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Floating point format for Intel math coprocessors
On Fri, 27 Jun 2003 14:23:22 GMT, Jack Crenshaw
wrote: I've run across a peculiarity concerning the format used inside the Intel math coprocessor. I have always thought that the format used was in accordance with the IEEE 794 Unless IEEE 794 is something new, I think you mean IEEE 754. standard, and every reference I've seen on the web seems to imply that. But, as nearly as I can tell, it's not the same. The IEEE standard for 32-bit floats says the format should be sign -- 1 bit exponent -- 8 bits, power of 2, split on 127 With the proviso that the values of 0 and 255 for the exponent are special cases reserved for 0, Inf, denormals, and NaN. mantissa -- 23 bits + phantom bit in bit 24. The Intel processor seems to use the following: sign -- 1 bit exponent -- _SEVEN_ bits, power of _FOUR_ mantissa -- sometimes 23 bits, sometimes 24. Sometimes phantom bit, sometimes not. When it's there, it's in bit _TWENTY_THREE_ ! I don't think so. Are you mistaking the lsb of the exponent for the "visible" phantom bit? [...] You'll see that 1 -- 3f800000 (high bit is visible) seee eeee emmm mmmm mmmm mmmm mmmm mmmm 0011 1111 1000 0000 0000 0000 0000 0000 s = 0, e = 127, m = 0 (-1)^s * 2^(e-127) * (1+m/(2^23)) = 1*1*1 = 1.0 but 2 -- 40000000 (high bit is not) seee eeee emmm mmmm mmmm mmmm mmmm mmmm 0100 0000 0000 0000 0000 0000 0000 0000 s = 0, e = 128, m = 0 (-1)^s * 2^(e-127) * (1+m/(2^23)) = 1*2*1 = 2.0 Try a few others and see what you get. Some will surprise you. I'm not finding any surprises. 1.5 - 3fc00000 seee eeee emmm mmmm mmmm mmmm mmmm mmmm 0011 1111 1100 0000 0000 0000 0000 0000 s = 0, e = 127, m = 0x400000 (-1)^s * 2^(e-127) * (1+m/(2^23)) = 1*1*1.5 = 1.5 2.5 - 40200000 seee eeee emmm mmmm mmmm mmmm mmmm mmmm 0100 0000 0020 0000 0000 0000 0000 0000 s = 0, e = 128, m = 0x400000 (-1)^s * 2^(e-127) * (1+m/(2^23)) = 1*2*1.25 = 2.5 Perhaps I'm misunderstanding your point? Regards, -=Dave -- Change is inevitable, progress is not. |
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In article , Jack Crenshaw wrote:
I've run across a peculiarity concerning the format used inside the Intel math coprocessor. If you're talking about the format that IA32 FPs store values in memory, then I doubt it. If you're talking about the 80-bit internal format, I don't know. I've never tried to use that format externally. I've been exchanging float data between IA32 systems and at least a half-dozen other architectures since the 8086/8087 days. I never saw any format problems. I have always thought that the format used was in accordance with the IEEE 794 standard, It is IEEE something though 794 doesn't sound right... -- Grant Edwards grante Yow! .. this must be what at it's like to be a COLLEGE visi.com GRADUATE!! |
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Jack Crenshaw wrote:
.... snip ... The IEEE standard for 32-bit floats says the format should be sign -- 1 bit exponent -- 8 bits, power of 2, split on 127 mantissa -- 23 bits + phantom bit in bit 24. The Intel processor seems to use the following: sign -- 1 bit exponent -- _SEVEN_ bits, power of _FOUR_ mantissa -- sometimes 23 bits, sometimes 24. Sometimes phantom bit, sometimes not. When it's there, it's in bit _TWENTY_THREE_ ! .... snip ... You'll see that 1 -- 3f800000 (high bit is visible) but 2 -- 40000000 (high bit is not) Try a few others and see what you get. Some will surprise you. I know that there must be people out there to whom this is old, old news. Even so, I've never seen a word about it, and didn't find anything in a Google search. I'd appreciate any comments. What chips does this format appear in? I expect the presence or absence of normalization depends on the oddness of the exponent byte. It makes sense for byte addressed memory based systems, since zero (ignoring denormalization) can be detected in a single byte. -- Chuck F ) ) Available for consulting/temporary embedded and systems. http://cbfalconer.home.att.net USE worldnet address! |
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On Fri, 27 Jun 2003 17:58:50 GMT, Jonathan Kirwan
wrote: Hmm. Are you *THE* Jack Crenshaw? The "Let's Build A Compiler" and "Math Toolkit for Real-Time Programming" Jack _W._ Crenshaw? Actually, I think I've answered my own question, here. You really *are* that Jack. What clues me in is your use of "phantom" he mantissa -- 23 bits + phantom bit in bit 24. The same term used on page 50 in "Math toolkit..." In my own experience, even that predating the Intel 8087 or the IEEE standardization, it was called a "hidden bit" notation. I don't know where "phantom" comes from, as my own reading managed to completely miss it. So, a hearty "Hello" from me! Jon |
