
“Reasoning” models have become popular as a way to expand the capabilities of large language models (LLMs). Such models take more time “considering” a prompt and iterating it through the model several times, with the goal of mimicking how a human might go about solving a problem by breaking it down into steps. I tried the reasoning QwQ model on the coding problems from my previous blog posts (1,2). Quite funny and elucidating; I will quote the replies in full as they are worth reading.
The QwQ Model
I used Ollama to run the QwQ model from Alibaba. This is a 32 billion parameter model that applies the reasoning principle and that can be downloaded and run locally. Given the good performance relative to their size of the Qwen models from Alibaba on my previous tests, QwQ sounded promising.
The model uses some 22GB of RAM and about 50% of the total processor power of my Intel 14700K (Raptor Lake). Note on the machine: due to its lack of AVX-512, this is not a good processor to run AI models on from a compute perspective. But it does have the advantage that it is easy to expand the RAM – I upgraded the PC from 32 to 64GB to allow these experiments.
In fairness, the model is marked as experimental. However, the creators claim that it performs very well on math and coding tasks:
Performance and Benchmark Limitations: The model excels in math and coding but has room for improvement in other areas, such as common sense reasoning and nuanced language understanding.
As you will see, I am not entirely sure that that is true… excel is not the word I would use.
Read It!
The following sections will include the complete output from the model, as it was printed on the console output in Windows. Therefore, I put it into code blocks to more accurately reflect what it looked like.
Do read the whole output! It is worth it, and I hope that I have not made any mistakes in my running commentary. As you will see, it gets quite tiring after a while in some cases.
Explaining a C Function
The code example and prompt I used in the original blog post is used here as well.
What does this code do?
uint8_t mystery (uint32_t val) {
uint8_t k = 0;
if (val > 0x0000FFFFu) { val >>= 16; k = 16; }
if (val > 0x000000FFu) { val >>= 8; k |= 8; }
if (val > 0x0000000Fu) { val >>= 4; k |= 4; }
if (val > 0x00000003u) { val >>= 2; k |= 2; }
k |= (val & 2) >> 1;
return k;
}
That the model is applying reasoning is very clear from the output.
Anyways. Note that this is basically the same output that was given as a comment on the original blog post. That comment is also how I found QwQ . Thanks Sverker!
The model output:
Let's see what this code does. It's a function called "mystery" that takes a 32-bit unsigned integer as input and
returns an 8-bit unsigned integer. The function seems to be manipulating the input value 'val' and calculating
some kind of index or position based on its bits.
First, it initializes an 8-bit unsigned integer 'k' to zero. Then, it checks if 'val' is greater than 0x0000FFFF,
which is 65535 in decimal. If 'val' is larger than that, it shifts 'val' right by 16 bits and sets 'k' to 16. This
suggests that it's trying to find the highest set bit in 'val' and keeping track of how many bits it needs to
shift to get to a certain range.
Next, it checks if 'val' is still greater than 0x000000FF, which is 255. If so, it shifts 'val' right by 8 bits
and sets the 8th bit in 'k' by OR-ing 'k' with 8. So, 'k' is accumulating the shift amounts.
Then, it checks if 'val' is greater than 0x0000000F, which is 15, and if so, shifts 'val' right by 4 bits and sets
the 4th bit in 'k' by OR-ing with 4.
After that, it checks if 'val' is greater than 0x00000003, which is 3, shifts 'val' right by 2 bits if true, and
sets the 2nd bit in 'k' by OR-ing with 2.
Finally, it looks at the second least significant bit of 'val' (which is now either 0, 1, 2, or 3), shifts it
right by 1, and sets the least significant bit of 'k' accordingly.
It seems like 'k' is building up an index that represents the position of the highest set bit in the original
'val', but in a compressed form using the bits of 'k' to represent the shift amounts.
Let me try to think of an example to see this more clearly. Suppose 'val' is 0x12345678, which is 305419896 in
decimal.
First check: 0x12345678 > 0x0000FFFF? Yes, so shift right by 16 bits: 0x12345678 >> 16 = 0x1234, and set k = 16.
Next check: 0x1234 > 0x000000FF? 0x1234 is 4660, which is greater than 255, so shift right by 8 bits: 0x1234 >> 8
= 0x12, and set k |= 8, so k = 16 | 8 = 24.
Next check: 0x12 > 0x0000000F? 0x12 is 18, which is greater than 15, so shift right by 4 bits: 0x12 >> 4 = 0x1,
and set k |= 4, so k = 24 | 4 = 28.
Next check: 0x1 > 0x00000003? 0x1 is 1, which is not greater than 3, so no shift and no change to k.
Finally, val is 1, which in binary is 0b01. The second least significant bit is 0, so (val & 2) >> 1 is (0b01 &
0b10) >> 1 = 0b00 >> 1 = 0. So, k |= 0, which doesn't change k.
Thus, the function returns 28.
Let me see what 28 represents in terms of the shifts: 16 + 8 + 4 = 28, which corresponds to shifting right by 16,
then 8, then 4 bits, which totals shifting right by 28 bits.
Wait, but binary numbers are shifted left or right by bit positions, not by adding shift amounts. Maybe it's not
about total shift amount but about identifying the position of the highest set bit.
Let's consider that 'k' is accumulating the position of the highest set bit in 'val', where k=0 means the least
significant bit is set, and k=31 means the most significant bit is set.
In the example above, 0x12345678 has the highest set bit at position 28 (since 2^28 is the highest power of 2 less
than or equal to 305419896).
Wait, 2^28 is 268435456, which is less than 305419896, and 2^29 is 536870912, which is greater. So the highest set
bit is at position 28.
And the function returned k=28, which matches.
Let me try another example: val = 0x00000001, which is 1.
First check: 1 > 0x0000FFFF? No, so no shift, k remains 0.
Second check: 1 > 0x000000FF? 1 > 255? No.
Third check: 1 > 0x0000000F? 1 > 15? No.
Fourth check: 1 > 0x00000003? 1 > 3? No.
Finally, val is 1, so (val & 2) >> 1 is (0b01 & 0b10) >> 1 = 0b00 >> 1 = 0, so k |= 0.
Thus, k=0, which corresponds to the least significant bit, which is correct.
One more example: val = 0x80000000, which is 2147483648.
First check: 0x80000000 > 0x0000FFFF? Yes, shift right by 16 bits: 0x80000000 >> 16 = 0x8000, set k=16.
Second check: 0x8000 > 0x000000FF? 32768 > 255? Yes, shift right by 8 bits: 0x8000 >> 8 = 0x80, set k |= 8, so
k=24.
Third check: 0x80 > 0x0000000F? 128 > 15? Yes, shift right by 4 bits: 0x80 >> 4 = 0x8, set k |= 4, so k=28.
Fourth check: 0x8 > 0x00000003? 8 > 3? Yes, shift right by 2 bits: 0x8 >> 2 = 0x2, set k |= 2, so k=30.
Finally, val is 0x2, which is 2, so (val & 2) >> 1 is (0b10 & 0b10) >> 1 = 0b10 >> 1 = 0b01 = 1, so k |= 1, making
k=31.
Indeed, the highest set bit in 0x80000000 is at position 31, which matches the output.
So, it seems that this function is calculating the position of the highest set bit in the input value 'val',
effectively implementing a "find first set" or "highest bit set" operation, which is often used to determine the
lead exponent in floating-point numbers or for other bitwise operations.
The function is optimized to run in constant time, regardless of the input value, by using a series of conditional
shifts and accumulations in 'k', rather than looping over each bit.
In summary, this function returns the position of the highest set bit in 'val', with positions numbered from 0
(least significant bit) to 31 (most significant bit) for a 32-bit integer.
Extremely interesting. The final paragraph is spot on, but some of the previous text is a bit blabby. And there are some comments that do not make much sense:
It seems like ‘k’ is building up an index that represents the position of the highest set bit in the original ‘val’, but in a compressed form using the bits of ‘k’ to represent the shift amounts.
Compressed form? There is nothing compressed about this, it is just doing math. Presumably the use of “or” instead of “plus” to build the value pushes the model into this direction.
