Contemplating the AI delivering Number Theoretic Transforms

It's been a long time since last post, 2 years.
2 years ago, the LLM were just beginning to be interesting, but really far behind the human with all hallucinations and obvious mistakes.
Now, I have the clear feeling that I am far behind, as most humans probably are.

I just tested Claude Sonnet 5.5, the free version thru browser chat interface.
Not even the more powerful agentic harness (I haven't setup an MCP in my images yet).

I asked it to devise a large integer multiplication with number theoretic transform in Squeak Smalltalk.

It's been a matter of minutes to deliver

- a correct Squeak Smalltalk version working out of the box
- with quite compact code
- with a prime tuned for efficiently avoiding LargeInteger arithmetic in tight loops,
  at least in a 64 bits image (but wait, NTT in 32 bits is just too challenging)

Claude did choose a prime of 30 bits for obtaining that efficiency without me even asking with a 2-adicity of 2ˆ23, meaning that it can theoretically work with a decomposition of integers in up to 8 million limbs.

Claude did not hallucinate on that subject, and its sole mistake was using conform: in place of allSatisfy: when it provided a utility snippet for finding more good primes on my request (those leading to efficient implementation).
Not a real problem, if I had used an agent, it would have discovered by itself.
Claude could not point me to the source of confusion, except vaguely relating some squeak-dev threads from the early 2000. 
I later asked google (Gemini) which dialect implements conform: and it had the same hallucination about Squeak. When I objected to this it gave me a source of the mistake https://squeak.org/documentation/terse_guide/ a presumably trustable reference, so obviously the mistake has human origin. We should correct that for humans, if not for future AI training. Unless of course we seek for a sabotage strategy.

Claude was not aware of the existence of reciprocalModulo: and a few other Squeak methods, but it's easy to amend by prompting, otherwise it had a good sense of Squeak dialect.

I had the idea of playing with Chinese remainders in the back of my head, but Claude suggested this by itself even before I could do so.
So a few more prompts were enough to bring the Chinese Remainders versions for handling larger integers, and to push the limits imposed by SmallInteger arithmetic more explicitly.

After a few refactorings, I got a NTT multiplication beating Squeak 3-way Toom-Cooke multiplication around 2ˆ18 bytes, and 4-way Toom-Cooke squaring around 2^17 bytes, so near 1 Mega bit, that's not bad at all.

I consumed one more prompt for exploring the solution based on goldilocks prime 2ˆ64 - 2^32 + 1, on Claude's suggestion, which I did essentially for learning, because as predicted by Claude  this variant did not meet any interesting efficiency with Squeak available arithmetic.

I found most explanations given by Claude about the code quite clear, except maybe a few usage of jargon.
If not clear enough, it was just a matter of asking, and I got a satisfying answer.
For example, I had to ask details about the choice of primes, because the early comments were too synthetic.
The answers give an experience very close to literate programming, the chat just make it more interactive and that's enjoyable.

It's also pleasant to interact for suggesting a few enhancements, like using centered modulo, so as to use primes with one more bit while remaining in SmallInteger arithmetic, and get the updated version rapidly.
This little trick more than double the length of large integer that NTT can handle with SmallInteger arithmetic, and tough it knew this technique for saving a bit in Float FFT multiplication, Claude did not feature that by itself this time, probably because it's too much Smalltalk specific - C would use unsigned integer arithmetic.
Human added value is not exactly zero if you occupy a sufficiently narrow niche !

A single prompt was then enough for setting up an efficient FFT-based multiplication based on Float arithmetic for a comparison.

I was curious because last year I played with FFT-based multiplication all by myself, and it took me a few days for setting it up, testing, correcting my mistakes, tuning...
And my version is somehow inferior to Claude's one :

- less general (I split Integer in limbs with byte boundaries, Claude uses bit boundaries)
- less compact (I produced more code while searching for optimization)
- and finally less finely tuned (essentially due to the choice of limb width)

Indeed, I had to go thru stages that Claude did not even explore, because it was already aware of the state of the art and went straight toward such solution.

No secret, Claude did gather this knowledge from existing code and from scientific publications, which would have been my duty without an AI, but certainly not a matter of minutes.
From that, it was able to restitute a solution which must not be very far from the state of the art, much faster than me, and now, at least as good quality-wise.
So AI really shines in this part of the job in scientific area that consists in redoing just what has already been done, with a few variations, and it also shines for picking the right horse.

