I remember that many years ago, when floating point computation was expensive for Intel CPUs to do, there were multiple ways that programmers used integer trickery to work around this.
Chuck Moore of Forth fame demonstrated taking the value, say 1.6 multiplied by 4.1 and doing all the intermediate calculations via integers (16 * 41) and then formatting the output by putting the decimal point back in the "right place"; this worked as long as the range of floating point values was within a range that multiplying by 10 didn't exceed 65536 (16 bit integers), for instance. For embedded chips where for instance, you have an analog reading with 10 bits precision to quickly compute multiple times per second, this worked well.
I also recall talking many years ago with a Microsoft engineer who had worked with the Microsoft Streets and Trips program (https://archive.org/details/3135521376_qq_CD1 for a screenshot) and that they too had managed to fit what would normally be floating point numbers and the needed calculations into some kind of packed integer format with only the precision that was actually needed, that was faster on the CPUs of the day as well as more easily compressed to fit on the CDROM.
What you're describing is called fixed point arithmetic, a super cool technique I wish more programmers knew about.
Proper finance related code should use it, but in my experience in that industry it doesn't seem very common unless you're running mainframes.
Funnily enough, I've seen a lot more fixed point arithmetic in software rasterizers than anywhere else. FreeType, GDI, WPF, WARP (D3D11 reference rasterizer) all use it heavily.
I have worked on firmware that has plenty of fixed point arithmetic. The firmware usually runs on processors without hardware floating point units. For example certain Tesla ECUs use 32-bit integers where they divide it into four bits of integer part and 28 bits of fractional part. So values are scaled by 2^28.
>> The firmware usually runs on processors without hardware floating point units.
I'm working on control code one an ARM cortex-M4f. I wrote it all in fixed point because I don't trust an FPU to be faster, and I also like to have a 32bit accumulator instead of 24bit. I recently converted it all to floating point since we have the M4f part (f indicate FPU), and it's a little slower now. I did get to remove some limit checking since I can rely on the calculations being inside the limits but it's still a little slower than my fixed point implementation.
The other great thing about going fixed point is that it doesn't expose you to device specific floating point bugs, making your embedded code way more portable and easier to test.
32b float on your embedded device doesn't necessary match your 32b float running on your dev machine.
32b float can match your desktop. Really just takes a few compiler flags(like avoiding -funsafe-math), setting rounding modes, and not using the 80bit Intel mode(largely disused after 64bit transition).
You aren't guaranteed that your microcontrollers float is going to match your desktop. Microcontrollers are riddled with bugs, unless you need floats and fixedpoint is fast enough. My recommendation is still to use fixedpoint if application is high reliability.
Esp if your code needs to be portable across arm, risc-v, etc.
Many microcontrollers today, including ARM, RISC-V, and Xtensa have IEEE compliant FPUs or libms available. Same numeric format, same rounding, same result.
Fixed point isn't bad at all, just often slower when a compliant FPU is available.
> IEEE compliant FPUs or libms available. Same numeric format, same rounding, same result.
IEEE only mandates results within ½ ULP (= best possible) for basic operations such as addition, subtraction, multiplication, division, and reciprocal.
For many other ones such as trigonometric functions, exponential and logarithms, results can (and do) vary between conforming implementations.
“The IEEE standard does not require transcendental functions to be exactly rounded because of the table maker's dilemma. To illustrate, suppose you are making a table of the exponential function to 4 places. Then exp(1.626) = 5.0835. Should this be rounded to 5.083 or 5.084? If exp(1.626) is computed more carefully, it becomes 5.08350. And then 5.083500. And then 5.0835000. Since exp is transcendental, this could go on arbitrarily long before distinguishing whether exp(1.626) is 5.083500...0ddd or 5.0834999...9ddd. Thus it is not practical to specify that the precision of transcendental functions be the same as if they were computed to infinite precision and then rounded. Another approach would be to specify transcendental functions algorithmically. But there does not appear to be a single algorithm that works well across all hardware architectures. Rational approximation, CORDIC,16 and large tables are three different techniques that are used for computing transcendentals on contemporary machines. Each is appropriate for a different class of hardware, and at present no single algorithm works acceptably over the wide range of current hardware.”
IEEE 754-2019 says for the transcendental functions (the ones in §9.2):
> A conforming operation shall return results correctly rounded for the applicable rounding direction for all operands in its domain.
so all of them are supposed to be correctly rounded. I think IEEE 754-2008 also requires correct rounding, but I don't have that spec in front of me right now.
In practice, they're not correctly rounded--the C specification explicitly disclaims the need for them to be (§F.3¶20), reserving the cr_ prefix for future mandatory correctly-rounded variants.
