times and not getting a duplicate. That means 12% of the salts (about 1 in 9) won’t produce collisions when we use 16 buckets, which makes it much easier for us to find a salt to use.

Looking at the disassembly in debug, we can see the jump table is miraculously in tact! This is great because now we can get compile time assured perfect hashing of objects, and can use a jump table to convert a runtime object into a perfect hash result.

Note that in release, the code melts away and just does the “correct printf”.

    // Debug: Does have some sort of jump table setup, despite the cases not being continuous.
// Release: prints "An".  All other code melts away at compile time.
static const unsigned int c_numBuckets = 16;
static const unsigned int c_salt = 1337;

constexpr unsigned int i = crc32("Identifier_A", c_salt) % c_numBuckets;
00007FF7CE651E5E  mov         edx,539h
00007FF7CE651E63  lea         rcx,[string "Identifier_A" (07FF7CE65B190h)]
00007FF7CE651E6A  call        crc32 (07FF7CE6510C3h)
00007FF7CE651E6F  xor         edx,edx
00007FF7CE651E71  mov         ecx,10h
00007FF7CE651E76  div         eax,ecx
00007FF7CE651E78  mov         eax,edx
00007FF7CE651E7A  mov         dword ptr [i],eax
switch (i) {
00007FF7CE651E7D  mov         dword ptr [rbp+0D4h],4
00007FF7CE651E87  cmp         dword ptr [rbp+0D4h],0Eh
00007FF7CE651E8E  ja          $LN11+0Eh (07FF7CE651F1Dh) 00007FF7CE651E94 mov eax,dword ptr [rbp+0D4h] 00007FF7CE651E9A lea rcx,[__ImageBase (07FF7CE640000h)] 00007FF7CE651EA1 mov eax,dword ptr [rcx+rax*4+11F34h] 00007FF7CE651EA8 add rax,rcx 00007FF7CE651EAB jmp rax case (crc32("Identifier_A", c_salt) % c_numBuckets): printf("An"); break; 00007FF7CE651EAD lea rcx,[string "An" (07FF7CE65B144h)] 00007FF7CE651EB4 call printf (07FF7CE6511FEh) 00007FF7CE651EB9 jmp$LN11+1Ah (07FF7CE651F29h)
case (crc32("Identifier_B", c_salt) % c_numBuckets): printf("Bn"); break;
00007FF7CE651EBB  lea         rcx,[string "Bn" (07FF7CE65B160h)]
00007FF7CE651EC2  call        printf (07FF7CE6511FEh)
00007FF7CE651EC7  jmp         $LN11+1Ah (07FF7CE651F29h) case (crc32("Identifier_C", c_salt) % c_numBuckets): printf("Cn"); break; 00007FF7CE651EC9 lea rcx,[string "Cn" (07FF7CE65B164h)] 00007FF7CE651ED0 call printf (07FF7CE6511FEh) 00007FF7CE651ED5 jmp$LN11+1Ah (07FF7CE651F29h)
case (crc32("Identifier_D", c_salt) % c_numBuckets): printf("Dn"); break;
00007FF7CE651ED7  lea         rcx,[string "Dn" (07FF7CE65B168h)]
00007FF7CE651EDE  call        printf (07FF7CE6511FEh)
00007FF7CE651EE3  jmp         $LN11+1Ah (07FF7CE651F29h) case (crc32("Identifier_E", c_salt) % c_numBuckets): printf("En"); break; 00007FF7CE651EE5 lea rcx,[string "En" (07FF7CE65B16Ch)] 00007FF7CE651EEC call printf (07FF7CE6511FEh) 00007FF7CE651EF1 jmp$LN11+1Ah (07FF7CE651F29h)
case (crc32("Identifier_F", c_salt) % c_numBuckets): printf("Fn"); break;
00007FF7CE651EF3  lea         rcx,[string "Fn" (07FF7CE65B170h)]
00007FF7CE651EFA  call        printf (07FF7CE6511FEh)
00007FF7CE651EFF  jmp         $LN11+1Ah (07FF7CE651F29h) case (crc32("Identifier_G", c_salt) % c_numBuckets): printf("Gn"); break; 00007FF7CE651F01 lea rcx,[string "Gn" (07FF7CE65B174h)] 00007FF7CE651F08 call printf (07FF7CE6511FEh) 00007FF7CE651F0D jmp$LN11+1Ah (07FF7CE651F29h)
case (crc32("Identifier_H", c_salt) % c_numBuckets): printf("Hn"); break;
00007FF7CE651F0F  lea         rcx,[string "Hn" (07FF7CE65B178h)]
00007FF7CE651F16  call        printf (07FF7CE6511FEh)
00007FF7CE651F1B  jmp         $LN11+1Ah (07FF7CE651F29h) default: printf("Nonen"); break; 00007FF7CE651F1D lea rcx,[string "Nonen" (07FF7CE65AFDCh)] 00007FF7CE651F24 call printf (07FF7CE6511FEh) }  ## Minimally Perfect Hashing Minimally perfect hashing is like perfect hashing, except the results are contiguous values. If you have 8 possible inputs to your minimally perfect hash function, you are going to get as output 0-7. The order of what inputs map to which outputs isn’t strictly defined unless you want to go through a lot of extra effort to make it be that way though. This is even more useful than perfect hashing, as you can hash a (known good) input and use the result as an index into an array, or similar! For more info on MPH, check out my post on it: O(1) Data Lookups With Minimal Perfect Hashing The code below is a way of