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On Fri, 27 Jun 2003 14:23:22 GMT, Jack Crenshaw
wrote: I've run across a peculiarity concerning the format used inside the Intel math coprocessor. I have always thought that the format used was in accordance with the IEEE 794 standard, and every reference I've seen on the web seems to imply that. But, as nearly as I can tell, it's not the same. The IEEE standard for 32-bit floats says the format should be sign -- 1 bit exponent -- 8 bits, power of 2, split on 127 mantissa -- 23 bits + phantom bit in bit 24. The Intel processor seems to use the following: sign -- 1 bit exponent -- _SEVEN_ bits, power of _FOUR_ mantissa -- sometimes 23 bits, sometimes 24. Sometimes phantom bit, sometimes not. The sign bit is there, as the highest order bit -- just as you note. This is followed, working left to right, by the exponent which is in "excess 127" format. It's not a signed format, but an excess 127 format. A couple of special values, 0 and 255, are reserved for the fancy stuff. But those correspond to exponent values of -127 and +128 and no one is supposed to miss them. The mantissa is quite simply *always* associated with a hidden bit, which always leads the value. Okay, the exception made to the above is for the exact value of zero, where the exponent is 0 and the mantissa is 0 and the hidden bit is assumed 0, as well. When it's there, it's in bit _TWENTY_THREE_ ! I don't agree. At least, not yet in my experience. And I haven't seen an example below which makes your point. The way this works is, if you have an exponent other than 2, you must shift by more than one bit to normalize. Well, I know how to normalize. After writing a few complex-FP FFT routines for integer processors, it gets to be kind of routine and hum-drum. So I'll skip the explanation. The old IBM 360 used base 16, so had to shift by four bits. That's why its f.p. precision was so awful. Interesting note about the 360. I only had a few opportunities to program in BAL and never got into the floating point formats. The Intel 32-bit format shifts by two, so sometimes the high bit is a one, sometimes a zero. High bit is *always* a 1, after normalizing (except for zero.) But, as you know, it is thrown away. Never kept. That causes the format to look really funky. Try this: typedef union{ float x; long n; }float_hack; void main(void){ float_hack num; while(1){ cin num.x; cout hex num.n endl; } } You'll see that 1 -- 3f800000 (high bit is visible) but 2 -- 40000000 (high bit is not) Which doesn't make your point, because it's quite correct to use those two values to represent 1 and 2. 3F800000 is: 1 -- hidden bit 0 01111111 00000000000000000000000 - -------- ----------------------- S exponent mantissa 40000000 is: 1 -- hidden bit 0 10000000 00000000000000000000000 - -------- ----------------------- S exponent mantissa In those two cases, the only difference is that the exponents are 1 apart from each other. Which is exactly what you'd expect for 1.0 and 2.0. The mantissa is the same for both. Try a few others and see what you get. Some will surprise you. I have, believe me. I know that there must be people out there to whom this is old, old news. Even so, I've never seen a word about it, and didn't find anything in a Google search. I'd appreciate any comments. Well, I hope that helps some. Jon |
#6
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In article , Jonathan Kirwan wrote:
The old IBM 360 used base 16, so had to shift by four bits. That's why its f.p. precision was so awful. Interesting note about the 360. I only had a few opportunities to program in BAL and never got into the floating point formats. IIRC, the Navy's UYK-44 processor (probably UYK-20 as well, though I'm not sure it did FP) also used base 16 for the exponent, so increasing the exponent by 1 shifted the mantissa by 4. I dare anybody to claim that's a useful bit of information to have retained for 15+ years.... -- Grant Edwards grante Yow! .. bleakness.... at desolation.... plastic visi.com forks... |
#7
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In article , Jonathan Kirwan wrote:
Interesting note about the 360. I only had a few opportunities to program in BAL and never got into the floating point formats. IIRC, the Navy's UYK-44 processor (probably UYK-20 as well, though I'm not sure it did FP) also used base 16 for the exponent, so increasing the exponent by 1 shifted the mantissa by 4. I dare anybody to claim that's a useful bit of information to have retained for 15+ years.... I think I've actually read about this, once. Been a while, though. And... I'm glad I was able to forget it. Now, you've gone and forced those poor brain cells to re-align on this and I'm probably going to forget something else important. Could be worse, I could've explained what BAM variables were... -- Grant Edwards grante Yow! YOW!! I am having at FUN!! visi.com |