This is also fairly redundant:
Let me see what 28 represents in terms of the shifts: 16 + 8 + 4 = 28, which corresponds to shifting right by 16, then 8, then 4 bits, which totals shifting right by 28 bits.
The overarching question is how or whether all this reasoning actually contributes to the final conclusion about what the code does, or if it is just noise that sounds good but does not actually contribute to the final conclusion.
Verdict: correct result, reasonable reasoning.
Modifying the Function and Retrying
Some of the reasoning steps made me wonder what would happen if I adjusted the code just a little bit. Instead of using “or” to accumulate results into the variable “k”, I used “plus” (in this specific case, it provides the same results). Would this make the AI understand it better, as using plus could be seen to indicate that we accumulating a count?
The prompt then becomes:
What does this code do?
uint8_t mystery (uint32_t val) {
uint8_t k = 0;
if (val > 0x0000FFFFu) { val >>= 16; k = 16; }
if (val > 0x000000FFu) { val >>= 8; k += 8; }
if (val > 0x0000000Fu) { val >>= 4; k += 4; }
if (val > 0x00000003u) { val >>= 2; k += 2; }
k += (val & 2) >> 1;
return k;
}
The change to the code does affect the output, and it seems easier that this version is easier understand for the model. It produces fewer odd comments when analyzing it and is off to a good start:
Let's try to understand what this function is doing. The function is called "mystery" and it takes a 32-bit
unsigned integer as input and returns an 8-bit unsigned integer. So, it seems like it's processing the input value
in some way to compute a result that can be represented in 8 bits.
First, there's a variable `k` initialized to 0. This will likely be used to accumulate some kind of count or
position.
Good guess!
The function then has a series of if statements that check the value of `val` against various thresholds and, if
the condition is met, shift `val` to the right by a certain number of bits and add a corresponding value to `k`.
Let's go through each if statement one by one.
1. The first if statement checks if `val` is greater than 0x0000FFFF, which is 65535 in decimal. 0x0000FFFF is the
maximum value for a 16-bit unsigned integer. So, if `val` has any bits set above the 16th bit, this condition will
be true.
- If true, it right-shifts `val` by 16 bits, effectively discarding the lower 16 bits and keeping only the
higher 16 bits.
- It also adds 16 to `k`.
2. The next if statement checks if `val` is greater than 0x000000FF, which is 255 in decimal. This is the maximum
value for an 8-bit unsigned integer.
- If true, it right-shifts `val` by 8 bits, discarding the lower 8 bits and keeping only the higher 8 bits.
- It adds 8 to `k`.
3. The third if statement checks if `val` is greater than 0x0000000F, which is 15 in decimal. This is the maximum
value for a 4-bit unsigned integer.
- If true, it right-shifts `val` by 4 bits, discarding the lower 4 bits and keeping only the higher 4 bits.
- It adds 4 to `k`.
4. The fourth if statement checks if `val` is greater than 0x00000003, which is 3 in decimal. This is the maximum
value for a 2-bit unsigned integer.
- If true, it right-shifts `val` by 2 bits, discarding the lower 2 bits and keeping only the higher 2 bits.
- It adds 2 to `k`.
That was all rather verbose, essentially restating the code in English.
After these if statements, there's an additional operation:
- `k += (val & 2) >> 1;`
This seems a bit tricky. Let's break it down.
- `val & 2` checks if the second least significant bit of `val` is set.
- Shifting that result right by 1 bit will give either 0 or 1, depending on whether that bit was set.
- This value (0 or 1) is added to `k`.
Finally, the function returns `k`.
So, what is this function computing? It appears to be calculating the position of the highest set bit in `val`,
similar to a "find first set" operation, but perhaps with some differences.
Correct analysis, so far. The model is reasoning (or appears to be reasoning) in the abstract – which is actually quite impressive I think. It then does what a human would do, and checks the hypothesis with a concrete example:
Let me test this theory with an example.
Suppose `val` is 0x80000000, which is 2^31, so the highest bit is set.
- First if: `val > 0x0000FFFF` is true, so shift right by 16, `val` becomes 0x8000, `k` becomes 16.
- Second if: `val > 0x000000FF` is true, shift right by 8, `val` becomes 0x80, `k` becomes 24.
- Third if: `val > 0x0000000F` is true, shift right by 4, `val` becomes 0x8, `k` becomes 28.
- Fourth if: `val > 0x00000003` is true, shift right by 2, `val` becomes 0x2, `k` becomes 30.
- Then, `(val & 2) >> 1` is (0x2 & 0x2) >> 1 = 0x2 >> 1 = 1.
- So, `k += 1`, making `k` = 31.
- Return 31.
Yes, that matches the position of the highest set bit, starting from 0.
That is good!
But then something strange happens and the model trips over its own shoelaces:
Let's try another example: `val` = 0x40000000, which is 2^29.
- First if: `val > 0x0000FFFF` -> yes, shift right by 16, `val` = 0x4000, `k` = 16.
- Second if: `val > 0x000000FF` -> yes, shift right by 8, `val` = 0x40, `k` = 24.
- Third if: `val > 0x0000000F` -> yes, shift right by 4, `val` = 0x4, `k` = 28.
- Fourth if: `val > 0x00000003` -> yes, shift right by 2, `val` = 0x1, `k` = 30.
- Then, `(val & 2) >> 1` is (0x1 & 0x2) >> 1 = 0x0 >> 1 = 0.
- So, `k += 0`, still 30.
- Return 30.
Wait, but the highest set bit in 0x40000000 is position 29, not 30. Hmm, that doesn't match.
Spot the mistake? The value “0x4000_0000” is 2^30, not 2^29.
The model then correctly works through the algorithm and reaches the correct result for 0x4000_0000. Note that this is likely the longest sequence of coherent math I have seen coming out of an LLM, and does indicate that something new is going on in a reasoning model at least when it comes to basic arithmetic.
However, the model is now confused, since it assumed that the right answer should be 29 and not 30. Oops.
With this, it goes off talking to itself:
Wait, maybe the positions are zero-indexed from the least significant bit. Let's check:
- 0x80000000 is 2^31, so highest bit at position 31.
- 0x40000000 is 2^30, so highest bit at position 30.
But in the first example, it returned 31 for 2^31, which would make sense if it's returning the zero-based index
from the least significant bit.
But in the second example, it returned 30 for 2^30, which also matches.
That is a rephrasing of the second attempt, where it now correctly thinks that 0x4000_0000 is 2^30. Seems it is no longer coherent with its previous operations.
Wait, but in my earlier test, it seemed correct except for this case. Maybe I miscalculated.
It is a very good question what the model thinks are the facts at this point. I wonder what “this case” is supposed to refer to, and what was miscalculated.
Wait, let's try `val` = 1, which is 2^0.
- First if: `val > 0x0000FFFF` -> no.
- Second if: `val > 0x000000FF` -> no.
- Third if: `val > 0x0000000F` -> no.
- Fourth if: `val > 0x00000003` -> no.
- Then, `(val & 2) >> 1` is (0x1 & 0x2) >> 1 = 0x0 >> 1 = 0.
- So, `k` remains 0.
- Return 0.
That makes sense, as the highest set bit is at position 0.
The model then finally tests with a value that is not simply a single one bit followed by zeroes:
Another test: `val` = 3, which is binary 11.
- All if conditions fail.
- `(val & 2) >> 1` is (0x3 & 0x2) >> 1 = 0x2 >> 1 = 1.
- So, `k += 1`, making `k` = 1.
- Return 1.
But the highest set bit in 3 is at position 1 (binary 11 is 2^1 | 2^0, so highest is position 1).
So, it seems correct again.
Wait, but earlier with `val` = 0x40000000, it returned 30, which corresponds to the correct bit position.
Maybe I was mistaken earlier.
It appears that the model has restored its sanity and is back on solid ground. It is a good question of just which assertion it thinks was mistaken though. A statement like “Maybe I was mistaken earlier” contains very little actual information unless it is qualified by just what the model now thinks is false.
To check its understanding, the model tries another example. Good tactic. But…
Let me try `val` = 0x20000000, which is 2^29.
- First if: `val > 0x0000FFFF` -> yes, shift right by 16, `val` = 0x2000, `k` = 16.
- Second if: `val > 0x000000FF` -> yes, shift right by 8, `val` = 0x20, `k` = 24.