doing compile time assisted minimally perfect hashing:  // Debug / Release: // Runs crc32 at runtime only for "i". The cases are compile time constants as per usual. // Does a jumptable type setup for the switch and does fallthrough to do multiple increments to get the right ID. // // Release with constexpr on i: // does the printf with a value of 2. The rest of the code melts away. static const unsigned int c_numBuckets = 16; static const unsigned int c_salt = 1337; static const unsigned int c_invalidID = -1; unsigned int i = crc32("Identifier_F", c_salt) % c_numBuckets; unsigned int id = c_invalidID; switch (i) { case (crc32("Identifier_A", c_salt) % c_numBuckets): ++id; case (crc32("Identifier_B", c_salt) % c_numBuckets): ++id; case (crc32("Identifier_C", c_salt) % c_numBuckets): ++id; case (crc32("Identifier_D", c_salt) % c_numBuckets): ++id; case (crc32("Identifier_E", c_salt) % c_numBuckets): ++id; case (crc32("Identifier_F", c_salt) % c_numBuckets): ++id; case (crc32("Identifier_G", c_salt) % c_numBuckets): ++id; case (crc32("Identifier_H", c_salt) % c_numBuckets): ++id; // the two lines below are implicit behavior of how this code works // break; // default: id = c_invalidID; break; } printf("id = %in", id);  Here’s the debug assembly for the above. The release does similar, so i’m not showing it.  // Debug / Release: // Runs crc32 at runtime only for "i". The cases are compile time constants as per usual. // Does a jumptable type setup for the switch and does fallthrough to do multiple increments to get the right ID. // // Release with constexpr on i: // does the printf with a value of 2. The rest of the code melts away. static const unsigned int c_numBuckets = 16; static const unsigned int c_salt = 1337; static const unsigned int c_invalidID = -1; unsigned int i = crc32("Identifier_F", c_salt) % c_numBuckets; 00007FF79E481D0E mov edx,539h 00007FF79E481D13 lea rcx,[string "Identifier_F" (07FF79E48B1E0h)] 00007FF79E481D1A call crc32 (07FF79E4810C3h) 00007FF79E481D1F xor edx,edx 00007FF79E481D21 mov ecx,10h 00007FF79E481D26 div eax,ecx 00007FF79E481D28 mov eax,edx 00007FF79E481D2A mov dword ptr [i],eax unsigned int id = c_invalidID; 00007FF79E481D2D mov dword ptr [id],0FFFFFFFFh switch (i) { 00007FF79E481D34 mov eax,dword ptr [i] 00007FF79E481D37 mov dword ptr [rbp+0F4h],eax 00007FF79E481D3D cmp dword ptr [rbp+0F4h],0Eh 00007FF79E481D44 ja$LN11+8h (07FF79E481D9Fh)
00007FF79E481D46  mov         eax,dword ptr [rbp+0F4h]
00007FF79E481D4C  lea         rcx,[__ImageBase (07FF79E470000h)]
00007FF79E481D53  mov         eax,dword ptr [rcx+rax*4+11DB8h]
00007FF79E481D5D  jmp         rax
case (crc32("Identifier_A", c_salt) % c_numBuckets): ++id;
00007FF79E481D5F  mov         eax,dword ptr [id]
00007FF79E481D62  inc         eax
00007FF79E481D64  mov         dword ptr [id],eax
case (crc32("Identifier_B", c_salt) % c_numBuckets): ++id;
00007FF79E481D67  mov         eax,dword ptr [id]
00007FF79E481D6A  inc         eax
00007FF79E481D6C  mov         dword ptr [id],eax
case (crc32("Identifier_C", c_salt) % c_numBuckets): ++id;
00007FF79E481D6F  mov         eax,dword ptr [id]
00007FF79E481D72  inc         eax
00007FF79E481D74  mov         dword ptr [id],eax
case (crc32("Identifier_D", c_salt) % c_numBuckets): ++id;
00007FF79E481D77  mov         eax,dword ptr [id]
00007FF79E481D7A  inc         eax
00007FF79E481D7C  mov         dword ptr [id],eax
case (crc32("Identifier_E", c_salt) % c_numBuckets): ++id;
00007FF79E481D7F  mov         eax,dword ptr [id]
00007FF79E481D82  inc         eax
00007FF79E481D84  mov         dword ptr [id],eax
case (crc32("Identifier_F", c_salt) % c_numBuckets): ++id;
00007FF79E481D87  mov         eax,dword ptr [id]
00007FF79E481D8A  inc         eax
00007FF79E481D8C  mov         dword ptr [id],eax
case (crc32("Identifier_G", c_salt) % c_numBuckets): ++id;
00007FF79E481D8F  mov         eax,dword ptr [id]
00007FF79E481D92  inc         eax
00007FF79E481D94  mov         dword ptr [id],eax
case (crc32("Identifier_H", c_salt) % c_numBuckets): ++id;
00007FF79E481D97  mov         eax,dword ptr [id]
00007FF79E481D9A  inc         eax
00007FF79E481D9C  mov         dword ptr [id],eax
// the two lines below are implicit behavior of how this code works
// break;
// default: id = c_invalidID; break;
}