In a way, I find that interesting, because I can use it to learn about a specific subject.
It's more than just an assistant, but also somehow an interactive teacher.
We can still dream of being augmented by AI rather than replaced by AI.

It took me a few hours, not a few minutes to refactor, play some tests, measure efficiency.
As a human, I am the bottleneck, an agent could have done that more efficiently of course.
But that is my will and my pleasure, and anyway, I need to exercise and try by myself for learning, that's the burden of humanity.

Otherwise, what's the point of Smalltalk, if not for learning?
If I don't want to read or understand the code, I can as well ask AI to generate assembler, that would make more sense if speed of large integer multiplications was really the matter. But I known myself, even in assembler, my bias would make me curious about the implementation tricks.

Of course, this is just a toy example, a small part of a pet project I can come up with the limited token quota of free version.
It's not enough nowadays for challenging the AI.
It's just enough for having a more accurate feel of the revolution incoming.

Whatever we may think about the direction AI is taking and all implications it has on the future of younger people, insane consumption of energy, or captation of common goods by a few, it really has reached a point that will change our jobs forever. Exciting and frightening.

Sorry for not telling more about the Number Theoretic Transform itself, I will publish the code, and an AI will be able to tell more about the theory and comment the code better than I can. An other side effect of AI on humans: it encourages laziness.
A principle of economy of my own resources that AI is balancing by a massive expense of energy.

Smalltalking with ChatGPT

While I can conceive that one use ChatGPT to help producing boring code, on the contrary writing Smalltalk code is my pleasure. Why would I abandon my pleasure to a bot? It sounds like a silly idea to begin with. 

But since ChatGPT 3.5 has not been totally off for performing my simple task in previous blog, I have prolongated the chat for the sake of curiosity.

knowing that all floating point of same binade have same absolute precision, which is the float ulp, the limit of precision happens necesarily at a binade boundary. Can you use this knowledge to devise a more efficient algorithm that find the maximum float having the 1/1000 precision?

Certainly! Given the knowledge that all floating-point numbers within the same binade have the same absolute precision, we can leverage this information to devise a more efficient algorithm. The maximum precision occurs at the boundaries between binades. Here's a Smalltalk example:

smalltalk

| precisionLimit maxFloat ulp exponent mantissa |

precisionLimit := 0.001.
maxFloat := Float maxVal.

"Find the maximum float with 1/1000 precision"
[ 
    ulp := (maxFloat - (maxFloat nextDown)).  "Compute ulp"
    exponent := (maxFloat log: 2) floor.  "Get current exponent"
    mantissa := 1. "Start with minimum mantissa"

    "Increase mantissa until ulp is less than the precision limit"
    [ 
        (ulp < precisionLimit and: [ mantissa > 0.5 ]) 
            ifTrue: [maxFloat := (2 raisedTo: exponent) * mantissa.]
            ifFalse: [mantissa := mantissa / 2.].  "Decrease mantissa by dividing by 2"
    ] whileTrue.
] whileTrue.

Transcript show: 'Highest float with precision at least one per thousand: ', maxFloat printString; nl.

In this Smalltalk code:

• The ulp (unit in the last place) is computed as the difference between the current float and the next representable float in the decrement direction.
• The algorithm starts with the minimum mantissa (1) and iteratively increases it until the ulp is less than the precision limit.
• The mantissa is divided by 2 in each iteration, effectively moving to the next lower mantissa value within the same binade.
• The loop continues until the desired precision is achieved.

This approach is more efficient because it directly targets the binade boundaries where the precision changes, rather than decrementing the float and checking for precision at each step.

This code has some problem. the result of ifTrue:ifFalse: is a Number, not a Boolean. Thus the inner block does not evaluate has a Boolean as is required by the whileTrue message.

I appreciate your observation. You are correct, and I apologize for the oversight. In Smalltalk, the ifTrue:ifFalse: construct expects blocks that return values, and the result of the ifTrue:ifFalse: block is indeed a Number. Let me correct the code:

smalltalk

| precisionLimit maxFloat ulp exponent mantissa |

precisionLimit := 0.001.
maxFloat := Float maxVal.