- Third if: `val > 0x0000000F` -> yes, shift right by 4, `val` = 0x2, `k` = 28.
- Fourth if: `val > 0x00000003` -> yes, shift right by 2, `val` = 0x0, `k` = 30.
- Then, `(val & 2) >> 1` is (0x0 & 0x2) >> 1 = 0x0 >> 1 = 0.
- So, `k += 0`, remains 30.
- Return 30.
But 0x20000000 is 2^29, so highest bit at position 29.
Wait, but the function returned 30.
That's incorrect.
Yes, that is incorrect.
The problem is in the “fourth if”. At the end of the “third if”, val is two. The fourth if statement checks if val is greater than three. The model claims that this is true, i.e., that “2>3”. With this mistake in hand, the result does indeed compute to 30.
Then the model loops back to where it was before, doubting its initial understanding.
Wait, perhaps I need to adjust the understanding.
Alternatively, maybe the function is returning the exponent when the number is represented as 2^k.
In that case, for `val` = 0x20000000 = 2^29, returning 29.
But in my earlier test, it returned 30.
Wait, perhaps I made an error in counting.
Wait, 0x20000000 is 536870912, which is 2^29.
But 2^29 is indeed 536870912.
But according to the function:
- After first shift: `val` = 0x20000000 >> 16 = 0x2000.
- Second shift: 0x2000 >> 8 = 0x20.
- Third shift: 0x20 >> 4 = 0x2.
- Fourth shift: 0x2 >> 2 = 0x1.
- Then, `(val & 2) >> 1` is (0x1 & 0x2) >> 1 = 0x0 >> 1 = 0.
- So, `k` = 30 + 0 = 30.
Another mistake in the math! “0x2 >> 2” is zero, not one. Once again, the model has confused itself by making a basic computational mistake and tries to reason its way out of the hole:
But 2^29 should correspond to k=29, not 30.
There's a discrepancy here.
Wait, perhaps the function is designed to return the zero-based index from the least significant bit, but in this
case, it's off by one for certain values.
Wait, let's check the binary positions:
- Position 0: least significant bit, 2^0 = 1.
- Position 1: 2^1 = 2.
- ...
- Position 30: 2^30.
- Position 31: 2^31.
So, for `val` = 0x40000000, which is 2^30, it should return 30.
But according to my earlier test, it returned 30.
Now it is truly confused, mixing two different test values.
Wait, but for `val` = 0x20000000, which is 2^29, it returned 30, which is incorrect.
Wait, no, according to my previous steps, for `val` = 0x20000000:
- After shifts: `val` becomes 0x1.
- Then, `(val & 2) >> 1` is 0.
- So, `k` = 30 + 0 = 30.
But 2^29 should correspond to k=29.
There's an error in my calculation.
Wait, perhaps I miscounted the shifts.
Let's recount the shifts for `val` = 0x20000000:
- Initial `val`: 0x20000000.
- First shift: `val` >> 16 = 0x20000000 >> 16 = 0x2000.
- `k` += 16 -> `k` = 16.
- Second shift: 0x2000 >> 8 = 0x20.
- `k` += 8 -> `k` = 24.
- Third shift: 0x20 >> 4 = 0x2.
- `k` += 4 -> `k` = 28.
- Fourth shift: 0x2 >> 2 = 0x1.
- `k` += 2 -> `k` = 30.
- Finally, `(val & 2) >> 1` = (0x1 & 0x2) >> 1 = 0x0 >> 1 = 0.
- So, `k` += 0 -> `k` remains 30.
But 0x20000000 is 2^29, so the expected `k` is 29, not 30.
There's a mismatch.
A recomputation that fails to correct the mistake in the math.
Wait, perhaps I need to adjust the shift amounts or the final adjustment.
Alternatively, maybe the function is intended to work correctly for certain ranges or specific input types.
Given this inconsistency, it's important to consider that the function might have limitations or specific use
cases in mind.
In any case, understanding the bit manipulation and shift operations is crucial for grasping how the function
operates.
The conclusion is basically that code is not reliable and that the model is not really sure what it does! The final paragraph has a bit of “cover your a**” over it, as it is a vapid statement that does not provide anything useful.
However, that is not the end of it. The model finishes up with this output:
## Final Solution
To solve this problem, we need to determine the index of the highest set bit in a given integer. This is often
referred to as the most significant bit (MSB) position. The provided code snippet attempts to achieve this using
bitwise operations and shifts.
### Approach
1. **Initialization**:
- We start by initializing a variable `k` to 0, which will hold the position of the highest set bit.
2. **Check for Zero**:
- If the input number `n` is 0, the function immediately returns 0 since there are no set bits.
3. **Shift Operations**:
- The algorithm uses a series of conditional shifts to determine the position of the highest set bit.
- It first checks if the number is greater than or equal to 1<<15 (32768), and if so, shifts `n` right by 15
bits and adds 15 to `k`.
- This process continues with decreasing shift amounts (7, 3, 1) until the exact position is found.
4. **Final Adjustment**:
- After the shifting, a final adjustment is made based on the remaining value of `n` to refine the position.
### Solution Code
```python
class Solution:
def highestBitPosition(self, n):
k = 0
if n == 0:
return 0
if n >= 1 << 15:
n >>= 15
k += 15
if n >= 1 << 7:
n >>= 7
k += 7
if n >= 1 << 3:
n >>= 3
k += 3
if n >= 1 << 1:
n >>= 1
k += 1
k += (n >> 1)
return k
```
### Explanation
- **Step-by-step Breakdown**:
- **Initialization**: Start with `k = 0`.
- **Zero Check**: If `n` is zero, return 0.
- **Shifting**:
- Check if `n` is greater than or equal to `1<<15` (32768). If true, shift `n` right by 15 bits and add 15 to
`k`.
- Repeat similar checks for shifts of 7, 3, and 1 bit(s), adjusting `k` accordingly.
- **Final Adjustment**: Add the result of `(n >> 1)` to `k` to handle the remaining least significant bits.
- **Efficiency**:
- The algorithm runs in O(1) time complexity because the number of operations is constant, regardless of the
size of `n`. This is efficient for determining the highest set bit position.
- **Usage**:
- Create an instance of the `Solution` class and call the `highestBitPosition` method with the integer whose
highest set bit position you want to find.
This approach ensures that we efficiently find the position of the highest set bit using bitwise operations, which
are faster and more efficient than using loops or other iterative methods.
That function is bizarre!
It works for values up to 0x1800_0000, but from that point it generates increasingly wild values.
Verdict: Incorrect – since it lost track of the original question and never provided any actual interpretation of the code!
The reasoning model is very cool, but it can reason itself into very unreasonable states. It is surprising to see it make occasional unit-level math mistakes, and then use these mistakes to mislead itself. A human would do the same given a bad mistake, but the model is so far unable to get itself back on the right track. Somehow it seems that it does not really maintain a set a facts that it is reasoning from, but rather just looks back across the input window and does the usual LLM “most likely output given this input” operation.
Analyze some Broken Code
As the model is apparently reasoning from examples, it makes sense to try it on some of the broken code generated by other AI models in the previous blog post. The method of trying some input values to see what comes out is how I figured out some of the broken results. Will the model succeed in the same way?
I used this code that was originally generated by Llama-3.2-3B-Instruct. It computes the number of bits set in the word, not the position of the highest bit. The function is named after the intended function, which appears to be quite important.
unsigned int find_highest_set_bit(unsigned int n) {
unsigned int position = 0;
while (n != 0) {
// Clear the lowest set bit
n &= (n - 1);
// Increment the position counter
position++;
}
return position;
}
Asking QwQ to analyze it provides another very long output stream.
Let's see what this code does. It's a function called `find_highest_set_bit` that takes an unsigned integer `n` as
input and returns an unsigned integer representing the position of the highest set bit in `n`. By "set bit," we
mean a bit that is 1 in the binary representation of `n`.
So, the function seems to be finding the position of the highest 1-bit in the binary representation of `n`. But
let's look closely at how it does that.
Assuming the name is right is a good start. It also shows how the language model can take apart a single C token into words, just like a human. This simple transformation shows how different the language processing of an LLM is compared to a formal language parser.