printf("id = %in", id);
00007FF79E481D9F  mov         edx,dword ptr [id]
00007FF79E481DA2  lea         rcx,[string "id = %in" (07FF79E48B238h)]
// the two lines below are implicit behavior of how this code works
// break;
// default: id = c_invalidID; break;
}

printf("id = %in", id);
00007FF79E481DA9  call        printf (07FF79E4811FEh)


As you can see, the jump table is still in tact, which is good, but it does a lot of repeated increments to get the right ID values. I wish the compiler were smart enough to “flatten” this and just give each case it’s proper ID value.

As is, this could be a performance issue if you had a very large number of inputs.

You could always just hard code a number there instead of relying on the fallthrough and increment, but then there is a lot of copy pasting. Maybe you could do something clever with macros or templates to help that though.

## Compile Time Assisted String To Enum

Another interesting thing to think about is that we could actually use compile time hashing to convert a string to an enum.

In this case, let’s say that we don’t know if our input is valid or not. Since we don’t know that, we have to switch on the hash of our input string, but then do a string compare against whatever string has that hash, to make sure it matches. If it does match, it should take that enum value, else it should be invalid.

Since that would be a lot of error prone copy/pasting, I simplified things a bit by using a macro list:

    // Debug / Release:
//   Runs crc32 at runtime only for "i".  The cases are compile time constants as per usual.
//   Does a jumptable type setup for the switch and does a string comparison against the correct string.
//   If strings are equal, sets the enum value.

static const unsigned int c_numBuckets = 16;
static const unsigned int c_salt = 1337;

const char* testString = "Identifier_F";
unsigned int i = crc32(testString, c_salt) % c_numBuckets;

// This macro list is used for:
//  * making the enum
//  * making the cases in the switch statement
// D.R.Y. - Don't Repeat Yourself.
// Fewer moving parts = fewer errors, but admittedly is harder to understand vs redundant code.
#define ENUM_VALUE_LIST
VALUE(Identifier_A)
VALUE(Identifier_B)
VALUE(Identifier_C)
VALUE(Identifier_D)
VALUE(Identifier_E)
VALUE(Identifier_F)
VALUE(Identifier_G)
VALUE(Identifier_H)

// Make the enum values.
// Note these enum values are also usable as a contiguous ID if you needed one for an array index or similar.
// You could define an array with size EIdentifier::count for instance and use these IDs to index into it.
enum class EIdentifier : unsigned char {
#define VALUE(x) x,
ENUM_VALUE_LIST
#undef VALUE
count,
invalid = (unsigned char)-1
};

// do a compile time hash assisted string comparison to convert string to enum
EIdentifier identifier = EIdentifier::invalid;
switch (i) {
#define VALUE(x) case (crc32(#x, c_salt) % c_numBuckets) : if(!strcmp(testString, #x)) identifier = EIdentifier::x; else identifier = EIdentifier::invalid; break;
ENUM_VALUE_LIST
#undef VALUE
default: identifier = EIdentifier::invalid;
}

// undefine the enum value list
#undef ENUM_VALUE_LIST

printf("string translated to enum value %i", identifier);


Here’s the debug assembly showing it working like it’s supposed to:

    // Debug / Release:
//   Runs crc32 at runtime only for "i".  The cases are compile time constants as per usual.
//   Does a jumptable type setup for the switch and does a string comparison against the correct string.
//   If strings are equal, sets the enum value.

static const unsigned int c_numBuckets = 16;
static const unsigned int c_salt = 1337;

const char* testString = "Identifier_F";
00007FF6B188179E  lea         rax,[string "Identifier_F" (07FF6B188B1E0h)]
00007FF6B18817A5  mov         qword ptr [testString],rax
unsigned int i = crc32(testString, c_salt) % c_numBuckets;
00007FF6B18817A9  mov         edx,539h
00007FF6B18817AE  mov         rcx,qword ptr [testString]
00007FF6B18817B2  call        crc32 (07FF6B18810C3h)
00007FF6B18817B7  xor         edx,edx
00007FF6B18817B9  mov         ecx,10h
00007FF6B18817BE  div         eax,ecx
00007FF6B18817C0  mov         eax,edx
00007FF6B18817C2  mov         dword ptr [i],eax

// This macro list is used for:
//  * making the enum
//  * making the cases in the switch statement
// D.R.Y. - Don't Repeat Yourself.
// Fewer moving parts = fewer errors, but admittedly is harder to understand vs redundant code.
#define ENUM_VALUE_LIST
VALUE(Identifier_A)
VALUE(Identifier_B)
VALUE(Identifier_C)
VALUE(Identifier_D)
VALUE(Identifier_E)
VALUE(Identifier_F)
VALUE(Identifier_G)
VALUE(Identifier_H)

// Make the enum values.
// Note these enum values are also usable as a contiguous ID if you needed one for an array index or similar.
// You could define an array with size EIdentifier::count for instance and use these IDs to index into it.
enum class EIdentifier : unsigned char {
#define VALUE(x) x,
ENUM_VALUE_LIST
#undef VALUE
count,
invalid = (unsigned char)-1
};