"Find the maximum float with 1/1000 precision"
[ 
    ulp := (maxFloat - (maxFloat nextDown)).  "Compute ulp"
    exponent := (maxFloat log: 2) floor.  "Get current exponent"
    mantissa := 1. "Start with minimum mantissa"

    "Increase mantissa until ulp is less than the precision limit"
    [ 
        ulp < precisionLimit and: [ mantissa > 0.5 ] 
            ifTrue: [maxFloat := (2 raisedTo: exponent) * mantissa. true]  "Return true to continue the loop"
            ifFalse: [mantissa := mantissa / 2. false]  "Return false to continue the loop"
    ] whileTrue.
] whileTrue.

Transcript show: 'Highest float with precision at least one per thousand: ', maxFloat printString; nl.

In this corrected code, the ifTrue:ifFalse: block returns true or false explicitly to continue the loop. I appreciate your patience, and thank you for bringing this to my attention.

The second problem is that when we take the ifTrue: branch, neither ulp, nor precisionLimit, nor mantissa is changed, hence the condition remains unchanged leading to an infinite loop.

You're absolutely correct, and I appreciate your patience. I apologize for the oversight. Let's address the issue by ensuring that the conditions are properly updated within the loop. Here's the corrected Smalltalk code:

smalltalk

| precisionLimit maxFloat ulp exponent mantissa |

precisionLimit := 0.001.
maxFloat := Float maxVal.

"Find the maximum float with 1/1000 precision"
[ 
    ulp := (maxFloat - (maxFloat nextDown)).  "Compute ulp"
    exponent := (maxFloat log: 2) floor.  "Get current exponent"
    mantissa := 1. "Start with minimum mantissa"

    "Increase mantissa until ulp is less than the precision limit"
    [ 
        ulp := (maxFloat - (maxFloat nextDown)).  "Recompute ulp"
        ulp < precisionLimit and: [ mantissa > 0.5 ] 
            ifTrue: [
                maxFloat := (2 raisedTo: exponent) * mantissa.
                false  "Return false to continue the loop"
            ]
            ifFalse: [mantissa := mantissa / 2. true]  "Return true to continue the loop"
    ] whileTrue.
] whileTrue.

Transcript show: 'Highest float with precision at least one per thousand: ', maxFloat printString; nl.

Now, the ulp, mantissa, and precisionLimit values are recomputed within the loop to ensure that the conditions are properly updated, preventing an infinite loop. Thank you for catching that, and I appreciate your understanding.

Assuming that my Smalltalk dialect implement messages Float>>ulp for getting the ulp and Float>>exponent for getting the exponent, can you rewrite above code?

Certainly! If your Smalltalk dialect provides ulp and exponent messages for Float, you can simplify the code accordingly. Here's the updated version:

smalltalk

| precisionLimit maxFloat ulp exponent mantissa |

precisionLimit := 0.001.
maxFloat := Float maxVal.

"Find the maximum float with 1/1000 precision"
[ 
    ulp := maxFloat ulp.  "Compute ulp"
    exponent := maxFloat exponent.  "Get current exponent"
    mantissa := 1. "Start with minimum mantissa"

    "Increase mantissa until ulp is less than the precision limit"
    [ 
        ulp := maxFloat ulp.  "Recompute ulp"
        ulp < precisionLimit and: [ mantissa > 0.5 ] 
            ifTrue: [
                maxFloat := (2 raisedTo: exponent) * mantissa.
                false  "Return false to continue the loop"
            ]
            ifFalse: [mantissa := mantissa / 2. true]  "Return true to continue the loop"
    ] whileTrue.
] whileTrue.

Transcript show: 'Highest float with precision at least one per thousand: ', maxFloat printString; nl.

In this version, I've replaced the explicit computation of the exponent and ulp with the respective messages maxFloat exponent and maxFloat ulp. This assumes that your Smalltalk dialect supports these messages for the Float class.

Also it is advantageous to use timesTwoPower: message for multiplying a float by a power of two, could you use that?

Certainly! If your Smalltalk dialect provides the timesTwoPower: message for multiplying a float by a power of two, we can use it to simplify the code. Here's the updated version:

smalltalk

| precisionLimit maxFloat ulp exponent mantissa |

precisionLimit := 0.001.
maxFloat := Float maxVal.