#8
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In article , Everett M. Greene wrote:
IIRC, the Navy's UYK-44 processor (probably UYK-20 as well, though I'm not sure it did FP) also used base 16 for the exponent, so increasing the exponent by 1 shifted the mantissa by 4. I dare anybody to claim that's a useful bit of information to have retained for 15+ years.... I think I've actually read about this, once. Been a while, though. And... I'm glad I was able to forget it. Now, you've gone and forced those poor brain cells to re-align on this and I'm probably going to forget something else important. Could be worse, I could've explained what BAM variables were... BAM is neat! BTW: Wasn't it AYK-20? -20 didn't have any FP. I've never been hear a -44. I _think_ it was UYK, since everybody prounouced it "yuck". The 44 was a small version of the same architecture that was done by, um, Sperry (I think). Originally it was designed for use on submarines (A 44 chassis would fit (barely) through a submarine's loading hatch). A '20, OTOH, was more of a standard computer-room VAX-sized thing -- you'd have to build a sub hull around it. A '44 consisted of a backplane full of very expensive little boards (about 3x6 inches). It took several of the boards for the CPU, and then there were memory and I/O modules. The whole thing, including power supply was the size of a small suitcase. The CPU was built out of AM2901 bit-slice processors, and executed a superset of the 20's instruction set. The '44 was "standardized" as the Navy's official embedded computer. It was about as powerful as decent 8086 single-board-computer, only 100X larger and 1000X more expensive. It did have plug in cards for all the oddball USN-specific serial/parallel interfaces, which gave it a leg-up on commercial stuff. The '44 had FP, and the ones I played with used EEPROM/RAM instead of core (though core memory was available for it, IIRC). It was sort of cool that it could do polar-rectangular coordinate transforms in a single machine instruction. For the project I worked on, we would have embedded a couple 8086's and done C programs given our 'druthers, but NAVSEA insisted that we use '44s and write in CMS/2 or CMS-2 or whatever it was called. The also wanted us to use some OS or other from the '20. But, there was no way it could deal with the real-time requirements we had, so they let us write out own simple kernel. The whole project was cancelled after a couple years (never even got a prototype working). A few years later it was revived and redesigned using "commercial" processors before being cancelled again. Sure glad I'm out of defense work... -- Grant Edwards grante Yow! My forehead feels at like a PACKAGE of moist visi.com CRANBERRIES in a remote FRENCH OUTPOST!! |
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Jonathan Kirwan wrote:
On Fri, 27 Jun 2003 17:58:50 GMT, Jonathan Kirwan wrote: Hmm. Are you *THE* Jack Crenshaw? The "Let's Build A Compiler" and "Math Toolkit for Real-Time Programming" Jack _W._ Crenshaw? Actually, I think I've answered my own question, here. You really *are* that Jack. Grin! Yep, I really am. What clues me in is your use of "phantom" he mantissa -- 23 bits + phantom bit in bit 24. The same term used on page 50 in "Math toolkit..." In my own experience, even that predating the Intel 8087 or the IEEE standardization, it was called a "hidden bit" notation. I don't know where "phantom" comes from, as my own reading managed to completely miss it. So, a hearty "Hello" from me! Hello. Re the term, phantom bit: I've been using that term since I can remember -- and that's a looooonnnngggg time. Then again, I still sometimes catch myself saying "cycles" or "kilocycles," or "B+". I first heard the term in 1975. Not sure when it became Politically Incorrect. Maybe someone objected to the implied occult nature of the term, "phantom"? Who knows? but as far as I'm concerned the term "hidden bit" is a Johnny-come-lately on the scene. Back to the point. I want to thank you and everyone else who responded (except the guy who said "stop it") for helping to straighten out my warped brain. It's nice that you have my book. Thanks for buying it. As a matter of fact, I first ran across this "peculiarity" three years ago, when I was writing it. I needed to twiddle the components of the floating-point number -- separate the exponent from mantissa -- to write the fp_hack structure for the square root algorithm. I looked at the formats for float, double, and long