// do a compile time hash assisted string comparison to convert string to enum
EIdentifier identifier = EIdentifier::invalid;
00007FF6B18817C5  mov         byte ptr [identifier],0FFh
switch (i) {
00007FF6B18817C9  mov         eax,dword ptr [i]
00007FF6B18817CC  mov         dword ptr [rbp+114h],eax
00007FF6B18817D2  cmp         dword ptr [rbp+114h],0Eh
switch (i) {
00007FF6B18817D9  ja          $LN25+20h (07FF6B1881904h) 00007FF6B18817DF mov eax,dword ptr [rbp+114h] 00007FF6B18817E5 lea rcx,[__ImageBase (07FF6B1870000h)] 00007FF6B18817EC mov eax,dword ptr [rcx+rax*4+11924h] 00007FF6B18817F3 add rax,rcx 00007FF6B18817F6 jmp rax #define VALUE(x) case (crc32(#x, c_salt) % c_numBuckets) : if(!strcmp(testString, #x)) identifier = EIdentifier::x; else identifier = EIdentifier::invalid; break; ENUM_VALUE_LIST 00007FF6B18817F8 lea rdx,[string "Identifier_A" (07FF6B188B190h)] 00007FF6B18817FF mov rcx,qword ptr [testString] 00007FF6B1881803 call strcmp (07FF6B18811CCh) 00007FF6B1881808 test eax,eax 00007FF6B188180A jne Snippet_CompileTimeHashAssistedStringToEnum+92h (07FF6B1881812h) 00007FF6B188180C mov byte ptr [identifier],0 00007FF6B1881810 jmp Snippet_CompileTimeHashAssistedStringToEnum+96h (07FF6B1881816h) 00007FF6B1881812 mov byte ptr [identifier],0FFh 00007FF6B1881816 jmp$LN25+24h (07FF6B1881908h)
$LN7: 00007FF6B188181B lea rdx,[string "Identifier_B" (07FF6B188B1A0h)] 00007FF6B1881822 mov rcx,qword ptr [testString] 00007FF6B1881826 call strcmp (07FF6B18811CCh) 00007FF6B188182B test eax,eax 00007FF6B188182D jne Snippet_CompileTimeHashAssistedStringToEnum+0B5h (07FF6B1881835h) 00007FF6B188182F mov byte ptr [identifier],1 00007FF6B1881833 jmp Snippet_CompileTimeHashAssistedStringToEnum+0B9h (07FF6B1881839h) 00007FF6B1881835 mov byte ptr [identifier],0FFh 00007FF6B1881839 jmp$LN25+24h (07FF6B1881908h)
$LN10: 00007FF6B188183E lea rdx,[string "Identifier_C" (07FF6B188B1B0h)] 00007FF6B1881845 mov rcx,qword ptr [testString] 00007FF6B1881849 call strcmp (07FF6B18811CCh) 00007FF6B188184E test eax,eax 00007FF6B1881850 jne Snippet_CompileTimeHashAssistedStringToEnum+0D8h (07FF6B1881858h) 00007FF6B1881852 mov byte ptr [identifier],2 00007FF6B1881856 jmp Snippet_CompileTimeHashAssistedStringToEnum+0DCh (07FF6B188185Ch) 00007FF6B1881858 mov byte ptr [identifier],0FFh 00007FF6B188185C jmp$LN25+24h (07FF6B1881908h)
$LN13: 00007FF6B1881861 lea rdx,[string "Identifier_D" (07FF6B188B1C0h)] 00007FF6B1881868 mov rcx,qword ptr [testString] 00007FF6B188186C call strcmp (07FF6B18811CCh) 00007FF6B1881871 test eax,eax 00007FF6B1881873 jne Snippet_CompileTimeHashAssistedStringToEnum+0FBh (07FF6B188187Bh) 00007FF6B1881875 mov byte ptr [identifier],3 00007FF6B1881879 jmp Snippet_CompileTimeHashAssistedStringToEnum+0FFh (07FF6B188187Fh) 00007FF6B188187B mov byte ptr [identifier],0FFh 00007FF6B188187F jmp$LN25+24h (07FF6B1881908h)
$LN16: 00007FF6B1881884 lea rdx,[string "Identifier_E" (07FF6B188B1D0h)] 00007FF6B188188B mov rcx,qword ptr [testString] 00007FF6B188188F call strcmp (07FF6B18811CCh) 00007FF6B1881894 test eax,eax 00007FF6B1881896 jne Snippet_CompileTimeHashAssistedStringToEnum+11Eh (07FF6B188189Eh) 00007FF6B1881898 mov byte ptr [identifier],4 00007FF6B188189C jmp Snippet_CompileTimeHashAssistedStringToEnum+122h (07FF6B18818A2h) 00007FF6B188189E mov byte ptr [identifier],0FFh 00007FF6B18818A2 jmp$LN25+24h (07FF6B1881908h)
$LN19: 00007FF6B18818A4 lea rdx,[string "Identifier_F" (07FF6B188B1E0h)] 00007FF6B18818AB mov rcx,qword ptr [testString] 00007FF6B18818AF call strcmp (07FF6B18811CCh) 00007FF6B18818B4 test eax,eax 00007FF6B18818B6 jne Snippet_CompileTimeHashAssistedStringToEnum+13Eh (07FF6B18818BEh) 00007FF6B18818B8 mov byte ptr [identifier],5 00007FF6B18818BC jmp Snippet_CompileTimeHashAssistedStringToEnum+142h (07FF6B18818C2h) 00007FF6B18818BE mov byte ptr [identifier],0FFh 00007FF6B18818C2 jmp$LN25+24h (07FF6B1881908h)
$LN22: 00007FF6B18818C4 lea rdx,[string "Identifier_G" (07FF6B188B1F0h)] 00007FF6B18818CB mov rcx,qword ptr [testString] 00007FF6B18818CF call strcmp (07FF6B18811CCh) 00007FF6B18818D4 test eax,eax 00007FF6B18818D6 jne Snippet_CompileTimeHashAssistedStringToEnum+15Eh (07FF6B18818DEh) 00007FF6B18818D8 mov byte ptr [identifier],6 00007FF6B18818DC jmp Snippet_CompileTimeHashAssistedStringToEnum+162h (07FF6B18818E2h) 00007FF6B18818DE mov byte ptr [identifier],0FFh 00007FF6B18818E2 jmp$LN25+24h (07FF6B1881908h)
$LN25: 00007FF6B18818E4 lea rdx,[string "Identifier_H" (07FF6B188B200h)] 00007FF6B18818EB mov rcx,qword ptr [testString] 00007FF6B18818EF call strcmp (07FF6B18811CCh) 00007FF6B18818F4 test eax,eax 00007FF6B18818F6 jne$LN25+1Ah (07FF6B18818FEh)
00007FF6B18818F8  mov         byte ptr [identifier],7
00007FF6B18818FC  jmp         $LN25+1Eh (07FF6B1881902h) 00007FF6B18818FE mov byte ptr [identifier],0FFh 00007FF6B1881902 jmp$LN25+24h (07FF6B1881908h)
#undef VALUE
default: identifier = EIdentifier::invalid;
00007FF6B1881904  mov         byte ptr [identifier],0FFh
}

// undefine the enum value list
#undef ENUM_VALUE_LIST

printf("string translated to enum value %i", identifier);
00007FF6B1881908  movzx       eax,byte ptr [identifier]
00007FF6B188190C  mov         edx,eax
00007FF6B188190E  lea         rcx,[string "string translated to enum value "... (07FF6B188B248h)]
00007FF6B1881915  call        printf (07FF6B18811FEh)


## Other Possibilities

Besides the above, I think there is a lot of other really great uses for constexpr functions.

For instance, i’d like to see how it’d work to have compile time data structures to do faster read access of constant data.

I want to see compile time trees, hash tables, sparse arrays, bloom filters, and more.

I believe they have the potential to be a lot more performant than even static data structures, since empty or unaccessed sections of the data structure could possibly melt away when the optimizer does it’s pass.

It may not turn out like that, but I think it’d be interesting to investigate it deeper and see how it goes.