Even with that and ignoring C’s “we don’t support that”), it still can be hard to write C code that provides identical results on all platforms. For example, I don’t think much code uses float_t or double_t or checks FLT_EVAL_METHOD (https://en.cppreference.com/w/c/types/limits/FLT_EVAL_METHOD)
So the things you mention aren't useful for getting consistent numerical results. You really have to start getting into obscure platforms like mainframes to find stuff where float and double aren't IEEE 754 single and double precision, respectively. FLT_EVAL_METHOD is largely only relevant if you're working on 32-bit x86 code, and even then, you can sidestep those problems if you're willing to require that hardware be newer than 20 years old or so.
The actual thing you need to do for consistency is to be extremely vigilant in the command line options you use, and also bring your own math library implementations rather than using the standard library. You also need vigilance in your dependencies, for somebody deciding to enable denormal flushing screws everybody in the same process.
Ah, I've had a slightly different task many times: porting a high level algorithm from MATLAB or labview or keras to C.
As part of this I construct a series of test inputs, and confirm that they are bitwise equivalent to the high level language. It's usually as simple as aligning the rounding mode, disabling fused MAC, and a few other compiler flags that shouldn't be project defaults.
The other fun part is using the vector unit - for that we have to define IEEE arithmetic in the order the embedded device does it(usually 4x or 8x interleaved), port that back up, and verify.
Never did use a whole lot of transcendentals - maybe due to the domains I worked in.
Just look at the instruction set for your particular CPU. Every CPU is different, but in most architectures I've seen, floating point operations are 2-3 times slower for the same word size.
Single float adds are usually 2 or 3 CPU cycles while single-word integer adds are usually 1 cycle.
Again, this is extremely dependent on the particular CPU you have. Some architectures do have single-cycle FPU operations, but it's not very common in microcontrollers as far as I can tell.
That would vary wildly with the ARM chip you are talking about. I would say figure out which ARM you’re interested in and go down the rabbit hole from there.
What do they use? Not float I hope.
Plus given that some currencies have different precisions...
Don't tell me it's rounding errors over trillion monies?! :o)
As I indicate in another post, I work in finance and I use binary floats. So do a lot of others who work in the industry. I sympathize with people who think that IEEE floating points are some weird or error prone representation and that fixed point arithmetic solves every problem, but in my professional experience that isn't true and systems that start by using fixed point arithmetic eventually end up making a half-assed error prone and slow version of floating point arithmetic as soon as they need to handle more sophisticated use cases like handling multiple currencies, doing calculations involving percentages such as interest rates, etc etc...
The IEEE 754 floating point standard is a very well thought out standard that is suitable for representing money as-is. If you have requirements such as compliance/legal/regulatory needs that mandate a minimum precision, then you can either opt to use decimal floating point or use binary floating point where you adjust the decimal place up to whatever legally required precision you are required to handle.
For example the common complaint about binary floating point is that $1.10 can't be represented exactly so you should instead use a fixed integer representation in terms of cents and represent it as 110. But if your requirement is to be able to represent values exactly to the penny, then you can simply do the same thing but using a floating point to represent cents and represent $1.10 as the floating point 110.0. The fixed integer representation conveys almost no benefit over the floating point representation, and once you need to work with and mix currencies that are significantly out of proportion to one another, you begin to really appreciate the nuances and work that went into IEEE 754 for taking into account a great deal of corner cases that a fixed integer representation will absolutely and spectacularly fail to handle.
I build cash registers, and I avoid floats like the plague.
I think the difference is where you need an exact result. Auditors have forced me to go through a years transactions to find an 1 cent error. They were right - at one point we weren't handling the fractional cents correctly. After finding that the bug was fixed. Had we been using floating point our answer would have been "shrug, if it's a problem chose another vendor".
You are working in finance so I suspect a 0.00001% error doesn't matter to you. Usually it doesn't. But occasionally, proofs of correctness are important. The can demonstrate for example one of your programmers isn't ripping you off by rounding (0, 0.5) to zero instead of (0, 0.5] and stealing the resulting cents. People have gone to jail for doing exactly that. Which is why, a good auditor can get very picky finding a 1 cent error. He doesn't care about value of that 1c any more that you do. What he cares about greatly is a machine whose job is to add up numbers reliably apparently can't get basic arithmetic right.
Programmer with battle scars from working in that environment are sick and tired of being told by others how much easier floats are to use 99.9999% of the time. Believe me, they know.
There are more problems with using floating-point for exact monetary quantities than just the inexact representations of certain quantities which are exact in base 10. For example, integers have all of the following advantages over floats:
Integer arithmetic will never return NaN or infinity.
Integer (a*b)*c will always equal a*(b*c).
Integer (a+b)%n will always equal (a%n+b%n)%n, i.e. low-order bits are always preserved.
IEEE 754 is not bad and shouldn't be feared, but it is not a universal solution to every problem.