double, and found the second two formats easy enough to grok. But when I looked at the format for floats, I sort of went, "Gag!" and quickly decided to use doubles for the book. It's funny how an idea, once formed, can persist. Lo those many years ago, I didn't have a lot of time to think about it -- had to get the chapter done. I just managed to convince myself that the format used this peculiar convention, what with base-4 exponents, and all. I had no more need of it at the time, so never went back and revisited the impression. It's persisted ever since. All of the folks who responded are absolutely right. Once I got my head screwed on straight, it was quite obvious that the format has no mysteries. It is indeed the IEEE 754 format, plain and simple. The thing that had me confused was the exponents: 3f8, 400, 408, etc. With one bit for the sign and eight for the exponent, it's perfectly obvious that the exponent has to bleed down one bit into the next lower hex digit. That's what I was seeing, but somehow in my haste, I didn't recognize it as such, and formed this "theory" that it was using a base-4 exponent. Wanna hear the funny part? After tinkering with it for awhile, I worked out the rules for my imagined format, that worked just fine. At work, I've got a Mathcad file that takes the hex number, shifts it two bits at a time, diddles the "phantom" bit, and produces the right results. I can go from integer to float and back nicely, using this cockamamie scheme. Needless to say, the conversion is a whole lot easier if one uses the real format! My Mathcad file just got a lot shorter. Thanks again to everyone who responded, and my apologies for bothering y'all with this imaginary problem. Jack |
#10
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On Tue, 01 Jul 2003 13:03:19 GMT, Jack Crenshaw
wrote: Jonathan Kirwan wrote: On Fri, 27 Jun 2003 17:58:50 GMT, Jonathan Kirwan wrote: Hmm. Are you *THE* Jack Crenshaw? The "Let's Build A Compiler" and "Math Toolkit for Real-Time Programming" Jack _W._ Crenshaw? Actually, I think I've answered my own question, here. You really *are* that Jack. Grin! Yep, I really am. Hehe. Nice to know one of my antennas is still sharp. What clues me in is your use of "phantom" he mantissa -- 23 bits + phantom bit in bit 24. The same term used on page 50 in "Math toolkit..." In my own experience, even that predating the Intel 8087 or the IEEE standardization, it was called a "hidden bit" notation. I don't know where "phantom" comes from, as my own reading managed to completely miss it. So, a hearty "Hello" from me! Hello. Re the term, phantom bit: I've been using that term since I can remember -- and that's a looooonnnngggg time. I think my first exposure to hidden-bit as a term dates to about 1974. But I could be off, by a year, either way. Then again, I still sometimes catch myself saying "cycles" or "kilocycles," or "B+". Hehe. Now those terms aren't so "hidden" to me. I learned my early electronics on tube design manuals. One sticking point I remember bugging me for a long time was exactly, "How do they size those darned grid leak resistors?" I just couldn't figure out where they got the current from which to figure their magnitude. So even B+ is old hat to me. I first heard the term in 1975. Well, that's about the time for "hidden bit," too. Probably, at that time the term was still in a state of flux. I just got my hands on different docs, I imagine. Not sure when it became Politically Incorrect. Oh, it's fine to me, anyway. I knew what was meant the moment I saw the term. It's pretty clear. I just think time has settled more on one term than another. But to take your allusion and run with it a bit... I don't know of anyone part of some conspiracy to set the term -- in any case, setting terms usually is propagandistic, designed for setting agendas in peoples' minds and here is a case where everyone would want the same agenda. Maybe someone objected to the implied occult nature of the term, "phantom"? Oh, geez. I've never known a geek to care about such things. I suppose they must exist, somwhere. I've just never met one willing to let me know they thought like that. But that's an interesting thought. It would fit the weird times in the US we live in, with about 30% aligning themselves as fundamentalists. Nah... it just can't be. Who knows? I really think it was more the IEEE settling on a term. But then, this isn't my area so I could be dead wrong about that -- I'm only guessing. but as far as I'm concerned the term "hidden bit" is a Johnny-come-lately on the scene. Hehe. I've no problem if that's true. Back to the point. I want to thank you and everyone else who responded (except the guy who said "stop it") for helping to straighten out my warped brain. No problem. It was really pretty easy to recall the details. Like learning to ride a bicycle, I suppose. It's nice that you have my book. Thanks for buying it. Oh, there was no question. I've a kindred interest in physics and engineering, I imagine. I'm currently struggling through Robert Gilmore's books, one on lie groups and algebras and the other on catastrophe theory for engineers as well as polytropes, packing spheres, and other delights. There were some nice insights in your book, which helped wind me on just enough of a different path to stretch me without losing me. By the way!! I completely agree with you about MathCad! What a piece of *&!