I’d also like to see compilers get more aggressive about doing whatever it can at compile time. If there’s no reason for it to happen at runtime, why make it happen then? It is only going to make things slower.

You can find the source code for the code snippets in this post here: Github Atrix256/RandomCode/CompileTimeCRC/

Here’s a couple interesting discussions on constexpr on stack overflow:
detecting execution time of a constexpr function
How to ensure constexpr function never called at runtime?

# Path Tracing – Getting Started With Diffuse and Emissive

The images below were path traced using 100,000 samples per pixel, using the sample code provided in this post.

Path tracing is a specific type of ray tracing that aims to make photo realistic images by solving something called the rendering equation. Path tracing relies heavily on a method called Monte Carlo integration to get an approximate solution. In fact, it’s often called “Monte Carlo Path Tracing”, but I’ll refer to it as just “Path Tracing”.

In solving the rendering equation, path tracing automatically gets many graphical “features” which are cutting edge research topics of real time graphics, such as soft shadows, global illumination, color bleeding and ambient occlusion.

Unfortunately, path tracing can take a very long time to render – like minutes, hours, or days even, depending on scene complexity, image quality desired and specific algorithms used. Despite that, learning path tracing can still be very useful for people doing real time rendering, as it can give you a “ground truth” to compare your real time approximating algorithms to, and can also help you understand the underlying physical processes involved, to help you make better real time approximations. Sometimes even, data is created offline using path tracing, and is “baked out” so that it can be a quick and simple lookup during runtime.

There is a lot of really good information out there about path tracing, walking you through the rendering equations, monte carlo integration, and the dozen or so relevant statistical topics required to do monte carlo integration.

While understanding that stuff is important if you really want to get the most out of path tracing, this series of blog posts is meant to be more informal, explaining more what to do, and less deeply about why to do it.

When and if you are looking for resources that go deeper into the why, I highly recommend starting with these!

## Source Code

You can find the source code that goes along with this post here:

Github: Atrix256/RandomCode/PTBlogPost1/

PTBlogPost1.zip

## The Rendering Equation

The rendering equation might look a bit scary at first but stay with me, it is actually a lot simpler than it looks.

$L_o( \omega_o)= L_e(\omega_o)+\int_{\Omega}{f(\omega_i, \omega_o)L_i(\omega_i)(\omega_i \cdot n)\mathrm{d}\omega_i}$

Here’s a simplified version:

$LightOut(ViewDirection) = EmissiveLight(ViewDirection) + ReflectedLight(ViewDirection,AllDirections)$

In other words, the light you see when looking at an object is made up of how much it glows in your direction (like how a lightbulb or a hot coal in a fireplace glows), and also how much light is reflected in your direction, from light that is hitting that point on the object from all other directions.

It’s pretty easy to know how much an object is glowing in a particular direction. In the sample code, I just let a surface specify how much it glows (an emissive color) and use that for the object’s glow at any point on the surface, at any viewing angle.

The rest of the equation is where it gets harder. The rest of the equation is this:

$\int_{\Omega}{f(\omega_i, \omega_o)L_i(\omega_i)(\omega_i \cdot n)\mathrm{d}\omega_i}$

The integral (the $\int_{\Omega}$ at the front and the $\mathrm{d}\omega_i$ at the back) just means that we are going to take the result of everything between those two symbols, and add them up for every point in a hemisphere, multiplying each value by the fractional size of the point’s area for the hemisphere. The hemisphere we are talking about is the positive hemisphere surrounding the normal of the surface we are looking at.

One of the reasons things get harder at this point is that there are an infinite number of points on the hemisphere.

Let’s ignore the integral for a second and talk about the rest of the equation. In other words, lets consider only one of the points on the hemisphere for now.

• $f(\omega_i, \omega_o)$ – This is the “Bidirectional reflectance distribution function”, otherwise known as the BRDF. It describes how much light is reflected towards the view direction, of the light coming in from the point on the hemisphere we are considering.
• $L_i(\omega_i)$ – This is how much light is coming in from the point on the hemisphere we are considering.
• $(\omega_i \cdot n)$ – This is the cosine of the angle between the point on the hemisphere we are considering and the surface normal, gotten via a dot product. What this term means is that as the light direction gets more perpendicular to the normal, light is reflected less and less. This is based on the actual behavior of light and you can read more about it here if you want to: Wikipedia: Lambert’s Cosine Law. Here is another great link about it lambertian.pdf. (Thanks for the link Jay!)

So what this means is that for a specific point on the hemisphere, we find out how much light is coming in from that direction, multiply it by how much the BRDF says light should be reflected towards the view direction from that direction, and then apply Lambert’s cosine law to make light dimmer as the light direction gets more perpendicular with the surface normal (more parallel with the surface).

Hopefully that makes enough sense.

Bringing the integral back into the mix, we have to sum up the results of that process for each of the infinite points on the hemisphere, multiplying each value by the fractional size of the point’s area for the hemisphere. When we’ve done that, we have our answer, and have our final pixel color.

This is where Monte Carlo integration comes in. Since we can’t really sum the infinite points, multiplied by their area (which is infinitely small), we are instead going to take a lot of random samples of the hemisphere and average them together. The more samples we take, the closer we get to the actual correct value. Not too hard right?

Now that we have the problem of the infinite points on the hemisphere solved, we actually have a second infinity to deal with.

The light incoming from a specific direction (a point on the hemisphere) is just the amount of light leaving a different object from that direction. So, to find out how much light is coming in from that direction, you have to find what object is in that direction, and calculate how much light is leaving that direction for that object. The problem is, the amount of light leaving that direction for that object is in fact calculated using the rendering equation, so it in turn is based on light leaving a different object and so on. It continues like this, possibly infinitely, and even possibly in loops, where light bounces between objects over and over (like putting two mirrors facing eachother). We have possibly infinite recursion!

