[swift-evolution] Pre-proposal: Safer Decimal Calculations

[Michael Gottesman]

> On Mar 18, 2016, at 3:42 PM, Rainer Brockerhoff via swift-evolution <swift-evolution at swift.org> wrote:
> 
> First draft towards a tentative pre-proposal:
> https://gist.github.com/rbrockerhoff/6874a5698bb479886e83
> ------
> 
> Pre-proposal: Safer Decimal Calculations
> Proposal: TBD
> Author(s): Rainer Brockerhoff
> Status: TBD
> Review manager: TBD
> 
> Quoting the “The Swift Programming Language” book: “Swift adopts safe
> programming patterns…”; “Swift is friendly to new programmers”. The
> words “safe” and “safety” are found many times in the book and in online
> documentation. The usual rationale for safe features is, to quote a
> typical sentence, “…enables you to catch and fix errors as early as
> possible in the development process”.
> 
> One frequent stumbling point for both new and experienced programmers
> stems from the vagaries of binary floating-point arithmetic. This
> tentative pre-proposal suggests one possible way to make the dangers
> somewhat more clear.
> 
> My intention here is to start a discussion on this to inform the ongoing
> (and future) reasoning on extending and regularising arithmetic in Swift.
> 
> Motivation
> 
> Floating-point hardware on most platforms that run Swift — that is,
> Intel and ARM CPUs — uses the binary representation forms of the IEEE
> 754-2008 standard. Although some few mainframes and software libraries
> implement the decimal representations this is not currently leveraged by
> Swift. Apple's NSDecimal and NSDecimalNumber implementation is awkward
> to use in Swift, especially as standard arithmetic operators cannot be
> used directly.
> 
> Although it is possible to express floating-point constants in
> hexadecimal (0x123.AB) with an optional binary exponent (0x123A.Bp-4),
> decimal-form floating-point constants (123.45 or 1.2345e2) are extremely
> common in practice.
> 
> Unfortunately it is tempting to use floating-point arithmetic for
> financial calculations or other purposes such as labelling graphical or
> statistical data. Constants such as 0.1, 0.01, 0.001 and variations or
> multiples thereof will certainly be used in such applications — and
> almost none of these constant can be precisely represented in binary
> floating-point format.
> 
> Rounding errors will therefore be introduced at the outset, causing
> unexpected or outright buggy behaviour down the line which will be
> surprising to the user and/or the programmer. This will often happen at
> some point when the results of a calculation are compared to a constant
> or to another result.
> 
> Current Solution
> 
> As things stand, Swift's default print() function, Xcode playgrounds
> etc. do some discreet rounding or truncation to make the problem less
> apparent - a Double initialized with the literal 0.1 prints out as 0.1
> instead of the exact value of the internal representation, something
> like 0.100000000000000005551115123125782702118158340454101562.
> 
> This, unfortunately, masks this underlying problem in settings such as
> “toy” programs or educational playgrounds, leading programmers to be
> surprised later when things won't work. A cursory search on
> StackOverflow reveals tens of thousands of questions with headings like
> “Is floating point math broken?".
> 
> Warning on imprecise literals
> 
> To make decimal-format floating-point literals safe, I suggest that the
> compiler should emit a warning whenever a literal is used that cannot be
> safely represented as an exact value of the type expected. (Note that
> 0.1 cannot be represented exactly as any binary floating-point type.)
> 
> The experienced programmer will, however, be willing to accept some
> imprecision under circumstances that cannot be reliably determined by
> the compiler. I suggest, therefore, that this acceptance be indicated by
> an annotation to the literal; a form such as ~0.1 might be easiest to
> read and implement, as the prefix ~ operator currently has no meaning
> for a floating-point value. A “fixit” would be easily implemented to
> insert the missing notation.
> 
> Conversely, to avoid inexperienced or hurried programmers to strew ~s
> everywhere, it would be useful to warn, and offer to fix, if the ~ is
> present but the literal does have an exact representation.
> 
> Tolerances
> 
> A parallel idea is that of tolerances, introducing an ‘epsilon’ value to
> be used in comparisons. Unfortunately an effective value of the epsilon
> depends on the magnitude of the operands and there are many edge cases.
> 
> Introducing a special type along the lines of “floating point with
> tolerances” — using some accepted engineering notation for literals like
> 100.5±0.1 — might be useful for specialised applications but will not
> solve this specific problem. Expanding existing constructs to accept an
> optional tolerance value, as has been proposed elsewhere, may be useful
> in those specific instances but not contribute to raise programmer
> awareness of unsafe literals.
> 
> Full Decimal type proposal
> 
> There are cogent arguments that prior art/habits and the already complex
> interactions between Double, Float, Float80 and CGFloat are best left alone.
> 
> However, there remains a need for a precise implementation of a workable
> Decimal value type for financial calculations. IMHO repurposing the
> existing NSDecimalNumber from Objective-C is not the best solution.
> 
> As most experienced developers know, the standard solution for financial
> calculations is to internally store fixed-point values — usually but not
> always in cents — and then print the “virtual” point (or decimal comma,
> for the rest of us) on output.
> 
> I propose, therefore, an internal data layout like this:
> 
> UInt16 - position of the “virtual” point, starting at 0
> UInt16 - data array size - 1
> [Int32] - contiguous data array, little-endian order, grown as needed.
> Note that both UInt16 fields being zero implies that the number is
> reduced to a 32-bit Integer. Number literals in Swift can be up to 2048
> bits in size, so the maximum data array size would be 64, although it
> could conceivably grow beyond that. The usual cases of the virtual point
> position being 0 or 2 could be aggressively optimized for normal
> arithmetic operators.
> 
> Needless to say such a Decimal number would accept and represent
> literals such as 0.01 with no problems. It would also serve as a BigNum
> implementation for most purposes.
> 
> No doubt implementing this type in the standard library would allow for
> highly optimized implementations for all major CPU platforms. In
> particular, the data array should probably be [Int64] for 64-bit platforms.
> 
> Acknowledgement
> 
> Thanks to Erica Sadun for their help with an early version of this
> pre-proposal.
> 
> Some references
> 
> http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html
> https://docs.python.org/2/tutorial/floatingpoint.html
> https://en.wikipedia.org/wiki/IEEE_floating_point
> https://randomascii.wordpress.com/category/floating-point/
> http://code.jsoftware.com/wiki/Essays/Tolerant_Comparison

Rainer: I quickly skimmed this. Just to make sure I am understanding 100%: you are proposing a fixed point decimal calculation or a floating point decimal calculation. The former, no?

> 
> -- 
> Rainer Brockerhoff  <rainer at brockerhoff.net>
> Belo Horizonte, Brazil
> "In the affairs of others even fools are wise
> In their own business even sages err."
> http://brockerhoff.net/blog/
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