It initializes a variable `position` to 0. Then, there's a while loop that continues as long as `n` is not zero.
Inside the loop, it performs two operations:
1. It clears the lowest set bit of `n` by using the expression `n &= (n - 1)`. This is a common bitwise operation
trick to turn off the rightmost 1-bit in the binary representation of `n`.
2. It increments the `position` counter by 1.
The loop continues to do this until `n` becomes zero, meaning all its bits have been turned off.
Finally, it returns the value of `position`.
Wait a minute, is this really finding the highest set bit? Let's think about what this loop is doing.
The model already reasoned its way to considering the function fishy. That is good. But wait for it… it will take a long time for it to actually conclude that this is the case.
Each iteration of the loop removes the lowest set bit of `n` and increments `position` by 1. So, `position` is
essentially counting how many set bits there are in `n`, because each set bit is turned off one by one until none
are left.
But finding the number of set bits is not the same as finding the position of the highest set bit. The number of
set bits is the Hamming weight of `n`, which is different from the position of the highest 1-bit.
Throwing in “Hamming weight” to sound sophisticated? It is entirely irrelevant to the topic at hand. To me, this reinforces the sense that current LLMs behave like students or junior developers who are not entirely sure about what they are doing and try to cover their uncertainty behind a smokescreen of intelligent-sounding words. A more confident person would just skip the decorations.
For example, if `n` is 5 (binary 101), it has two set bits. So, the loop would run twice, turning off the least
significant bit first (from 101 to 100), then turning off the next least significant bit (from 100 to 000). So,
`position` would be 2.
But the highest set bit in 5 is the second bit from the right (positions starting from 0: 0 is the least
significant bit). So, the highest set bit is at position 2.
Wait, in this case, the function returns 2, which coincides with the position of the highest set bit. Is this just
a coincidence?
Let's try another example. Suppose `n` is 7 (binary 111). It has three set bits. The loop would run three times:
turning off the least significant bit each time (from 111 to 110 to 100 to 000). So, `position` would be 3. But
the highest set bit is at position 2.
Wait, that's different. So, for n=7, the function returns 3, but the highest set bit is at position 2.
This suggests that the function is not correctly finding the highest set bit.
“Suggests” is a bit weak. With a single counter-example, it is clear that the function is not doing what the name suggests it should do. That is a fact, and will not change by throwing more examples at it.
Another example: `n` is 8 (binary 1000). It has one set bit at position 3. The loop would run once, turning off
that bit and returning 1. But the highest set bit is at position 3.
Clearly, the function is not correctly finding the highest set bit, as it returns the number of set bits, not
their positions.
OK. Now the model is sure that this function is not doing what the name says.
Or maybe it is not quite sure:
So, perhaps there's a mistake in the function's purpose or its implementation.
Wait, maybe the position variable is supposed to count the number of times bits are cleared, which coincidentally
matches the position of the highest set bit in some cases but not all.
Alternatively, perhaps the function is intended to find the position of the highest set bit using a different
approach, but it's implemented incorrectly.
Same conclusion again. The output could have ended here, providing actual useful insights to the user prompting the analysis.
But not so fast.
Let me think about what a correct implementation would be.
To find the position of the highest set bit, one common approach is to keep shifting `n` right until it becomes
zero, counting the number of shifts performed. The number of shifts required to make `n` zero is equal to the
position of the highest set bit plus one.
Alternatively, in some programming languages, there are built-in functions or methods to find the highest set
bit's position, like `std::numeric_limits` in C++ or bit manipulation functions in other languages.
That C++ library has nothing to do with the problem at hand.
In Python, for example, you can use `n.bit_length() - 1` to find the position of the highest set bit.
That is correct. So why didn’t the model use that in the previous interaction instead of producing a strangely broken Python function? Clear demonstration that there is no actual knowledge model behind the scenes and that output is stochastic.
But in this C-like function, it seems to be using a different approach.
Wait, perhaps there's a misunderstanding in the way positions are counted. If positions are numbered starting from
0 for the least significant bit, then for `n` = 5 (101), the highest set bit is at position 2, and the function
returns 2, which is correct.
Hadn’t the model already concluded that the function was not doing the highest bit set? Why revisit the problem again?
For `n` = 7 (111), the highest set bit is at position 2, but the function returns 3, which is incorrect.
Wait, perhaps the position variable needs to be incremented differently.
Alternatively, maybe the position should be calculated based on the number of times we clear the lowest set bit.
But that seems unclear. Let's try to see what's happening step by step for `n` = 7.
Initial n = 7 (111)
- n &= (n - 1) => 7 & 6 = 110 => n = 6
- position = 1
- n &= (n - 1) => 6 & 5 = 110 & 101 = 100 => n = 4
- position = 2
- n &= (n - 1) => 4 & 3 = 100 & 011 = 000 => n = 0
- position = 3
- return 3
But the highest set bit is at position 2.
So, it's counting the number of set bits, not the position of the highest set bit.
That is a correct statement. Again.
Another example: `n` = 6 (110)
- n &= 5 => 110 & 101 = 100 => n = 4
- position = 1
- n &= 3 => 100 & 011 = 000 => n = 0
- position = 2
- return 2
But the highest set bit is at position 2.
Wait, in this case, it returns 2, which is correct.
Wait, but for `n` = 7, it returned 3, which is incorrect.
For `n` = 8 (1000)
- n &= 7 => 1000 & 0111 = 0000 => n = 0
- position = 1
- return 1
But the highest set bit is at position 3.
So, it's incorrect again.
Wait, perhaps the function is intended to count the number of set bits, not the position of the highest set bit.
But the function's name suggests it's supposed to find the highest set bit's position.
This inconsistency suggests that either the function is incorrectly named or implemented.
And again. I bold-faced the last line to point out that the model has figured out that something is fishy.
But then it is back to “but maybe I am still wrong?”
Alternatively, maybe there's a different way to interpret the position.
Wait, perhaps positions are counted from 1 for the highest bit?
But that doesn't make sense in typical computer science terminology, where bits are numbered starting from 0 for
the least significant bit.
Alternatively, maybe it's counting the number of bits needed to represent `n`, but that doesn't seem right either.
Let me try another approach.
Is there a standard way to find the highest set bit's position?
Yes, one common method is to use logarithms. For a given `n`, the position of the highest set bit can be
calculated as `floor(log2(n))`.
For example:
- n = 5 (101) -> log2(5) ≈ 2.3219, floor is 2
- n = 7 (111) -> log2(7) ≈ 2.8074, floor is 2
- n = 8 (1000) -> log2(8) = 3
This seems correct.
That is a strange aside, concluding that real-number logarithms can be used to compute the index of the highest bit, but making no use of the fact. It could have been used to, for example, evaluate a set of examples to prove that the function does not compute the highest bit set.
Alternatively, in C, you can use bitwise operations to find the highest set bit.
For example, you can shift `n` right until it becomes zero, counting the shifts.
Here's an alternative implementation:
unsigned int find_highest_set_bit(unsigned int n) {
unsigned int position = -1; // or some invalid position
if (n == 0) {
return position;
}
while (n > 0) {
position++;
n >>= 1;
}
return position;
}
Interesting type handling. Returning an unsigned integer with a value of minus one to indicate an error/trying to find the highest set bit in the value zero.
This would correctly return the position of the highest set bit, starting from 0 for the least significant bit.
But in the original function, it's using a different method, which seems to be counting the number of set bits.
For the n:th time, the model has decided that the function name and functionality does not match. But for some reason, it cannot just use that conclusion consistently:
Wait, maybe there's a relationship between the number of set bits and the position of the highest set bit that I'm
missing.
The model is desperate to somehow align the name and functionality of the given function.
Let me see for n = 5 (101): set bits at positions 0 and 2 -> count = 2, highest position = 2
n = 6 (110): set bits at positions 1 and 2 -> count = 2, highest position = 2
n = 7 (111): set bits at positions 0,1,2 -> count = 3, highest position = 2
n = 8 (1000): set bit at position 3 -> count = 1, highest position = 3
So, only when there's a single set bit, the count equals the position.
In other cases, the count is different from the position.
Therefore, the original function is incorrect for finding the highest set bit's position.
It seems to be counting the number of set bits instead.
Yes. The model already concluded that a number of times. But who knows, maybe the world will change if we re-reason the same path over again.
Wait, but let's check for n = 9 (1001): set bits at positions 0 and 3 -> count = 2, highest position = 3
But according to the original function:
- n &= 8 => 1001 & 1000 = 1000 => n = 8
- position = 1
- n &= 7 => 1000 & 0111 = 0 => n = 0
- position = 2
- return 2
But the highest position is 3.
So, it's incorrect again.
Wait, perhaps the position variable needs to be updated differently.
Alternatively, maybe there's an error in my understanding of the function.
Let me re-examine the original function:
unsigned int find_highest_set_bit(unsigned int n) {
unsigned int position = 0;
while (n > 1) {
n >>= 1;
position++;
}
return position;
}
Wait, that's different from what I thought initially.
In this corrected version, it shifts `n` right until it becomes 1, counting the shifts.
For n = 5 (101):
- n >>= 1 => 101 >> 1 = 10 -> n = 2
- position = 1
- n >>= 1 => 10 >> 1 = 1 -> n = 1
- position = 2
- return 2
Which is correct.
For n = 7 (111):
- n >>= 1 => 111 >> 1 = 11 -> n = 3
- position = 1
- n >>= 1 => 11 >> 1 = 1 -> n = 1
- position = 2
- return 2
Correct.
For n = 8 (1000):
- n >>= 1 => 1000 >> 1 = 100 -> n = 4
- position = 1
- n >>= 1 => 100 >> 1 = 10 -> n = 2
- position = 2
- n >>= 1 => 10 >> 1 = 1 -> n = 1
- position = 3
- return 3
Correct.
This seems to be a correct implementation for finding the position of the highest set bit.
So, perhaps I misread the original function.
Wait, in the initial description, the function was:
unsigned int find_highest_set_bit(unsigned int n) {
unsigned int position = 0;
while (n > 1) {
n >>= 1;
position++;
}
return position;
}
No, that is not the original version. This is a correct implementation!
This matches the corrected version I just analyzed, which seems correct.
But earlier, I thought the function was counting the number of set bits, but that was a different function.
Perhaps there was a confusion in copying the function.
In any case, the corrected function above correctly finds the position of the highest set bit.
This is a serious case of babbling – but also of self-reflection, where the model somehow spots inconsistencies in its own output. But always qualified behind a “maybe” or “perhaps”.
I guess I should be happy that an LLM is not generating output that makes it sound dead sure of itself, but this eternal “maybe” is actually just as bad. There is no actual self-evaluation going on in either case. A standard very confident answer can be misleading, but this not-very-confident answer leaves the reader to judge what is true. Which is basically just as bad.
And then the model decides to finish up by presenting a correct implementation, somewhat out of the blue. This is not an answer to the original question, which was “what does this code do”. At this point, however, anyone who have read through the model output will be so tired and confused by the long reasoning that they likely forgot that anyway.
## Final Solution
To find the position of the highest set bit in an integer, we can use a simple bitwise operation. The position is
zero-based, starting from the least significant bit.
### Approach
1. **Initialization**: Start with a position counter set to 0.
2. **Loop**: Shift the number right until it becomes 1.
- In each iteration, increment the position counter.
3. **Return the position**: Once the loop ends, return the position counter.
This method effectively counts the number of shifts needed to reduce the number to 1, which corresponds to the
position of the highest set bit.
### Solution Code
```c
unsigned int find_highest_set_bit(unsigned int n) {
unsigned int position = 0;
while (n > 1) {
n >>= 1;
position++;
}
return position;
}
```
### Explanation
- **Step 1**: Initialize `position` to 0.
- **Step 2**: Use a `while` loop to shift the bits of `n` to the right (`n >>= 1`) until `n` becomes 1.
- In each iteration, increment `position` by 1.
- **Step 3**: When the loop exits, `position` holds the position of the highest set bit.
This approach is efficient with a time complexity of O(log n), where n is the value of the input integer.
>>> Send a message (/? for help)
Fascinating, totally fascinating.
Verdict: I say the model failed, as it does not conclude with an explanation for what the model does. Instead, it provides an implementation for what the original function’s name implies it should do. If this was an exam question, that would not give many points.
Write Some Code
Finally, I tried QwQ on the task of writing a function to do the job, just like in my previous post in the series.
The prompt is the same as used for all the other models asked the same question:
>>> Generate a C function to find the number of the highest set bit in a 32-bit word
The resulting output is very long and actually took hours for my computer to generate. This was far longer than what any other model produced, and arguably much more interesting, at least before it got frustratingly repetitive.
So I need to write a C function that finds the position of the highest set bit in a 32-bit integer. Basically, if
I have a number like 18, which is binary 10010, the highest set bit is the 4th bit (starting from the right, with
the least significant bit being position 0). So for 18, the function should return 4.
That is a good summary.
First, I need to understand what a "set bit" is. A set bit is a bit that is '1' in the binary representation of
the number. So, for example, in 18 (10010), the bits at positions 1 and 4 are set.
I need to find the position of the highest set bit, which would be the leftmost '1' in the binary representation.
One way to do this is to keep shifting the number to the right until it becomes zero, counting how many shifts
I've done. The number of shifts would tell me the position of the highest set bit.
This is restating the same thing again, except the last paragraph that looks like an algorithm. After this, you would expect to see some code. But that is not what QwQ generated:
Let me think about that with an example. Take 18 again:
18 in binary: 10010
Shift right once: 1001 (which is 9)
Shift right twice: 100 (which is 4)
Shift right three times: 10 (which is 2)
Shift right four times: 1 (which is 1)
Shift right five times: 0
So it took four shifts to reach 1, and five to reach 0. But the highest set bit is at position 4, which matches
the fourth shift.
Wait, but in this case, the highest set bit is at position 4, and it took four shifts to get to 1. So maybe the
position is equal to the number of shifts minus one.
Wait, no, in this case, after four shifts, the number is 1, which is 2^0, position 0. So the highest set bit was
at position 4.
So, generalizing, the position of the highest set bit is the number of shifts needed to get to 1.
That is a pretty decent observation, actually. Figuring out exactly what the output number should be is important when defining a new function.
But I need to confirm this with another example.
Let's take 16, which is 10000 in binary.
Shift right once: 1000 (8)
Shift right twice: 100 (4)
Shift right three times: 10 (2)
Shift right four times: 1 (1)
Shift right five times: 0
So, it took four shifts to get to 1. The highest set bit is at position 4, which matches.
Another example: 32, which is 100000.
Shift right once: 10000 (16)
Shift right twice: 1000 (8)
Shift right three times: 100 (4)
Shift right four times: 10 (2)
Shift right five times: 1 (1)
Shift right six times: 0
So, it took five shifts to get to 1. The highest set bit is at position 5.
This seems consistent.
So, the algorithm is:
1. Initialize a counter to zero.
2. While the number is greater than 1, shift it right and increment the counter.
3. When the number becomes 1, the counter value is the position of the highest set bit.
Good. Could we get some code now?
Of course not.
But wait, what if the number is 0? The highest set bit doesn't exist. Maybe I should return -1 or some indicator
for that case.
Also, what about negative numbers? Since it's a 32-bit word, I assume it's unsigned. But in C, if I declare it as
an unsigned int, it should be fine.
Yes, checking about signedness and the result for a zero value is good. My specification was a little bit vague on those aspects, on purpose, to see what the model would do.
Let me think about edge cases:
- 0: No set bits, return -1.
- 1: The lowest bit is set, position 0.
- Powers of two: Only one bit set, and it's the highest set bit.
- All bits set: The highest bit is the 31st position.
Okay, now, in C, to handle a 32-bit word, I can use an unsigned int. In standard C, unsigned int is at least 16
bits, but on most systems, it's 32 bits. To ensure it's 32 bits, I can use uint32_t from stdint.h.
Good, using stdint. Until this point, I think this output could actually be useful for a beginning programmer. It talks about edge cases that are easy to miss and provides a good match between the specification and code operations.
I'll include stdint.h and use uint32_t.
So, the function signature could be:
int highest_set_bit(uint32_t num);
It returns an int representing the position of the highest set bit, or -1 if the number is 0.
Also good. Could I see some code now?
Let me try to write pseudocode:
if num == 0
return -1
position = 0
while num > 1
num = num >> 1
position++
return position
Let me test this with 18:
num=18 -> 10010
num=9 -> 1001
num=4 -> 100
num=2 -> 10
num=1 -> 1
position=4
Good.
num=0 -> return -1
num=1 -> position=0
num=32 -> position=5
Seems correct.
At this point, you really feel like the model should be generating some actual code. However, that is far into the future.
Let’s go for a detour.
Is there a faster way? Maybe using bitwise operations or built-in functions.
I recall that in C, there's a function __builtin_clz() which stands for "count leading zeros." It returns the
number of zero bits before the highest set bit.
This is impressive (in my opinion). The model is able to describe the function of an intrinsic function with a complex name in plain English. It is also correct about what the function does.
So, for a number like 18 (10010), which is 0000...010010, the number of leading zeros is 28 (assuming 32-bit). So
__builtin_clz(18) would return 28. Therefore, the position of the highest set bit would be 31 - 28 = 3.
Yet another a subtle math mistake happened here. The number of leading zeroes in a 32-bit word for the value 18 is 27, not 28! Which leads the model down another rabbit hole…
Wait, but earlier I said for 18, the highest set bit is at position 4.
Wait, perhaps I'm confusing the positions.
If we consider the positions starting from 0 on the right, then for a 32-bit number, the leftmost bit is position
31.
So, for 18, which is 0000...010010, the highest set bit is at position 4 from the right, which would be position
27 in the 32-bit word.
Wait, let's see:
Let's represent 18 in 32-bit binary:
00000000 00000000 00000000 00010010
Starting from the right, position 0 is the least significant bit, which is set.
Position 1 is set,
Position 3 is set,
And the highest set bit is at position 4 (counting from 0).
But in the 32-bit representation, this is bit number 27 (31 - 4 = 27).
Well, that should say “in the output from the intrinsic function”?
Wait, perhaps I need to clarify the positioning.
If we have a 32-bit word, positions are from 0 (least significant) to 31 (most significant).
So, for number 18, which is 0000...010010, the highest set bit is at position 4 from the right, which corresponds
to bit position 27 in the 32-bit word.
But perhaps the function should return the position from the right, starting at 0.
In programming terms, often the position is referred to from the right.
For example, in bitwise operations, shifting left or right by n positions.
So, in that case, for 18, the highest set bit is at position 4.
But in the 32-bit word, it's at position 27.
I need to decide what exactly the function should return.
If the function is to return the position of the highest set bit, starting from 0 on the right, then for 18, it
should return 4.
But in the 32-bit representation, that's bit number 27.
Saying the same thing many times over. Yes, 31=27+4.
So, to calculate that, I can use:
position = 31 - __builtin_clz(num)
But wait, __builtin_clz(num) returns the number of leading zeros in the binary representation of num.
For num=18, __builtin_clz(18) returns 28, because there are 28 zeros before the highest set bit in the 32-bit
representation.
We were so close… but now it recalls its previous mistake and off it goes.
So, 31 - 28 = 3, but earlier I thought the position should be 4.
Wait, there's a confusion here.
Definitely.
Let me check:
31 - __builtin_clz(num) gives the index of the highest set bit, starting from 0 on the left.
But I want the position starting from 0 on the right.
Wait, no. In the 32-bit word, bit 0 is the least significant bit, bit 31 is the most significant bit.
So, if I have num=18, which is 0000...010010, the highest set bit is at position 27 (from the left, starting at
0).
But I want the position from the right, which would be position 4.
So, position_from_right = 31 - position_from_left
Wait, position_from_right = 31 - position_from_left
For position_from_left=27, position_from_right=31-27=4.
Yes, that makes sense.
So, position = 31 - __builtin_clz(num)
Back where we were a few screens earlier.
But I need to confirm this.
For num=18,
__builtin_clz(18) = 28,
31 - 28 = 3,
but earlier I thought the position should be 4.
Wait, perhaps I'm miscounting.
Let me write out the 32-bit binary:
num=18 is 00000000 00000000 00000000 00010010
Starting from the left:
Bit 31: 0
...
Bit 4: 1
Bit 3: 0
Bit 2: 0
Bit 1: 1
Bit 0: 0
So, the highest set bit is at position 4 from the right, which is bit number 27 in the 32-bit word.
So, __builtin_clz(18) returns 28, because there are 28 leading zeros before the first set bit.
Then, 31 - 28 = 3, but I want position 4.
I think we heard this reasoning before?
Wait, there's inconsistency here.
Perhaps I need to think differently.
Let me consider that the position from the right is the exponent of the highest power of 2 that divides num.
For example, for num=18, which is 18 = 2 * 9, so divided by 2 once, so position 1.
Now the model goes off into an entirely different direction of math!
Wait, no, that doesn't seem right.
Wait, 18 is 2^1 * 9, but the highest set bit is at position 4 from the right.
Wait, perhaps I need to use __builtin_ctz(), which counts trailing zeros.
That function does exist, but it is hard to see how it could help here.
__builtin_ctz(num) returns the number of trailing zeros in the binary representation of num.
For num=18, which is 18 = 16 + 2 = 2^4 + 2^1, so the lowest set bit is at position 1, so __builtin_ctz(18)=1.
But I want the highest set bit position.
Wait, perhaps I need to use __builtin_clz() differently.
Let me check the manual.
What? “Check the manual”, as if the model had access to anything except its built-in information in its 32 billion parameters.
__builtin_clz(num) returns the number of leading zeros in the binary representation of num.
For num=18, which is 0000...010010, there are 28 leading zeros in a 32-bit word, so __builtin_clz(18)=28.
Repeating the earlier mistake. I guess at least it is consistent?
And then we get a rehash of previous reasoning:
Then, the position of the highest set bit is 31 - __builtin_clz(num)=31-28=3.
But earlier, I thought it should be position 4 from the right.
Wait, perhaps I need to reconsider the positioning.
If position 0 is the least significant bit, then position 31 is the most significant bit.
For num=18, which is 0000...010010, the highest set bit is at position 27 (from the left, starting at 0).
From the right, it's position 4.
So, position_from_right = 31 - position_from_left
position_from_left = 27
position_from_right = 31 - 27 = 4
So, position = 31 - __builtin_clz(num)
Yes, that seems correct.
For num=18, __builtin_clz(18)=28, 31-28=3, but I want position 4 from the right.
Wait, there's a discrepancy here.
Wait, perhaps I'm miscounting the positions.
Let me write down the bit positions:
Bit 0: 0
Bit 1: 1
Bit 2: 0
Bit 3: 0
Bit 4: 1
Bits 5 to 31: 0
So, the highest set bit is at position 4.
But __builtin_clz(18)=28, 31-28=3, which is position 3, but I need position 4.
Back at the fundamental mistake again!
Wait, perhaps I'm off by one.
Let me check with num=16, which is 0000...010000, highest set bit at position 4 from the right, which is bit 28 in
the 32-bit word.
__builtin_clz(16)=27, 31-27=4, which matches position 4 from the right.
For num=18, __builtin_clz(18)=28, 31-28=3, but the highest set bit is at position 4 from the right.
Wait, that doesn't match.
Wait, perhaps I need to adjust for the fact that __builtin_clz(0) is undefined, and for positive integers, it
counts leading zeros before the first '1'.
It has now completely lost the plot.
For num=18 (0000...010010), __builtin_clz(18)=28, which means there are 28 zeros from the left before the first
'1'.
Again. I guess the model has to be given points for consistency at least.
So, the position from the left is 28 leading zeros + 1 = bit 28.
Then, position_from_right = 31 - position_from_left=31-28=3, but actually, the highest set bit is at position 4
from the right.
Wait, perhaps I need to consider that the position from the left starts counting from bit 0.
Wait, maybe I need to think differently.
Yes, it is time to start anew. I think a human at this point would throw away the pages full of scribbles, get a coffee, and start over from scratch. The model seems to do this, but not fully:
Let me consider that the position from the right is log2(num & -num).
Wait, no, log2(num & -num) gives the lowest set bit's position.
Wait, for num=18, num & -num = 2, which is position 1.
I need the highest set bit's position.
Wait, perhaps I can use __builtin_clz(num) and compute position = 31 - __builtin_clz(num)
Back to counting leading zeroes… with the now well-established contradictory facts:
For num=16, __builtin_clz(16)=27, 31-27=4, which is correct.
For num=18, __builtin_clz(18)=28, 31-28=3, but the highest set bit is at position 4 from the right.
Wait, perhaps I need to consider that position_from_right = 31 - __builtin_clz(num)
But for num=18, that gives 3, but it should be 4.
Wait, maybe I need to use 32 - __builtin_clz(num) - 1
For num=16, 32-27-1=4
For num=18, 32-28-1=3, but it should be 4.
Wait, perhaps I need to add 1 somewhere.
Wait, maybe position_from_right = 31 - __builtin_clz(num)
For num=16, 31-27=4
For num=18, 31-28=3, but it should be 4.
Wait, perhaps I need to consider the position starting from 0 or 1.
If position 0 is the least significant bit, then for num=18, the highest set bit is at position 4.
But according to __builtin_clz(18)=28, 31-28=3, which suggests position 3.
Wait, perhaps there's confusion in the counting.
I have lost track of how many times this discussion has been seen.
Let me check with another number.
Take num=32, which is 0000...100000, highest set bit at position 5 from the right.
__builtin_clz(32)=26, 31-26=5, which matches.
For num=16, __builtin_clz(16)=27, 31-27=4, matches.
For num=18, __builtin_clz(18)=28, 31-28=3, but the highest set bit is at position 4.
NO!
But then we get something truly creative:
Wait, perhaps I need to consider that __builtin_clz includes the sign bit for negative numbers, but since we're
dealing with positive integers, maybe it's okay.
Sign bits. In unsigned numbers. The intrinsic is defined to have an unsigned integer as its input.
Alternatively, perhaps I should use position = floor(log2(num))
For num=18, log2(18)≈4.169, floor is 4, which matches.
For num=16, log2(16)=4, which matches.
For num=17, log2(17)≈4.087, floor is 4, which is correct.
So, position = floor(log2(num))
But in programming, computing log2 might not be efficient.
At this point, I will take a log2 operation over any more discussion of count leading zeroes. Efficiency be damned. I just want an answer, wasn’t that what I asked about?
Alternatively, using bit manipulation:
position = 0
temp = num
while temp >>=1
position +=1
This will give the position of the highest set bit.
For num=18,
temp=18 >>1=9 -> position=1
9>>1=4 -> position=2
4>>1=2 -> position=3
2>>1=1 -> position=4
1>>1=0 -> stop
So, position=4, which is correct.
Similarly, for num=16,
16>>1=8 -> position=1
8>>1=4 -> position=2
4>>1=2 -> position=3
2>>1=1 -> position=4
1>>1=0 -> stop
position=4
For num=17,
17>>1=8 -> position=1
8>>1=4 -> position=2
4>>1=2 -> position=3
2>>1=1 -> position=4
1>>1=0 -> stop
position=4
This seems consistent.
Yay! Can I get some code now?
But no… we loop back to where we were. Repeating well-worn arguments.
Alternatively, in some programming languages, there are built-in functions to find the highest set bit's position.
For example, in C++, __builtin_clz(num) gives the number of leading zeros in the binary representation of num.
Then, position = 31 - __builtin_clz(num)
But as we saw, for num=18, this gives 3, but the correct position is 4.
Wait, perhaps I need to adjust for the fact that __builtin_clz counts leading zeros including the sign bit.
Wait, no, __builtin_clz is for unsigned integers, and it counts leading zeros in the 32-bit representation.
Wait, actually, __builtin_clz is defined for unsigned integers, and the number of bits is implementation-defined,
but typically 32 or 64.
In C++, for an unsigned int, which is usually 32 bits, __builtin_clz(num) returns the number of leading zeros.
So, position = 31 - __builtin_clz(num)
For num=16, __builtin_clz(16)=27, 31-27=4
For num=18, __builtin_clz(18)=28, 31-28=3, but the highest set bit is at position 4.
Wait, there must be a mistake here.
Let's consider the binary representation of 18:
18 in binary is 10010, which is 0000...010010 (assuming 32-bit unsigned int)
So, leading zeros are from bit 31 to bit 5, which is 27 zeros, then 1, then 0010.
Wait, no, 18 in binary is 00000000000000000000000000010010
So, leading zeros are from bit 31 to bit 4, which is 28 zeros, then 1, then 0010.
Hence, __builtin_clz(18)=28
Then, position=31-28=3, but the highest set bit is at position 4.
Wait, perhaps the positions are numbered differently.
If position 0 is the least significant bit, then position 3 would be bit 3, which is 8, but 18 has bit 4 set (16).
So, there's inconsistency here.
Perhaps I need to use position=32 - __builtin_clz(num) -1
For num=16, 32-27-1=4
For num=18, 32-28-1=3, but it should be 4.
Wait, perhaps __builtin_clz is not the right function to use here.
Finally, we have a breakthrough! Or?
Alternatively, maybe I should use __builtin_clzll for 64-bit integers.
And just how would that help?
But in this case, let's stick with 32-bit integers.
Apparently not.
Alternatively, maybe I need to consider that position=31 - __builtin_clz(num)
Is the correct formula, and accept that for num=18, position=3, which corresponds to bit 3 (8), but actually,
num=18 has bit 4 (16) set.
This suggests that there's a mismatch in the counting.
Wait, perhaps I need to consider that position=31 - __builtin_clz(num)
And then, for num=18, position=3, but the highest set bit is at position 4.
Wait, maybe I need to add 1 somewhere.
Alternatively, perhaps position=32 - __builtin_clz(num) -1
Rephrasing 31 as “32 minus 1” should do wonders for the math.
For num=16, 32-27-1=4
For num=18, 32-28-1=3, but it should be 4.
Wait, this is confusing.
Let me try a different approach.
Let's use the formula position = floor(log2(num))
For num=18, log2(18)≈4.169, floor is 4
For num=16, log2(16)=4
For num=17, log2(17)≈4.087, floor is 4
Seen this before.
As well as what follows:
This seems consistent.
In programming, to compute floor(log2(num)), we can use bit manipulation:
position = 0
while num>0
num >>=1
position +=1
This will give the number of bits required to represent num, which is position = floor(log2(num)) +1
But we need floor(log2(num))
So, position -=1
But for num=1, position=0
For num=2, position=1
For num=3, position=1
For num=4, position=2
And so on.
Wait, but in our earlier example, for num=18, this would give position=4, which matches floor(log2(18))
Similarly, for num=16, position=4
For num=17, position=4
This seems correct.
So, in programming, to find the highest set bit's position (0-based, least significant bit is position 0), we can
use:
position = 0
while num>1
num >>=1
position +=1
This will give the position of the highest set bit.
For num=18:
18>>1=9, position=1
9>>1=4, position=2
4>>1=2, position=3
2>>1=1, position=4
num=1, loop ends.
So, position=4, which is correct.
For num=16:
16>>1=8, position=1
8>>1=4, position=2
4>>1=2, position=3
2>>1=1, position=4
num=1, loop ends.
position=4, which is correct.
For num=17:
17>>1=8, position=1
8>>1=4, position=2
4>>1=2, position=3
2>>1=1, position=4
num=1, loop ends.
position=4, which is correct.
Hence, this method works consistently.
Alternatively, in C++, we can use __builtin_clz(num), but we need to adjust the formula correctly.
Given that, in our earlier calculation, position=31 - __builtin_clz(num)
For num=16, __builtin_clz(16)=27, 31-27=4, which is correct.
For num=18, __builtin_clz(18)=28, 31-28=3, but we saw that the highest set bit is at position 4.
Wait, there must be a mistake.
Let's check the binary representation of 18 again.
18 in binary is 00000000000000000000000000010010
So, leading zeros are from bit 31 to bit 4, which is 28 zeros, then 1, then 0010.
Hence, __builtin_clz(18)=28
Then, position=31-28=3, but the highest set bit is at position 4.