It's also not hard to multiply by fractions in fixed-point. You do a widening multiplication by the numerator followed by a narrowing division by the denominator. For percentages and interest rates etc., you can represent them using percentage points, basis points, or even parts-per-million depending on the precision you need.
>Integer arithmetic will never return NaN or infinity.
I use C++ and what integer arithmetic will do in situations where floating point returns NaN is undefined behavior.
I prefer the NaN over undefined behavior.
>Integer (ab)c will always equal a(bc).
In every situation where an integer will do that, a floating point will do that as well. Floating point numbers behave like integers for integer values, the only question is what do you do for non-integer values. My argument is that in many if not most cases you can apply the same solution you would have applied using integers to floating points and get an even more robust, flexible, and still high performance solution.
>For percentages and interest rates etc., you can represent them using percentage points, basis points, or even parts-per-million depending on the precision you need.
And this is precisely when people end up reimplementing their own ad-hoc floating point representation. You end up deciding and hardcoding what degree of precision you need to use depending on assumptions you make beforehand and having to switch between different fixed point representations and it just ends up being a matter of time before someone somewhere makes a mistake and mixes two close fixed point representations and ends up causing headaches.
With floating point values, I do hardcode a degree of precision I want to guarantee, which in my case is 6 decimal places, but in certain circumstances I might perform operations or work with data that needs more than 6 decimal places and using floating point values will still accommodate that to a very high degree whereas the fixed arithmetic solution will begin to fail catastrophically.
C++ is no excuse; it has value types and operator overloading. You can write your own types and define your own behavior, or use those already provided by others. Even if you insist on using raw ints (or just want a safety net), there's compiler flags to define that undefined behavior.
Putting everything into floats as integers defeats the purpose of using floats. Obviously you will want some fractions at some point and then you will have to deal with that issue, and the denominator of those fractions being a power of 2 and not a power of 10. Approximation is good enough for some things, but not others. Accounts and ledgers are definitely in the latter category, even if lots of other financial math isn't.
You need always be mindful of your operating precision and scale. Even double-precision floats have finite precision, though this won't be a huge issue until you've compounded the results of many operations. If you use fixed-point and have different denominators all over the place, then it's probably time to break out rational numbers or use the type system to your advantage. You will know the precision and scale of types called BasisPoints or PartsPerMillion or Fixed6 because it's in the name and is automatically handled as part of the operations between types.
>I use C++ and what integer arithmetic will do in situations where floating point returns NaN is undefined behavior. I prefer the NaN over undefined behavior.
Really? IME it's much more difficult to debug where a NaN value came from, since it's irreversible and infectious. And although the standard defines which integer operations should have undefined behavior, usually the compiler just generates code that behaves reasonably. Like, you can take INT_MAX and then increment and decrement it and get INT_MAX back.
(That does mean that you're left with a broken program that works by accident, but hey, the program works.)
Integer division by zero will raise an exception in most modern languages.
Integer overflow is more problematic. While some languages in some situations will raise exceptions, most don't. While it's easier to detect overflow that has already occurred with floats (though you'll usually have lost low-order bits long before you get infinity), it's easier to avoid overflow in the first place with integers.
It really depends on your need. In some countries e.g. VAT calculations used to specify rounding requirements that were a pain to guarantee with floats. I at one point had our CFO at the time breathing down my neck while I implemented the VAT calculations while clutching a printout of the relevant regulations on rounding because in theory he could end up a defendant in a court case if I got it wrong (in practice not so much, but it spooked him enough that it was the one time he paid attention to what I was up to). Many tax authorities are now more relaxed, as long as your results average out in their favour, but there's a reason for this advice.
> if your requirement is to be able to represent values exactly to the penny, then you can simply do the same thing but using a floating point to represent cents and represent $1.10 as the floating point 110.0.
Not if you need to represent more than about 170 kilo dollars.
The industry standard in finance is decimal floating point. C# for example has 'decimal', with 128 bits of precision.
On occasion I've seen people who didn't know any better use floats. One time I had to fix errors of single satoshis in a customer's database because their developer used 1.0 to represent 1 BTC.
I recall playing with FRACTINT, which was a fractal generator that existed before floating point coprocessors were common, that used fixed point math to calculate and display fractals. That was back when fractals were super cool and everyone wanted to be in the business of fractals, and all the Nobel Prizes were given out to fractal researchers.
Nothing to do with perf is a strong claim. If you genuinely don't care about performance you can use an arbitrary-precision rational number representation.
But performance often matters, so you trade off precision for performance. I think people are wrong to dismiss floating point numbers in favor of fixed point arithmetic, and I've seen plenty of fixed point arithmetic that has failed spectacularly because people think if you use it, it magically solves all your problems...
Whatever approach you take other than going all in with arbitrary precision fractions, you will need to have a good fundamental understanding of your representation and its trade-offs. For me personally I use floating point binary and adjust the decimal point so I can exactly represent any value to 6 decimal places. It's a good trade-off between performance, flexibility, and precision.
