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/* -*- Mode: C++; tab-width: 8; indent-tabs-mode: nil; c-basic-offset: 2 -*-
* vim: set ts=8 sts=2 et sw=2 tw=80:
* This Source Code Form is subject to the terms of the Mozilla Public
* License, v. 2.0. If a copy of the MPL was not distributed with this
#include "jit/ReciprocalMulConstants.h"
#include "mozilla/Assertions.h"
using namespace js::jit;
ReciprocalMulConstants ReciprocalMulConstants::computeDivisionConstants(
uint32_t d, int maxLog) {
MOZ_ASSERT(maxLog >= 2 && maxLog <= 32);
// In what follows, 0 < d < 2^maxLog and d is not a power of 2.
MOZ_ASSERT(d < (uint64_t(1) << maxLog) && (d & (d - 1)) != 0);
// Speeding up division by non power-of-2 constants is possible by
// calculating, during compilation, a value M such that high-order
// bits of M*n correspond to the result of the division of n by d.
// No value of M can serve this purpose for arbitrarily big values
// of n but, for optimizing integer division, we're just concerned
// with values of n whose absolute value is bounded (by fitting in
// an integer type, say). With this in mind, we'll find a constant
// M as above that works for -2^maxLog <= n < 2^maxLog; maxLog can
// then be 31 for signed division or 32 for unsigned division.
//
// The original presentation of this technique appears in Hacker's
// Delight, a book by Henry S. Warren, Jr.. A proof of correctness
// for our version follows; we'll denote maxLog by L in the proof,
// for conciseness.
//
// Formally, for |d| < 2^L, we'll compute two magic values M and s
// in the ranges 0 <= M < 2^(L+1) and 0 <= s <= L such that
// (M * n) >> (32 + s) = floor(n/d) if 0 <= n < 2^L
// (M * n) >> (32 + s) = ceil(n/d) - 1 if -2^L <= n < 0.
//
// Define p = 32 + s, M = ceil(2^p/d), and assume that s satisfies
// M - 2^p/d <= 2^(p-L)/d. (1)
// (Observe that p = CeilLog32(d) + L satisfies this, as the right
// side of (1) is at least one in this case). Then,
//
// a) If p <= CeilLog32(d) + L, then M < 2^(L+1) - 1.
// Proof: Indeed, M is monotone in p and, for p equal to the above
// value, the bounds 2^L > d >= 2^(p-L-1) + 1 readily imply that
// 2^p / d < 2^p/(d - 1) * (d - 1)/d
// <= 2^(L+1) * (1 - 1/d) < 2^(L+1) - 2.
// The claim follows by applying the ceiling function.
//
// b) For any 0 <= n < 2^L, floor(Mn/2^p) = floor(n/d).
// Proof: Put x = floor(Mn/2^p); it's the unique integer for which
// Mn/2^p - 1 < x <= Mn/2^p. (2)
// Using M >= 2^p/d on the LHS and (1) on the RHS, we get
// n/d - 1 < x <= n/d + n/(2^L d) < n/d + 1/d.
// Since x is an integer, it's not in the interval (n/d, (n+1)/d),
// and so n/d - 1 < x <= n/d, which implies x = floor(n/d).
//
// c) For any -2^L <= n < 0, floor(Mn/2^p) + 1 = ceil(n/d).
// Proof: The proof is similar. Equation (2) holds as above. Using
// M > 2^p/d (d isn't a power of 2) on the RHS and (1) on the LHS,
// n/d + n/(2^L d) - 1 < x < n/d.
// Using n >= -2^L and summing 1,
// n/d - 1/d < x + 1 < n/d + 1.
// Since x + 1 is an integer, this implies n/d <= x + 1 < n/d + 1.
// In other words, x + 1 = ceil(n/d).
//
// Condition (1) isn't necessary for the existence of M and s with
// the properties above. Hacker's Delight provides a slightly less
// restrictive condition when d >= 196611, at the cost of a 3-page
// proof of correctness, for the case L = 31.
//
// Note that, since d*M - 2^p = d - (2^p)%d, (1) can be written as
// 2^(p-L) >= d - (2^p)%d.
// In order to avoid overflow in the (2^p) % d calculation, we can
// compute it as (2^p-1) % d + 1, where 2^p-1 can then be computed
// without overflow as UINT64_MAX >> (64-p).
// We now compute the least p >= 32 with the property above...
int32_t p = 32;
while ((uint64_t(1) << (p - maxLog)) + (UINT64_MAX >> (64 - p)) % d + 1 < d) {
p++;
}
// ...and the corresponding M. For either the signed (L=31) or the
// unsigned (L=32) case, this value can be too large (cf. item a).
// Codegen can still multiply by M by multiplying by (M - 2^L) and
// adjusting the value afterwards, if this is the case.
ReciprocalMulConstants rmc;
rmc.multiplier = (UINT64_MAX >> (64 - p)) / d + 1;
rmc.shiftAmount = p - 32;
return rmc;
}