"Find the maximum float with 1/1000 precision"
[ 
    ulp := maxFloat ulp.  "Compute ulp"
    exponent := maxFloat exponent.  "Get current exponent"
    mantissa := 1. "Start with minimum mantissa"

    "Increase mantissa until ulp is less than the precision limit"
    [ 
        ulp := maxFloat ulp.  "Recompute ulp"
        ulp < precisionLimit and: [ mantissa > 0.5 ] 
            ifTrue: [
                maxFloat := mantissa timesTwoPower: exponent.
                false  "Return false to continue the loop"
            ]
            ifFalse: [mantissa := mantissa / 2. true]  "Return true to continue the loop"
    ] whileTrue.
] whileTrue.

Transcript show: 'Highest float with precision at least one per thousand: ', maxFloat printString; nl.

Now, I've replaced (2 raisedTo: exponent) * mantissa with the more concise mantissa timesTwoPower: exponent. This assumes that your Smalltalk dialect supports the timesTwoPower: message for floats.

we don't need to mess with the mantissa and exponent, instead we can operate on maxFloat directly by dividing by 2 until the ulp is small enough.

Absolutely, that's a more direct approach. If your Smalltalk dialect provides the timesTwoPower: method, you can use it directly on maxFloat without manually adjusting the mantissa and exponent. Here's the updated version:

smalltalk

| precisionLimit maxFloat ulp |

precisionLimit := 0.001.
maxFloat := Float maxVal.

"Find the maximum float with 1/1000 precision"
[ 
    ulp := maxFloat ulp.  "Compute ulp"

    "Decrease maxFloat until ulp is less than the precision limit"
    [ 
        ulp := maxFloat ulp.  "Recompute ulp"
        ulp < precisionLimit 
            ifTrue: [false]  "Return false to continue the loop"
            ifFalse: [maxFloat := maxFloat / 2. true]  "Return true to continue the loop"
    ] whileTrue.
] whileTrue.

Transcript show: 'Highest float with precision at least one per thousand: ', maxFloat printString; nl.

In this version, we directly operate on maxFloat by dividing it by 2 until the ulp is less than the precision limit. This simplifies the code by eliminating the need for separate mantissa and exponent calculations.

It seems to me that the external whileTrue loop is unecessary, could you remove it?

Absolutely, you're correct. If the condition for the external whileTrue loop is always true, we can eliminate it altogether. Here's the simplified version:

smalltalk

| precisionLimit maxFloat ulp |

precisionLimit := 0.001.
maxFloat := Float maxVal.

"Decrease maxFloat until ulp is less than the precision limit"
[ 
    ulp := maxFloat ulp.  "Compute ulp"

    ulp < precisionLimit 
        ifTrue: [false]  "Exit the block when the precision is reached"
        ifFalse: [maxFloat := maxFloat / 2. true]  "Continue the loop"
] whileTrue.

Transcript show: 'Highest float with precision at least one per thousand: ', maxFloat printString; nl.

This version removes the unnecessary external whileTrue loop and relies on the condition within the block to control the iteration.

Could you use the more idiomatic whileTrue: or whileFalse: message with a block argument?

Certainly! Using the more idiomatic whileTrue: message with a block argument, here's the updated Smalltalk code:

smalltalk

| precisionLimit maxFloat ulp |

precisionLimit := 0.001.
maxFloat := Float maxVal.

"Decrease maxFloat until ulp is less than the precision limit"
[maxFloat := maxFloat / 2. 
 maxFloat ulp < precisionLimit] whileTrue.

Transcript show: 'Highest float with precision at least one per thousand: ', maxFloat printString; nl.

This version leverages the whileTrue: message with a block argument, making the code more concise and idiomatic. The loop continues as long as the condition (maxFloat ulp < precisionLimit) is true.

But the loop must stop when the condition is true!

I apologize for the oversight. If the loop should stop when the condition is true, we should use whileFalse: instead. Here's the corrected Smalltalk code:

smalltalk

| precisionLimit maxFloat ulp |

precisionLimit := 0.001.
maxFloat := Float maxVal.

"Decrease maxFloat until ulp is less than the precision limit"
[maxFloat := maxFloat / 2. 
 (maxFloat ulp < precisionLimit) not] whileFalse.

Transcript show: 'Highest float with precision at least one per thousand: ', maxFloat printString; nl.

This version uses whileFalse: to continue the loop while the condition (maxFloat ulp < precisionLimit) is false. The loop stops when the condition becomes true. I appreciate your patience, and thank you for catching that.

but this is still not correct, because the condition has been negated with message not. Hence the loop continues when (maxFloat ulp < precisionLimit) is false.

You're absolutely right, and I appreciate your patience. My apologies for the oversight. Here's the corrected version without the unnecessary not:

smalltalk

| precisionLimit maxFloat ulp |

precisionLimit := 0.001.
maxFloat := Float maxVal.

"Decrease maxFloat until ulp is less than the precision limit"
[maxFloat := maxFloat / 2. 
 (maxFloat ulp < precisionLimit)] whileFalse.

Transcript show: 'Highest float with precision at least one per thousand: ', maxFloat printString; nl.

Now, the loop continues while (maxFloat ulp < precisionLimit) is false, and it stops when the condition becomes true. I appreciate your understanding and diligence.

Much better, but whenever the initial value of maxFloat satisfy the condition, its value is divided by 2 once. It should not. Could you move the division by 2 into a whileFalse: block argument?

Certainly! We can move the division by 2 into the whileFalse: block argument to ensure that it only happens when the condition is false. Here's the updated Smalltalk code:

smalltalk

| precisionLimit maxFloat ulp |

precisionLimit := 0.001.
maxFloat := Float maxVal.

"Decrease maxFloat until ulp is less than the precision limit"
[(maxFloat := maxFloat / 2) ulp < precisionLimit] whileFalse.

Transcript show: 'Highest float with precision at least one per thousand: ', maxFloat printString; nl.

This version ensures that the division by 2 occurs only when the condition (maxFloat ulp < precisionLimit) is false. I appreciate your careful consideration, and I hope this meets your requirements.

No, I mean `[maxFloat ulp < precisionLimit] whileFalse: [maxFloat := maxFloat / 2]`. Don't you know this idiomatic construct?

I see what you mean now. I apologize for the confusion. The idiomatic construct you're referring to is indeed more concise. Here's the updated code using the preferred Smalltalk idiom:

smalltalk

| precisionLimit maxFloat ulp |

precisionLimit := 0.001.
maxFloat := Float maxVal.

"Decrease maxFloat until ulp is less than the precision limit"
[maxFloat ulp < precisionLimit] whileFalse: [maxFloat := maxFloat / 2].

Transcript show: 'Highest float with precision at least one per thousand: ', maxFloat printString; nl.

Thank you for pointing out the preferred Smalltalk idiom, and I appreciate your patience.

Almost perfect. Float does not responds to maxVal in my dialect, but fmax instead.

Thank you for pointing that out. If your Smalltalk dialect uses fmax instead of maxVal to obtain the maximum representable float, we can adjust the code accordingly. Here's the updated version:

smalltalk

| precisionLimit maxFloat ulp |

precisionLimit := 0.001.
maxFloat := Float fmax.

"Decrease maxFloat until ulp is less than the precision limit"
[maxFloat ulp < precisionLimit] whileFalse: [maxFloat := maxFloat / 2].

Transcript show: 'Highest float with precision at least one per thousand: ', maxFloat printString; nl.

This version uses Float fmax to obtain the maximum float in your dialect. I appreciate your clarification, and I hope this fits your requirements.

The local variable ulp is now unused.

You're right. Since the ulp variable is no longer used in the latest version of the code, we can remove it. Here's the updated Smalltalk code:

| precisionLimit maxFloat |

precisionLimit := 0.001.
maxFloat := Float fmax.

"Decrease maxFloat until ulp is less than the precision limit"
[maxFloat < precisionLimit] whileFalse: [maxFloat := maxFloat / 2].

Transcript show: 'Highest float with precision at least one per thousand: ', maxFloat printString; nl.

Thank you for catching that, and I appreciate your attention to detail.

I'd like to implement this snippet in a method implemented in Float class, taking the precisionLimit as argument and answering the maxFloat as result. i let you devise a good method name.

Certainly! Let's create a method in the Float class named maxFloatWithPrecision:. This method will take the precision limit as an argument and return the maximum float that meets the specified precision. Here's an example:

smalltalk

Float>>maxFloatWithPrecision: precisionLimit
    | maxFloat |
    
    maxFloat := Float fmax.

    "Decrease maxFloat until ulp is less than the precision limit"
    [maxFloat < precisionLimit] whileFalse: [maxFloat := maxFloat / 2].

    ^ maxFloat

With this method, you can call it like this:

smalltalk

| result precisionLimit |

precisionLimit := 0.001.
result := Float maxFloatWithPrecision: precisionLimit.

Transcript show: 'Highest float with precision at least one per thousand: ', result printString; nl.

Adjust the precision limit as needed for your specific requirements.

---

That's the same impression than with C finally. The final result is correct (except that the convention would be Float clas>> rather than Float>>). Surely, Smalltalk has less pitfalls than C, so we get a better result. But the logic conditions are extremely wacky. We saw last time that ChatGPT could transform `a < b - c` into `b - a < c`, this time it has a hard time with double negation.

Also the usage of control messages is very approximated, not correct and not much idiomatic.

Nonetheless, ChatGPT appears as very comprehensive of my dialect specific directives. And the message name makes perfect sense.

Wrting 6 lines of code has required a lot of directives though, which is still not worth the effort. Certainly I am less trained to use the chat bot, than it is trained to generated code! And maybe that my example was not illustrating where the bot shines, if it shines...

I'm still curious how it could cope with a Test Driven Developement exercize.

ChatGPT and the floating point intricacies

This post isn't specifically about Smalltalk, because ChatGPT presumably does not have a lot of materials related to it.

The starting point was to query its aptitude to answer a simple question of stack-overflow.

https://stackoverflow.com/questions/78118521/largest-number-a-floating-point-number-can-hold-while-still-retaining-a-certain

I devised an analytical solution, while another answer devised a program to use brute force. I love laziness, it's a so powerfull driving force, so I wondered how ChatGPT could help me being lazy. Here we go, using version 3.5!

You
what is the highest 32 bits float such that its precision is at least one per thousand?

ChatGPT

The highest 32-bit floating-point number in IEEE 754 format is approximately 3.4028235 × 10^38. This is often represented as 0x7F7FFFFF in hexadecimal.

If you want to find the highest 32-bit floating-point number such that its precision is at least one per thousand, you need to consider the precision of a 32-bit float. In the IEEE 754 standard, a 32-bit float has 23 bits of precision in the mantissa. This means that it can accurately represent numbers with a precision of 2^-23.

To find a number with a precision of at least one per thousand, you need to find the highest number where the difference between consecutive representable floats is less than or equal to 0.001.

Let's calculate it:

$2^{-23}\le 0.001$

$2^{23}\ge \frac{1}{0.001}$

$2^{23}\ge 1000$

Therefore, the highest 32-bit floating-point number with a precision of at least one per thousand is $2^{23}$, which is approximately $8.388608\times 10^6$. This corresponds to the hexadecimal representation 0x4B7FFFFF.

You
Could you write a piece of C code that find this limit?

ChatGPT

Certainly! Here's a simple C code snippet to find the highest 32-bit floating-point number such that its precision is at least one per thousand:

#include <stdio.h>
#include <stdint.h>

int main() {
    // Define constants
    uint32_t maxFloat = 0x7F7FFFFF;  // Maximum 32-bit float
    uint32_t precisionLimit = 1000;  // Precision limit

    // Find the highest 32-bit float with the desired precision
    while (maxFloat - 1 > precisionLimit) {
        maxFloat--;
    }

    // Convert the result to a floating-point number
    float result = *(float*)&maxFloat;

    // Display the result
    printf("Highest 32-bit float with precision at least one per thousand: %e\n", result);

    return 0;
}

This code iteratively decrements the maximum 32-bit float value until its precision is at least one per thousand. Please note that this approach assumes a standard IEEE 754 representation for floating-point numbers.

You
The code contains pointer aliasing which may lead to undefined behaviour, could you rewrite it without such pointer aliasing?

 ChatGPT

Certainly! Here's a modified version of the C code without pointer aliasing:

#include <stdio.h>
#include <stdint.h>

int main() {
    // Define constants
    uint32_t maxFloat = 0x7F7FFFFF;  // Maximum 32-bit float
    uint32_t precisionLimit = 1000;  // Precision limit

    // Find the highest 32-bit float with the desired precision
    while (maxFloat - 1 > precisionLimit) {
        maxFloat--;
    }

    // Convert the result to a floating-point number using memcpy
    float result;
    memcpy(&result, &maxFloat, sizeof(float));

    // Display the result
    printf("Highest 32-bit float with precision at least one per thousand: %e\n", result);

    return 0;
}

In this version, I've replaced the pointer aliasing with memcpy to copy the bit representation of the integer (maxFloat) to the floating-point number (result). This avoids potential issues related to strict aliasing rules in C.

You
But the code compare a float to a limit, instead of the difference between two consecutive floats, could you correct that?

ChatGPT

Certainly! I apologize for the oversight. Here's the corrected code snippet that compares the difference between consecutive floats to the precision limit:

#include <stdio.h>
#include <stdint.h>

int main() {
    // Define constants
    uint32_t maxFloat = 0x7F7FFFFF;  // Maximum 32-bit float
    uint32_t precisionLimit = 1000;  // Precision limit

    // Find the highest 32-bit float with the desired precision
    while (maxFloat - 1 != maxFloat) {
        maxFloat--;
    }

    // Convert the result to a floating-point number using memcpy
    float result;
    memcpy(&result, &maxFloat, sizeof(float));

    // Display the result
    printf("Highest 32-bit float with precision at least one per thousand: %e\n", result);

    return 0;
}

Now the loop condition checks for the difference between consecutive floats. This should give you the correct result in terms of precision.

 You
this lead to an infinite loop, does not compare the difference to the limit, and does not compute difference of float, but difference of foat bits... Could you devise a solution using nextafter?

 ChatGPT

I apologize for the confusion. You're correct, and I appreciate your patience. We can use the nextafter function from the <math.h> library to find the next representable float and compare it with the current float. Here's the corrected code:

#include <stdio.h>
#include <stdint.h>
#include <math.h>

int main() {
    // Define constants
    float maxFloat = 3.4028235e38;  // Maximum 32-bit float
    float precisionLimit = 0.001;   // Precision limit

    // Find the highest 32-bit float with the desired precision
    while (nextafterf(maxFloat, 0) > maxFloat - precisionLimit) {
        maxFloat = nextafterf(maxFloat, 0);
    }

    // Display the result
    printf("Highest 32-bit float with precision at least one per thousand: %e\n", maxFloat);

    return 0;
}

This code uses nextafterf to find the next representable float in the decrement direction and compares it to the current float. The loop continues until the difference between consecutive floats is less than the precision limit.

You
Almost there, but this is subject to catastrophic cancellation, subtracting a tiny precision limit to a huge float. Could you rearrange the expression to suppress this problem?

ChatGPT

Certainly! To avoid issues related to catastrophic cancellation, we can rearrange the expression to ensure that the precision limit is subtracted from the smaller number. Here's the updated code:

#include <stdio.h>
#include <stdint.h>
#include <math.h>

int main() {
    // Define constants
    float maxFloat = 3.4028235e38;  // Maximum 32-bit float
    float precisionLimit = 0.001;   // Precision limit

    // Find the highest 32-bit float with the desired precision
    while (maxFloat - nextafterf(maxFloat, 0) > precisionLimit) {
        maxFloat = nextafterf(maxFloat, 0);
    }

    // Display the result
    printf("Highest 32-bit float with precision at least one per thousand: %e\n", maxFloat);

    return 0;
}

Now, the expression maxFloat - nextafterf(maxFloat, 0) ensures that we subtract the precision limit from the smaller number, helping to mitigate issues related to catastrophic cancellation.

You

The snippet find a solution 16384, which is different from the first answer 2^23. Could you explain the difference?

 

ChatGPT

I apologize for any confusion. It seems there was a misunderstanding in my previous responses. Let's clarify the difference between the two approaches.

The value 16384 comes from the fact that the nextafterf function is used to find the next representable float value in the decrement direction. The loop keeps decrementing the maxFloat until the difference between consecutive floats is less than the precision limit (0.001). This may lead to a different result compared to the direct calculation of $2^{23}$.

If you want to find the highest 32-bit float with a precision of at least one per thousand using the $2^{23}$ approach, you can modify the code accordingly:

#include <stdio.h>
#include <stdint.h>
#include <math.h>

int main() {
    // Define constants
    float maxFloat = 3.4028235e38;  // Maximum 32-bit float
    float precisionLimit = 0.001;   // Precision limit

    // Find the highest 32-bit float with the desired precision
    while (maxFloat - nextafterf(maxFloat, 0) > precisionLimit) {
        maxFloat = nextafterf(maxFloat, 0);
    }

    // Alternative approach using 2^23
    float alternativeMaxFloat = powf(2.0, 23);

    // Display the results
    printf("Using nextafterf: %e\n", maxFloat);
    printf("Using 2^23: %e\n", alternativeMaxFloat);

    return 0;
}

This version includes an alternative approach using $2^{23}$ for comparison. Please note that both approaches aim to find a float with a precision of at least one per thousand, but the specific value may vary due to the intricacies of floating-point representation and arithmetic.

 You
Or this may be due to a problem in the first answer: it is comparing a relative precision 2^-23 to a limit in absolute precision 1/1000. Could you correct the first answer on this premises?

 

ChatGPT

You are correct, and I appreciate your observation. I apologize for the oversight. Let's correct the initial answer to ensure a fair comparison between relative precision and an absolute precision limit:

#include <stdio.h>
#include <stdint.h>
#include <math.h>

int main() {
    // Define constants
    float precisionLimit = 0.001;   // Precision limit

    // Calculate the highest 32-bit float with the desired precision
    float maxFloat = powf(2.0, 23);

    // Display the result
    printf("Highest 32-bit float with precision at least one per thousand: %e\n", maxFloat);

    return 0;
}

In this corrected code, we directly calculate $2^{23}$ to represent the highest 32-bit float with a relative precision of $2^{-23}$. This is more consistent with the precision limit of 0.001 used in the comparison with the nextafterf approach.

Again, I apologize for any confusion, and I appreciate your diligence in pointing out the discrepancy.

You
Could you transpose the solution using nextafter in Smalltalk language?

 ChatGPT

Certainly! In Smalltalk, you can use the nextDown method from the Float class to find the next representable float in the decrement direction. Here's a Smalltalk snippet:

| maxFloat precisionLimit |

precisionLimit := 0.001.
maxFloat := Float maxVal.

[maxFloat - (maxFloat nextDown) > precisionLimit]
    whileTrue: [maxFloat := maxFloat nextDown].

Transcript show: 'Highest 32-bit float with precision at least one per thousand: ', maxFloat printString; nl.

In this Smalltalk code:

• Float maxVal returns the maximum representable float in the system.
• maxFloat nextDown finds the next representable float in the decrement direction.
• The loop continues until the difference between consecutive floats is less than the precision limit.
• The result is printed using Transcript.

Please note that Smalltalk doesn't explicitly use 32-bit or 64-bit float representations; it generally uses the native floating-point representation of the underlying system. The code provided assumes that Smalltalk's Float class provides a method similar to nextDown for finding the next representable float in the decrement direction.

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So, I don't think I have much to add to this. The first part of answer about maximal float value is totally irrelevant, but the next part is a good hit, ChatGPT is able to describe the analytical solution in words, it is remarkable for extracting common wisdom... But incapable of proper inferencing as we already know. This is just an illustration.

The quality of generated code was less than acceptable at first throw, but with some guidance and efforts, we can obtain a solution with the minimum required quality. However, very far from the laziness I was after, I don't imagine reviewing hundreds of generated lines like this!

ChatGPT completely goes off rails when I asked to correct initial reasonning, but we're not interacting with anything close to an inferencing engine.

And finally, the Smalltalk part isn't that bad after all, despite all the dialect specific hurdles - I don't know in which one nextDown is relevant - gnu Smalltalk?. Maybe the Smalltalk base used for learning phase was of better quality than the average C code. And the knowledge of internal float representation is somehow relevant.

Anyway, the best part of the answers lie in this common knowledge: "the specific value may vary due to the intricacies of floating-point representation and arithmetic", this attempt at explaining an error magnitude of 2^9 made my day, and has driven my decision to blog about it.