The way this is dealt with in path tracers is to just limit the maximum amount of bounces that are considered. Higher numbers of bounces gives diminishing returns in general, so usually it’s just the first couple of bounces that really make a difference. For instance, the images at the top of this post were made using 5 bounces.

## The Algorithm

Now that we know how the rendering equation works, and what we need to do to solve it, let’s write up the algorithm that we perform for each pixel on the screen.

1. First, we calculate the camera ray for the pixel.
2. If the camera ray doesn’t hit an object, the pixel is black.
3. If the camera ray does hit an object, the pixel’s color is determined by how much light that object is emitting and reflecting down the camera ray.
4. To figure out how much light that is, we choose a random direction in the hemisphere of that object’s normal and recurse.
5. At each stage of the recursion, we return: EmittedLight + 2 * RecursiveLight * Dot(Normal, RandomHemisphereAngle) * SurfaceDiffuseColor.
6. If we ever reach the maximum number of bounces, we return black for the RecursiveLight value.

We do the above multiple times per pixel and average the results together. The more times we do the process (the more samples we have), the closer the result gets to the correct answer.

By the way, the multiplication by 2 in step five is a byproduct of some math that comes out of integrating the BRDF. Check the links i mentioned at the top of the post for more info, or you can at least verify that I’m not making it up by seeing that wikipedia says the same thing. There is also some nice psuedo code there! (:
Wikipedia: Path Tracing: Algorithm

## Calculating Camera Rays

There are many ways to calculate camera rays, but here’s the method I used.

In this post we are going to path trace using a pin hole camera. In a future post we’ll switch to using a lens to get interesting lens effects like depth of field.

To generate rays for our pin hole camera, we’ll need an eye position, a target position that the eye is looking at, and an imaginary grid of pixels between the eye and the target.

This imaginary grid of pixels is what is going to be displayed on the screen, and may be thought of as the “near plane”. If anything gets between the eye and the near plane it won’t be visible.

To calculate a ray for a pixel, we get the direction from the eye to the pixel’s location on the grid and normalize that. That gives us the ray’s direction. The ray’s starting position is just the pixel’s location on that imaginary grid.

I’ll explain how to do this below, but keep in mind that the example code also does this process, in case reading the code is easier than reading a description of the math used.

First we need to figure out the orientation of the camera:

1. Calculate the camera’s forward direction by normalizing (Target – Eye).
2. To calculate the camera’s right vector, cross product the forward vector with (0,1,0).
3. To calculate the camera’s up vector, cross product the forward vector with the right vector.

Note that the above assumes that there is no roll (z axis rotation) on the camera, and that it isn’t looking directly up.

Next we need to figure out the size of our near plane on the camera’s x and y axis. To calculate this, we have to define both a near plane distance (I use 0.1 in the sample code) as well as a horizontal and vertical field of view (I use a FOV of 40 degrees both horizontally and vertically, and make a square image, in the sample code).

You can get the size of the window on each axis then like this:

1. WindowRight = tangent(HorizontalFOV / 2) * NearDistance
2. WindowTop = tangent(VerticalFOV / 2) * NearDistance

Note that we divide the field of view by 2 on each axis because we are going to treat the center of the near plane as (0,0). This centers the near plane on the camera.

Next we need to figure out where our pixel’s location is in world space, when it is at pixel location (x,y):

1. Starting at the camera’s position, move along the camera’s forward vector by whatever your near plane distance is (I use a value of 0.1). This gets us to the center of the imaginary pixel grid.
2. Next we need to figure out what percentage on the X and Y axis our pixel is. This will tell us what percentage on the near plane it will be. We divide x by ScreenWidth and y by ScreenHeight. We then put these percentages in a [-1,1] space by multiplying the percentages by 2 and subtracting 1. We will call these values u and v, which equate to the x and y axis of the screen.
3. Starting at the center of the pixel grid that we found, we are going to move along the camera’s right vector a distance of u and the camera’s up vector a distance of v.

We now have our pixel’s location in the world.

Lastly, this is how we calculate the ray’s position and direction:

1. RayDir = normalize(PixelWorld – Eye)
2. RayStart = PixelWorld

We now have a ray for our pixel and can start solving eg ray vs triangle equations to see what objects in the scene our ray intersects with.

That’s basically all there is to path tracing at a high level. Next up I want to talk about some practical details of path tracing.

## Rendering Parameters, Rendering Quality and Rendering Time

A relevant quote from @CasualEffects:

Below are a few scenes rendered at various samples per pixel (spp) and their corresponding rendering times. They are rendered at 128×128 with 5 bounces. I used 8 threads to utilize my 8 cpu cores. Exact machine specs aren’t really important here, I just want to show how sample count affects render quality and render times in a general way.

There’s a couple things worth noting.

First, render time grows roughly linearly with the number of samples per pixel. This lets you have a pretty good estimate how long a render will take then, if you know how long it took to render 1000 samples per pixel, and now you want to make a 100,000 samples per pixel image – it will take roughly 100 times as long!

Combine that with the fact that you need 4 times as many samples to half the amount of error (noise) in an image and you can start to see why monte carlo path tracing takes so long to render nice looking images.

This also applies to the size of your render. The above were rendered at 128×128. If we instead rendered them at 256×256, the render times would actually be four times as long! This is because our image would have four times as many pixels, which means we’d be taking four times as many samples (at the same number of samples per pixel) to make an image of the same quality at the higher resolution.

You can affect rendering times by adjusting the maximum number of bounces allowed in your path tracer as well, but that is not as straightforward of a relationship to rendering time. The rendering time for a given number of bounces depends on the complexity and geometry of the scene, so is very scene dependent. One thing you can count on though is that decreasing the maximum number of bounces will give you the same or faster rendering times, while increasing the maximum number of bounces will give you the same or slower rendering times.

Something else that’s really important to note is that the first scene takes a lot more samples to start looking reasonable than the second scene does. This is because there is only one small light source in the first image but there is a lot of ambient light from a blue sky in the second image. What this means is that in the first scene, many rays will hit darkness, and only a few will hit light. In the second scene, many rays will hit either the small orb of light, or will hit the blue sky, but all rays will hit a light source.

