ISO/IEC JTC1 SC22 WG21 P0952R0

Date: 2018-02-12

To: LWG, SG6 (Numerics)

Thomas Köppe <tkoeppe@google.com>Davis Herring <herring@lanl.gov>

`std::generate_canonical`

This paper proposes a new specification for the function template
`generate_canonical`

[rand.util.canonical, 29.6.7.2]. The outcome is preserved
(namely a random number in the interval [0, 1)), but the algorithm and the constraints are
changed to ensure the desired statistical properties. The new specification
obsoletes LWG 2524.

- Problem statement
- Drive-by fixes
- DO NOT SUBMIT: Proposed naive wording
- Statistical flaws in the current specification
- A new specification
- Impact on the Standard
- Questions for LWG and SG6
- Proposed wording (not included)

The specification of `generate_canonical`

in C++17 and in the current working
paper (N4713) is effectively unimplementable since it is over-constrained. To be more
precise, it is *wrongly* constrained in terms of purely mathematical expressions
that ignore the reality of floating point rounding on real implementations. We will point
out two problems with the current specification. The first one is immediate, the second is
somewhat more subtle and will be discussed later in this paper. First, consider the the
following three current requirements.

- The result must lie in [0, 1).
- The algorithm is specified exactly and the underlying URBG must be invoked a specific, fixed number of times for a given set of parameters.
- The results must be uniformly distributed.

The immediate problem is that this is unimplementable for the following reasons: The
algorithm is currently specified exactly as a particular computation, which results in a
fraction
*S*/*R*^{k} that is mathematically guaranteed to be less
than 1. However, the value may be arbitrarily close to 1, and when expressed as a value of
a bounded-precision type in C++, the result may actually be exactly 1 due to rounding.
(This causes real bugs, e.g. when a computation divides by `(1 - x)`

, where
`x`

was obtained from `generate_canonical`

.) If we accept (2), the
algorithm as written, the rounding violates constraint (1). If we modify the result when
the algorithm results in 1, we violate (3), uniformity. If we want to preserve (1) and (3),
we need to rerun the algorithm in the case where it results in 1, which violates (2), the
precise computational prescription.

LWG decided at the 2017 Albuquerque meeting that the best solution is to change the
specification to rerun the algorithm until the result is not equal to 1. This means that
the complexity of the algorithm can no longer be stated precisely, but only in
expectation. If one round of the algorithm invokes the URBG *k* times, and the
result is 1 with probability (1 − p), then the expected number of
invocations of the URBG is now *k*/*p*. (In practice, *p* will be
very close to (and less than) 1.) This paper attempts to improve on that decision by
also addressing some

The original expression for *b* assumes that the radix is 2 when using
`numeric_limits<RealType>::digits`

as a bit count. We propose
to use the more general expression log_{2}(`numeric_limits<RealType>::radix`

)
× `numeric_limits<RealType>::digits`

.

If we simplify modified the specification to rerun the algorithm until it results in a value less than 1, we could use the following wording. However, this wording fails to address the deeper statistical problems that we will discuss below, so we do not want to keep the change as small as this. This section is included solely for historical interest because this wording has been discussed on the reflector before.

Modify [rand.util.canonical, 29.6.7.2] paragraphs 3 and 4 as follows.

template<class RealType, size_t bits, class URBG>
RealType generate_canonical(URBG& g);

3. *Complexity:* For each attempt (see below), exactly~~Exactly~~
*k* = max(1, ⌈*b* / log_{2}*R*⌉)
invocations of `g`

, where *b* is the lesser of
`numeric_limits<RealType>::digits`

and `bits`

and
log_{2}(`numeric_limits<RealType>::radix`

)
× `numeric_limits<RealType>::digits`

, and *R* is the value of
`g.max()`

− `g.min()`

+ 1.

4. *Effects:* For each attempt, invokes~~Invokes~~ `g()`

*k* times to obtain values *g*_{0}, …, *g*_{k−1},
respectively and calculates~~. Calculates~~ a quantity [*S* = …]
using arithmetic of type `RealType`

.
Attempts are repeated as long as the quantity *S*/*R*^{k}
has the value `1.0`

when expressed as type `RealType`

.

5. *Returns:* *S*/*R*^{k}.

6. *Throws:* What and when `g`

throws.

The currently specified algorithm does not always result in uniform output due to rounding. To see this, consider first a simple lemma.

**Claim.** *The restriction of a uniform distribution on a finite
set to a subset is uniform.*∎

We use this to look for statistical properties of `generate_canonical`

. If
the function generates uniformly distributed floats in the range [0, 1), then by throwing
away all numbers less than 0.5, we retain a uniform distribution on the set [0.5, 1). In
the popular IEEE-754 floating-point representation, numbers in this range have a fixed
exponent (of effective value −1), and so we get a uniform distribution of mantissas,
and thus each bit of the mantissa (or perhaps of a leading subset of significant bits, when
the algorithm is used with low precision) is independently uniformly distributed.
This is a property we can look for. More generally, similar statements should hold for restrictions
to intervals of the form [2^{−n}, 2^{−n + 1}).