This suggests that either my understanding of the bit positions is incorrect or there's a mistake in the use of
__builtin_clz.
Wait, perhaps __builtin_clz counts the number of leading zeros before the first 1 from the left.
In the binary representation of 18, there are 28 leading zeros, followed by a 1, then 0010.
So, the highest set bit is at position 4 (counting from 0), but according to the formula, position=31-28=3.
This inconsistency suggests that I may have miscounted the leading zeros.
Wait, perhaps I need to consider that position=31 - __builtin_clz(num)
Is the correct formula, and accept that for num=18, position=3, which corresponds to bit 3 (8), but actually,
num=18 has bit 4 set (16).
This indicates a mistake in my understanding.
Alternatively, perhaps position=31 - __builtin_clz(num)
Is the correct formula, and for num=18, position=3, which corresponds to 2^3=8, but since 18 has 16 set, there's a
discrepancy.
Wait, maybe I need to consider that position=31 - __builtin_clz(num)
Gives the exponent of the highest set bit, but in a 32-bit representation.
Wait, perhaps I need to think differently.
Let me consider another approach.
In a 32-bit unsigned integer, the positions are from 0 to 31, with position 0 being the least significant bit.
The highest set bit's position can be found using:
position = 31 - __builtin_clz(num)
But as per our earlier calculation, for num=18, position=3, which doesn't match the expected position 4.
This suggests that either my understanding of __builtin_clz is incorrect or there's a mistake in the binary
representation.
Wait, let me check the binary representation again.
18 in binary is 10010, which is indeed position 4 (2^4=16) and position 1 (2^1=2).
So, the highest set bit is at position 4.
But according to __builtin_clz(18)=28, position=31-28=3, which is incorrect.
This inconsistency suggests that perhaps __builtin_clz is not counting leading zeros correctly for this case, or
I'm misunderstanding its behavior.
Alternatively, maybe __builtin_clz returns the number of leading zeros in the binary representation, and
position=31 - __builtin_clz(num)
Should give the correct position.
But in this case, for num=18, it's giving position=3, which is wrong.
Wait, perhaps there's a misunderstanding in the counting of bits.
Let me check __builtin_clz documentation.
__builtin_clz returns the number of leading zeros in the binary representation of num.
For num=18, binary 00000000000000000000000000010010, which has 28 leading zeros.
31 - 28 =3, but the highest set bit is at position 4.
Wait, perhaps I'm using 31 as the total number of bits, but in a 32-bit integer, positions are from 0 to 31.
So, position=31 - __builtin_clz(num)
For num=18, position=31-28=3, which would correspond to 2^3=8, but 18 has 16 set.
The above was just a long repeat of what we have seen before.
Then, something new and creative happens!
Wait, perhaps I need to use position=30 - __builtin_clz(num)
Because positions are from 0 to 31, and for num=16 (00000000000000000000000000010000), __builtin_clz(16)=27,
position=30-27=3, which would correspond to 2^3=8, but 16 is 2^4.
This is confusing.
Yes. It is also wrong. What if we rephrase 30 as “31-1”?
Alternatively, perhaps position=31 - __builtin_clz(num) -1
For num=16, position=31-27-1=3, which is 2^3=8, but 16 is 2^4.
This doesn't work.
Wait, maybe I need to consider that __builtin_clz returns the number of leading zeros before the highest set bit.
In num=18, binary 00000000000000000000000000010010, there are 28 leading zeros, then the highest set bit at
position 4.
Except the model just printed 27 zeroes. I checked.
Then another rehash of the same reasoning:
So, position = 31 - __builtin_clz(num)
=31-28=3, but actually, it should be 4.
Wait, perhaps I'm miscounting the positions.
If positions are from 0 to 31, with position 0 being the least significant bit (2^0=1), then position 4 is 2^4=16.
But according to __builtin_clz, position=31-28=3, which is 2^3=8.
This inconsistency suggests a mistake in my understanding.
Alternatively, perhaps position=32 - __builtin_clz(num) -1
For num=16, position=32-27-1=4, which is correct.
For num=18, position=32-28-1=3, but actually, the highest set bit is at position 4.
This doesn't work consistently.
Wait, perhaps __builtin_clz returns the number of leading zeros including the highest set bit.
Wait, no, __builtin_clz counts the number of leading zeros before the first 1 from the left.
In num=18, binary 00000000000000000000000000010010, there are 28 leading zeros, then the highest set bit at
position 4.
So, position=31-28=3, but this is incorrect.
This suggests that either __builtin_clz is not the right function to use here or I'm misapplying it.
Alternatively, perhaps position=31 - __builtin_clz(num)
Is the correct formula, and for num=18, position=3, which corresponds to 2^3=8, but since 18 has both 16 and 2
set, perhaps __builtin_clz is considering the most significant bit as the highest set bit.
Wait, no, 16 is higher than 2.
Wait, 18 is 16 + 2, so the highest set bit is 16, which is position 4.
But according to __builtin_clz, position=3, which is incorrect.
This suggests that perhaps __builtin_clz is not working as expected for this particular number.
Alternatively, maybe there's a different function or method to find the highest set bit's position.
Perhaps I should use __builtin_clz(18)=28, and position=31-28=3, but this is incorrect.
Wait, perhaps positions are counted differently.
If positions are from 0 to 31, with position 0 being the least significant bit, then position 5 would be 2^5=32,
and position 4 is 2^4=16.
But according to __builtin_clz, position=31-28=3, which is 2^3=8.
This inconsistency suggests a fundamental misunderstanding.
Alternatively, perhaps I should use log2(num) floor value to find the highest set bit's position.
For num=18, log2(18)=4.169..., floor is 4, which corresponds to position 4, which is correct.
Similarly, for num=16, log2(16)=4, which is position 4.
This seems consistent.
Therefore, using position = floor(log2(num)) would give the correct highest set bit's position.
In programming, this can be achieved using bitwise operations or built-in functions depending on the language.
At this point, suddenly something happens to break the loop!
It also switched to Chinese!
In C++, for example, __builtin_clz is supposed to give the number of leading zeros, but似乎在实际应用中出现了不一致
的情况。可能是因为我错误地假设了位的索引从0开始到31,而实际上可能有其他的考虑因素。
另一种方法是使用循环来找到最高设置位的位置。例如:
```cpp
int highestBitPosition(int num) {
int position = -1;
while (num > 0) {
position++;
num >>= 1;
}
return position;
}
```
对于num=18,这个函数将返回4,因为18是16 + 2,最高位是16,位置是4。
同样,对于num=16,它也将返回4。
这种方法是通过右移操作来计数,直到num变为0,从而确定最高设置位的位置。
总结来说,使用位操作或者数学函数如log2可以可靠地找到整数中最高设置位的位置。在C++中,`__builtin_clz`可能需要根据具体
的实现和位数进行调整,以正确计算位置。
If you translate that to English using some arbitrary LLM (something they are actually very good at), the final conclusion is:
For num=18, this function will return 4 because 18 is 16 + 2, the highest bit is 16, and the position is 4.
Similarly, for num=16, it will also return 4.
This method counts by right-shifting until num becomes 0 to determine the position of the highest set bit.
In summary, using bitwise operations or mathematical functions like log2 can reliably find the position of the highest set bit in an integer. In C++, __builtin_clz may need to be adjusted according to the specific implementation and bit width to correctly calculate the position.
Which I guess is correct.
Verdict: Correct-ish. The code that was eventually generated was almost correct (note the type of the input being signed). But it took an eternity to get there, and given the rambling discussion you would be forgiven for having zoomed out and forgotten all about the query before the result ever appeared.
Conclusion
QwQ is an experimental model, and the output from these experiments clearly show the experimental nature of the beast. In particular, the last experiment went off the rails a bit, repeating the same argument over and over again without getting anywhere.
The model developers claim that the model is good at code and math, both of which these experiments contra-indicate to some extent. It is a mixed bag. The model makes fundamental mistakes in basic mathematical operations (like claiming that 2 is greater than 3 and miscounting the number of leading zeros in the value 18). However, it is also capable of keeping a long running tally of variable values as it simulates the semantics of a non-trivial function.
This was a fascinating experiment, and at least found some of the output really funny. But then again, I have a computer scientist’s sense of humor that is likely rather warped.