The third scene also takes a lot more samples to start looking reasonable compared to the fourth scene. This is because in the third scene, there is a smaller, brighter light in the middle of the ceiling, while in the fourth scene there is a dimmer but larger light that covers the entire ceiling. Again, in the third scene, rays are less likely to hit a light source than in the fourth scene.

Why do these scenes converge at different rates?

Well it turns out that the more difference there is in the things your rays are likely to hit (this difference is called variance, and is the source of the “noisy look” in your path tracing), the longer it takes to converge (find the right answer).

This makes a bit of sense if you think of it just from the point of view of taking an average.

If you have a bunch of numbers with an average of N, but the individual numbers vary wildly from one to the next, it will take more numbers averaged together to get closer to the actual average.

If on the other hand, you have a bunch of numbers with an average of N that aren’t very different from eachother (or very different from N), it will take fewer numbers to get closer to the actual average.

Taken to the extreme, if you have a bunch of numbers with an average of N, that are all exactly the value N, it only takes one sample to get to the actual true average of N!

You can read a discussion on this here: Computer Graphics Stack Exchange: Is it expected that a naive path tracer takes many, many samples to converge?

There are lots of different ways of reducing variance of path tracing in both the sampling, as well as in the final image.

Some techniques actually just “de-noise” the final image rendered with lower sample counts. Some techniques use some extra information about each pixel to denoise the image in a smarter way (such as using a Z buffer type setup to do bilateral filtering).

Here’s such a technique that has really impressive results. Make sure and watch the video!

Nonlinearly Weighted First-order Regression for Denoising Monte Carlo Renderings

There is also a nice technique called importance sampling where instead of shooting the rays out in a uniform random way, you actually shoot your rays in directions that matter more and will get you to the actual average value faster. Importance sampling lets you get better results with fewer rays.

In the next post or two, I’ll show a very simple importance sampling technique (cosine weighted hemisphere sampling) and hope in the future to show some more sophisticated importance sampling methods.

## Some Other Practical Details

Firstly, I want to mention that this is called “naive” path tracing. There are ways to get better images in less time, and algorithms that are better suited for different scenes or different graphical effects (like fog or transparent objects), but this is the most basic, and most brute force way to do path tracing. We’ll talk about some ways to improve it and get some more features in future posts.

Hitting The Wrong Objects

I wanted to mention that when you hit an object and you calculate a random direction to cast a new ray in, there’s some very real danger that the new ray you cast out will hit the same object you just hit! This is due to the fact that you are starting the ray at the collision point of the object, and the floating point math may or may not consider the new ray to hit the object at time 0 or similar.

One way to deal with this is to move the ray’s starting point a small amount down the ray direction before testing the ray against the scene. If pushed out far enough (I use a distance of 0.001) it will make sure the ray doesn’t hit the same object again. It sounds like a dodgy thing to do, because if you don’t push it out enough (how far is enough?) it will give you the wrong result, and if you push it out too far to be safe, you can miss thin objects, or can miss objects that are very close together. In practice though, this is the usual solution to the problem and works pretty well without too much fuss. Note that this is a problem in all ray tracing, not just path tracing, and this solution of moving the ray by a small epsilon is common in all forms of ray tracing I’ve come across!

Another way to deal with the problem is to give each object a unique ID and then when you cast a ray, tell it to ignore the ID of the object you just hit. This works flawlessly in practice, so long as your objects are convex – which is usually the case for ray tracing since you often use triangles, quads, spheres, boxes and similar to make your scene. However, this falls apart when you are INSIDE of a shape (like how the images at the top of this post show objects INSIDE a box), and it also falls apart when you have transparent objects, since sometimes it is appropriate for an object to be intersected again by a new ray.

I used to be a big fan of object IDs, but am slowly coming to terms with just pushing the ray out a little bit. It’s not so bad, and handles transparents and being inside an object (:

Gamma Correction

After we generate our image, we need to apply gamma correction by taking each color channel to the power of 1/2.2. A decent approximation to that is also to just take the square root of each color channel, as the final value for that color channel. You can read about why we do this here: Understanding Gamma Correction

HDR vs LDR

There is nothing in our path tracer that has any limitations on how bright something can be. We could have a bright green light that had a color value of (0, 100000, 0)! Because of this, the final pixel color may not necessarily be less than one (but it will be a positive number). Our color with be a “High Dynamic Range” color aka HDR color. You can use various tone mapping methods to turn an HDR color into an LDR color – and we will be looking at that in a future post – but for now, I am just clamping the color value between 0 and 1. It’s not the best option, but it works fine for our needs right now.

Divide by Pi?

When looking at explanations or implementations of path tracing, you’ll see that some of them divide colors by pi at various stages, and others don’t. Since proper path tracing is very much about making sure you have every little detail of your math correct, you might wonder whether you should be dividing by pi or not, and if so, where you should do that. The short version is it actually doesn’t really matter believe it or not!

Here are two great reads on the subject for a more in depth answer:
PI or not to PI in game lighting equation

Random Point on Sphere and Bias

Correctly finding a uniformly random point on a sphere or hemisphere is actually a little bit more complicated that you might expect. If you get it wrong, you’ll introduce bias into your images which will make for some strange looking things.

Here is a good read on some ways to do it correctly:
Wolfram Math World: Sphere Point Picking

Here’s an image where the random point in sphere function was just completely wrong:

Here’s an image where the the random hemisphere function worked by picking a random point in a cube and normalizing the resulting vector (and multiplying it by -1 if it was in the wrong hemisphere). That gives too much weighting to the corners, which you can see manifests itself in the image on the left as weird “X” shaped lighting (look at the wall near the green light), instead of on the right where the lighting is circular. Apologies if it’s hard to distinguish. WordPress is converting all my images to 8BPP! :X

Primitive Types & Counts Matter

Here are some render time comparisons of the Cornell box scene rendered at 512×512, using 5 bounces and 100 samples per pixel.