@&$^%$^ it is, now. I went through several iterations, loved at first the slant or approach in using it, but absolutely hate it now because, frankly, I can't run it for more than an hour before I don't have any memory left and it crashes out. Reboot time every hour is not my idea of a good thing. And that's only if I don't type and change things too fast. When I work quick on it, I can go through what's left with Win98 on a 256Mb RAM machine in a half hour! No help from them and two versions later I've simply stopped using it. I don't even want to hear from them, again. Hopefully, I'll be able to find an old version somewhere. For now, I'm doing without. As a matter of fact, I first ran across this "peculiarity" three years ago, when I was writing it. I needed to twiddle the components of the floating-point number -- separate the exponent from mantissa -- to write the fp_hack structure for the square root algorithm. I looked at the formats for float, double, and long double, and found the second two formats easy enough to grok. But when I looked at the format for floats, I sort of went, "Gag!" and quickly decided to use doubles for the book. Yes. But that's fine, I suspect. I've taught undergrad classes and most folks just go "barf" when confronted with learning floating point. In class evaluations, I think having to learn floating point was the bigger source of complaints about the classes. You probably addressed everything anyone "normal" could reasonably care about and more. It's funny how an idea, once formed, can persist. Lo those many years ago, I didn't have a lot of time to think about it -- had to get the chapter done. I just managed to convince myself that the format used this peculiar convention, what with base-4 exponents, and all. I had no more need of it at the time, so never went back and revisited the impression. It's persisted ever since. No problem. All of the folks who responded are absolutely right. Once I got my head screwed on straight, it was quite obvious that the format has no mysteries. It is indeed the IEEE 754 format, plain and simple. The thing that had me confused was the exponents: 3f8, 400, 408, etc. With one bit for the sign and eight for the exponent, it's perfectly obvious that the exponent has to bleed down one bit into the next lower hex digit. That's what I was seeing, but somehow in my haste, I didn't recognize it as such, and formed this "theory" that it was using a base-4 exponent. In any case, it's clear that your imagination is able to work overtime, here! Maybe that's a good thing. Wanna hear the funny part? After tinkering with it for awhile, I worked out the rules for my imagined format, that worked just fine. At work, I've got a Mathcad file that takes the hex number, shifts it two bits at a time, diddles the "phantom" bit, and produces the right results. I can go from integer to float and back nicely, using this cockamamie scheme. Hmm. Then you should be able to construct a function to map between these, proving the consistent results. I've a hard time believing there is one. But who knows? Maybe this is the beginning of a new facet of mathematics, like the investigation into fractals or something! Needless to say, the conversion is a whole lot easier if one uses the real format! My Mathcad file just got a lot shorter. Hehe!! When you get things right, they *do* tend to become a little more prosaic, too. Good thing for those of us with feeble minds, too. Thanks again to everyone who responded, and my apologies for bothering y'all with this imaginary problem. hehe. Best of luck. In the process, I did notice that you are entertaining thoughts on a revised "Let's build a compiler." Best of luck on that and if you feel the desire for unloading some of the work, I might could help a little. I've written a toy C compiler before, an assembler, several linkers, and a not-so-toy BASIC interpreter. I can, at least, be a little bit dangerous. Might be able to shoulder something, if it helps. Jon |
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