The problem in the current specification comes from the use of division combined with
floating point rounding. Whenever a larger range is used to derive a smaller range via
division (rather than just discarding bits, when that is an option), the floating point
rounding behaviour affects the results. To illustrate, consider the sequence 0.0, 0.5,
1.0, 1.5, 2.0, 2.5, 3.0, 3.5. If we *discard* the least significant bit, we obtain 0, 0, 1,
1, 2, 2, 3, 3, which is uniform. But if we employ the popular to-nearest-even rounding, we
obtain 0, 0, 1, 2, 2, 2, 3, 4, which biases even to odd numbers at a ratio of 3-to-1.
(This also shows another bias, namely that 0 only gets hit twice.)

To make this problem concrete, consider a typical 32-bit float with 24 mantissa digits.
For an extreme case, if the URBG returns 25 bits, then in the restricted range [0.5, 1.0)
the least significant bit results from rounding away the last digit of *g*_{0}
in just the same way as in the previous example, which leads to a 3-to-1 bias of zeros over ones
in the last bit. A 26-bit URBG would have to strip two bits in the range [0.5, 1.0) (leading to
a 5-to-3 bias), but only one bit in the range [0.25, 0.5), and so on. With the typical 32-bit URBG,
the bias in the last bit shows up when restricting to the range [2^{−8}, 2^{−7}) (or
`[0x1p-8, 0x1p-7)`

in code).

Other rounding modes may be fairer when rounding is needed for random number generation. However, the rounding mode is not in scope of the specification and thus not under our control. More importantly, the algorithm we will propose will use discard-and-retry rather than division, and thus avoid any rounding by design.

The problem with rounding to nearest-even was also discovered in
the
Java library’s `nextDouble`

function, which originally used a
non-trivial division and would thus experience a rounding-induced bias in the lowest bit.
(In the corrected, present version, the division by `1L << 53`

is just a
trivial exponent adjustment.)

We propose a new specification that provably results in uniform output, does not suffer from rounding problems, and is independent of the radix of the floating point implementation. The algorithm does not use non-trivial division to produce a limited output range. Rather, it will discard any result that falls outside the desired range and retry, so that the final division does not round. This ensures uniformity and avoids any dependency on floating-point rounding behaviour.

The set possible outcomes is not required to contain *every* representable
value of `RealType`

in the interval [0, 1). To do so would typically require
an exponential amount of retries, as smaller results would need to be increasingly
unlikely. Instead, the resulting values will be precisely the values 2^{−M}{0,
…, 2^{M} − 1}, where *M* is the value
*b*_{0} defined below. For example, for `bits = 2`

, the outcome is
(uniformly) 0, 0.25, 0.5 and 0.75. Note also that the mean of the resulting distribution
is smaller (by 2^{− M − 1}) than the ideal mean 0.5.

The proposed algorithm is as follows.

- Let
*r*be`numeric_limits<RealType>::radix`

, let*d*be`numeric_limits<RealType>::digits`

, and let*R*be`g.max()`

−`g.min()`

+ 1. - Let
*b*be the smallest integer such that*r*^{b}≥ 2^{bits}, but not larger than*d*. - Let
*k*be the smallest integer such that*R*^{k}≥*r*^{b}. - Let
*x*be the largest integer such that*x**r*^{b}≤*R*^{k}. Note that*x*need not be a power or*r*.

Now compute *S* = ∑_{i}
*g*_{i} *R*^{i}
in unbounded precision.
Whenever *S* ≥ *x* *r*^{b},
discard the result and retry. The return value is
*S* / (*x* *r*^{b}),
which can be computed without rounding, since *b* ≤ *d*.

For the edge case `bits = 0`

, we should always return 0 and not call the URBG;
the results have zero entropy.

Note that on the most common platforms *r* = 2
and *R* = 2^{n}
is a power of 2, so that the definitions simplify:
*b* = min(`bits`

, *d*),
*k* = ⌈*b* / n⌉,
and *x* = 2^{nk − b}.
The final return expression is *S* / 2^{nk} ≡ *S* / *R*^{k}.
(That expression is the same as in the current specification, but *S* is chosen more carefully.)

It is easy to see that this algorithm produces uniform outputs: The value
*S* < *x* *r*^{b} is obtained by restricting
the URBG to the range [0, *x* *r*^{b}). Since *b* does not exceed
the precision of `RealType`

, the final division
*S*’ / (*x* *r*^{b}) does not round and is representable
as a value of `RealType`

strictly less than 1.

This proposal changes the side effects, computational complexity and specific algorithmic
details of the library facility `std::generate_canonical`

. In particular, code
that depends on a specific sequence of results from repeated invocations, or on a particular
number of calls to the URBG argument, will be broken.

We would like clarification on the following details.

- Is the design intention that
`std::generate_canonical<RealType, bits>`

picks an integer uniformly from [0, 2^{M}) and returns the value divided by 2^{M}? This is in contrast to possible alternative interpretations such as “pick a uniform real number (mathematically) from [0, 1) and return its rounded`RealType`

representation”. The latter suffers from the round-to-1.0 bug, of course, but it could be amended to “round-down” semantics to avoid this. However, the requirement of a mathematically uniform real number requires highly variable (and potentially large) number of URBG invocations, so we believe that this is not the design intent. (This has been the subject of some reflector discussion.) - How much do we want to specify the algorithm? The above description is fairly detailed
and prescriptive, which we intend to result in identical results on conforming implementations.
However, for certain parameters our prescription requires more retries than strictly necessary.
An alternative would be to give implementations more freedom to use more efficient sampling
strategies; or we could consider
*mandating*more efficient sampling, at the cost of making the specification more complex.

To be supplied in a future revision.