It's also what the main Bitcoin implementation uses.
Huh? Bitcoin uses integers. The maximum supply of BTC in satoshis fits in 64 bits. JS implementations that need to handle BTC amounts use doubles, but only by necessity, since JS doesn't have an integer type. They still use the units to represent satoshis, which works because the maximum supply also fits in 53 bits, so effectively they're also using integers.
Anyone who uses binary floating point operations on monetary values doesn't know what they're doing and is asking for trouble.
So if I want to price a barrier in Bermudan rainbow via Monte Carlo I should take the speed hit for a few oddball double rounding problems that are pennies?
I mean, you do you. People generally don't complain if you're a couple hundred nanoseconds (if that) late. They do complain if your accounts don't add up by a single penny.
Sure, FRACTINT is called FRACTINT because it uses fixed-point ("integer") math. And fixed-point math is still standard in Forth; you can do your example in GForth like this:
: organize; gforth
Gforth 0.7.3, Copyright (C) 1995-2008 Free Software Foundation, Inc.
Gforth comes with ABSOLUTELY NO WARRANTY; for details type `license'
Type `bye' to exit
: %* d>s 10 m*/ ; : %. <# # [char] . hold #s #> type ; ok
1.6 4.1 %* %. 6.5 ok
Note that the correct answer is 6.56, so the result 6.5 is incorrectly rounded. Here's how this works.
(If you're not familiar with Forth, Forth's syntax is that words are separated by spaces. "ok" is the prompt, ":" defines a subroutine terminated with ";", and you use RPN, passing parameters and receiving results on a stack.)
In standard Forth, putting a decimal point in a number makes it a double-precision number, occupying two cells on the stack, and in most Forths the number of digits after the decimal point is stored (until the next number) in the non-standardized variable dpl, decimal point location. Here I've just decided that all my numbers are going to have one decimal place. This means that after a multiplication I need to divide by 10, so I define a subroutine called %* to do this operation. (Addition and subtraction can use the standard d+ and d- subroutines; I didn't implement division, but it would need to pre-multiply the dividend by the scale factor 10.)
"%*" is defined in terms of the standard subroutine m*/, which multiplies a double-precision number by a single-precision number and divides the result by a divisor, and the standard subroutine d>s, which converts a double-precision number to a single-precision number. (There's probably a better way to do %*. I'm no Forth expert.)
I also need to define a way to print out such numbers, so I define a subroutine called "%.", using Forth's so-called "pictured numeric output", which prints out an unsigned double-precision number inserting a decimal point in the right place with "hold", after printing out the least significant digit. (In PNO we write the format backwards, starting from the least significant digit.) The call to "type" types out the formatted number from the hold space used by PNO.
Then I invoked %* on 1.6 and 4.1 and %. on its result, and it printed out 6.5 before giving me the "ok" prompt.
If you want to adapt this to use two decimal places:
: %* d>s 100 m*/ ; : %. <# # # [char] . hold #s #> type ; redefined %* redefined %. ok
1.60 4.10 %* %. 6.56 ok
Note, however, that a fixed-point multiplication still involves a multiplication, requiring potentially many additions, not just an addition. The paper, which I haven't read yet, is about how to approximate a floating-point multiplication by using an addition, presumably because in multiplication you add the mantissas, or maybe using a table of logarithms.
Forth's approach to decimal numbers was a clever hack for the 01970s and 01980s on sub-MIPS machines with 8-bit and 16-bit ALUs, where you didn't want to be invoking 32-bit arithmetic casually, and you didn't have floating-point hardware. Probably on 32-bit machines it was already the wrong approach (a double-precision number on a 32-bit Forth is 64 bits, which is about 19 decimal digits) and clearly it is on 64-bit machines, where you don't even get out of the first 64-bit word until that many digits:
0 1 %. 184467440737095516.16 ok
GForth and other modern standard Forths do support floating-point, but for backward compatibility, they treat input with decimal points as double-precision integers.
Chuck Moore of Forth fame demonstrated taking the value, say 1.6 multiplied by 4.1 and doing all the intermediate calculations via integers (16 * 41) and then formatting the output by putting the decimal point back in the "right place"; this worked as long as the range of floating point values was within a range that multiplying by 10 didn't exceed 65536 (16 bit integers), for instance. For embedded chips where for instance, you have an analog reading with 10 bits precision to quickly compute multiple times per second, this worked well.
I also recall talking many years ago with a Microsoft engineer who had worked with the Microsoft Streets and Trips program (https://archive.org/details/3135521376_qq_CD1 for a screenshot) and that they too had managed to fit what would normally be floating point numbers and the needed calculations into some kind of packed integer format with only the precision that was actually needed, that was faster on the CPUs of the day as well as more easily compressed to fit on the CDROM.