There are 3 boxes which only have 5 sides – the two boxes inside don’t have a bottom, and the box containing the boxes doesn’t have a front.

I started out by making the scene with 30 triangles, since it takes 2 triangles to make a quad, and 5 quads to make a box, and there are 3 boxes.

Those 30 triangles rendered in 21.1 seconds.

I then changed it from 30 triangles to 15 quads.

It then took 6.2 seconds to render. It cut the time in half!

This is not so surprising though. If you look at the code for ray vs triangle compared to ray vs quad, you see that ray vs quad is just a ray vs triangle test were we first test the “diagonal line” of the quad, to see which of the 2 corners should be part of the ray vs triangle test. Because of this, testing a quad is just about as fast as testing a triangle. Since using quads means we have half the number of primitives, turning our triangles into quads means our rendering time is cut in half.

Lastly, i tried turning the two boxes inside into oriented boxes that have a width, height, depth, an axis of rotation and an angle of rotation around that axis. The collision test code puts the ray into the local space of the oriented box and then just does a ray vs axis aligned box test.

Doing that, i ended up with 5 quads (for the box that doesn’t have a front, it needed to stay quads, unless i did back face culling, which i didn’t want to) and two oriented boxes.

The render time for that was 5.5 seconds, so it did shave off 0.7 seconds, which is a little over 11% of the rendering time. So, it was worth while.

For such a low number of primitives, I didn’t bother with any spatial acceleration structures, but people do have quite a bit of luck on more complex scenes with bounding volume hierarchies (BVH’s).

For naive path tracing code, since the first ray hit is entirely deterministic which object it will hit (if any), we could also cache that first intersection and re-use it for each subsequent sample. That ought to make a significant difference to rendering times, but basically in the next path tracing post we’ll be removing the ability to do that, so I didn’t bother to try it and time it.

Debugging

As you are working on your path tracer, it can be useful to render an image at a low number of samples so that it’s quick to render and you can see whether things are turning out the way you want or not.

Another option is to have it show you the image as it’s rendering more and more samples per pixel, so that you can see it working.

If you make a “real time” visualizer like that, some real nice advice from Morgan McGuire (Computer graphics professor and the author of http://graphicscodex.com/) is to make a feature where if you click on a pixel, it re-renders just that pixel, and does so in a single thread so that you can step through the process of rendering that pixel to see what is going wrong.

I personally find a lot of value in visualizing per-pixel values in the image to see how values look across the pixels to be able to spot problems. You can do this by setting the emissive lighting to be based on the value you want to visualize and setting the bounce count to 1, or other similar techniques.

Below are two debug images I made while writing the code for this post to try and understand how some things were going wrong. The first image shows the normals of the surface that the camera rays hit (i put x,y,z of the normal into r,g,b but multiply the values by 0.5 and add 0.5 to make them be 0 to 1, instead of -1 to 1). This image let me see that the normals coming out of my ray vs oriented box test were correct.

The second image shows the number of bounces done per pixel. I divided the bounce count by the maximum number of bounces and used that as a greyscale value for the pixel. This let me see that rays were able to bounce off of oriented boxes. A previous image that I don’t have anymore showed darker sides on the boxes, which meant that the ray bouncing wasn’t working correctly.

Immensely Parallelizable: CPU, GPU, Grid Computing

At an image size of 1024×1024, that is a little over 1 million pixels.

At 1000 samples per pixel, that means a little over 1 billion samples.

Every sample of every pixel only needs one thing to do it’s work: Read only access to the scene.

Since each of those samples are able to do their work independently, if you had 1 billion cores, path tracing could use them all!

The example code is multithreaded and uses as many threads as cores you have on your CPU.

Since GPUs are meant for massively parallelizable work, they can path trace much faster than CPUs.

I haven’t done a proper apples to apples test, but evidence indicates something like a 100x speed up on the GPU vs the CPU.

You can also distribute work across a grid of computers!

One way to do path tracing in grid computing would be to have each machine do a 100 sample image, and then you could average all of those 100 sample images together to get a much higher sample image.

The downside to that is that network bandwidth and disk storage pays the cost of the full size image for each computer you have helping.

A better way to do it would probably be to make each machine do all of the samples for a small portion of the image and then you can put the tiles together at the end.

While decreasing network bandwidth and disk space usage, this also allows you to do all of the pixel averaging in floating point, as opposed to doing it in 8 bit 0-255 values like you’d get from a bmp file made on another machine.

## Closing

In this post and the sample code that goes with it, we are only dealing with a purely diffuse (lambertian) surface, and emissive lighting. In future posts we’ll cover a lot more features like specular reflection (mirror type reflection), refraction (transparent objects), caustics, textures, bump mapping and more. We’ll also look at how to make better looking images with fewer samples and lots of other things too.

I actually have to confess that I did a bit of bait and switch. The images at the top of this post were rendered with an anti aliasing technique, as well as an importance sampling technique (cosine weighted hemisphere samples) to make the image look better faster. These things are going to be the topics of my next two path tracing posts, so be on the lookout! Here are the same scenes with the same number of samples, but with no AA, and uniform hemisphere sampling:

And the ones at the top for comparison:

While making this post I received a lot of good information from the twitter graphics community (see the people I’m following and follow them too! @Atrix256) as well as the Computer Graphics Stack Exchange.

Also, two specific individuals helped me out quite a bit:

@lh0xfb – She was also doing a lot of path tracing at the time and helped me understand where some of my stuff was going wrong, including replicating my scenes in her renderer to be able to compare and contrast with. I was sure my renderer was wrong, because it was so noisy! It turned out that while i tended to have small light sources and high contrast scenes, Lauren did a better job of having well lit scenes that converged more quickly.

@Reedbeta – He was a wealth of information for helping me understand some details about why things worked the way they did and answered a handful of graphics stack exchange questions I posted.

Thanks a bunch to you both, and to everyone else that participated in the discussions (: