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957c61cbbf
This change upgrades to GCC 12.3 and GNU binutils 2.42. The GNU linker appears to have changed things so that only a single de-duplicated str table is present in the binary, and it gets placed wherever the linker wants, regardless of what the linker script says. To cope with that we need to stop using .ident to embed licenses. As such, this change does significant work to revamp how third party licenses are defined in the codebase, using `.section .notice,"aR",@progbits`. This new GCC 12.3 toolchain has support for GNU indirect functions. It lets us support __target_clones__ for the first time. This is used for optimizing the performance of libc string functions such as strlen and friends so far on x86, by ensuring AVX systems favor a second codepath that uses VEX encoding. It shaves some latency off certain operations. It's a useful feature to have for scientific computing for the reasons explained by the test/libcxx/openmp_test.cc example which compiles for fifteen different microarchitectures. Thanks to the upgrades, it's now also possible to use newer instruction sets, such as AVX512FP16, VNNI. Cosmo now uses the %gs register on x86 by default for TLS. Doing it is helpful for any program that links `cosmo_dlopen()`. Such programs had to recompile their binaries at startup to change the TLS instructions. That's not great, since it means every page in the executable needs to be faulted. The work of rewriting TLS-related x86 opcodes, is moved to fixupobj.com instead. This is great news for MacOS x86 users, since we previously needed to morph the binary every time for that platform but now that's no longer necessary. The only platforms where we need fixup of TLS x86 opcodes at runtime are now Windows, OpenBSD, and NetBSD. On Windows we morph TLS to point deeper into the TIB, based on a TlsAlloc assignment, and on OpenBSD/NetBSD we morph %gs back into %fs since the kernels do not allow us to specify a value for the %gs register. OpenBSD users are now required to use APE Loader to run Cosmo binaries and assimilation is no longer possible. OpenBSD kernel needs to change to allow programs to specify a value for the %gs register, or it needs to stop marking executable pages loaded by the kernel as mimmutable(). This release fixes __constructor__, .ctor, .init_array, and lastly the .preinit_array so they behave the exact same way as glibc. We no longer use hex constants to define math.h symbols like M_PI.
421 lines
16 KiB
C
421 lines
16 KiB
C
/*-*- mode:c;indent-tabs-mode:nil;c-basic-offset:4;tab-width:8;coding:utf-8 -*-│
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│ vi: set et ft=c ts=4 sts=4 sw=4 fenc=utf-8 :vi │
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╞══════════════════════════════════════════════════════════════════════════════╡
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│ Copyright (c) 2008-2016 Stefan Krah. All rights reserved. │
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│ │
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│ Redistribution and use in source and binary forms, with or without │
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│ modification, are permitted provided that the following conditions │
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│ are met: │
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│ │
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│ 1. Redistributions of source code must retain the above copyright │
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│ notice, this list of conditions and the following disclaimer. │
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│ │
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│ 2. Redistributions in binary form must reproduce the above copyright │
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│ notice, this list of conditions and the following disclaimer in │
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│ the documentation and/or other materials provided with the │
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│ distribution. │
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│ │
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│ THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS "AS IS" AND │
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│ ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE │
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│ IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR │
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│ PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS │
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│ BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, │
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│ OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT │
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│ OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR │
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│ BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, │
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│ WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE │
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│ OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, │
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│ EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. │
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╚─────────────────────────────────────────────────────────────────────────────*/
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#include "third_party/python/Modules/_decimal/libmpdec/fourstep.h"
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#include "third_party/python/Modules/_decimal/libmpdec/mpdecimal.h"
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#include "third_party/python/Modules/_decimal/libmpdec/numbertheory.h"
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#include "third_party/python/Modules/_decimal/libmpdec/sixstep.h"
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#include "third_party/python/Modules/_decimal/libmpdec/transpose.h"
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#include "third_party/python/Modules/_decimal/libmpdec/umodarith.h"
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__static_yoink("libmpdec_notice");
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/*
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Cache Efficient Matrix Fourier Transform
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for arrays of form 3×2ⁿ
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The Matrix Fourier Transform
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════════════════════════════
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In libmpdec, the Matrix Fourier Transform [1] is called four-step
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transform after a variant that appears in [2]. The algorithm requires
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that the input array can be viewed as an R*C matrix.
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All operations are done modulo p. For readability, the proofs drop all
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instances of (mod p).
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Algorithm four-step (forward transform)
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───────────────────────────────────────
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a := input array
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d := len(a) = R * C
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p := prime
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w := primitive root of unity of the prime field
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r := w**((p-1)/d)
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A := output array
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1) Apply a length R FNT to each column.
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2) Multiply each matrix element (addressed by j*C+m) by r**(j*m).
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3) Apply a length C FNT to each row.
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4) Transpose the matrix.
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Proof (forward transform)
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─────────────────────────
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The algorithm can be derived starting from the regular definition of
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the finite-field transform of length d:
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d-1
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,────
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\
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A[k] = | a[l] × r**(k × l)
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/
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`────
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l = 0
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The sum can be rearranged into the sum of the sums of columns:
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C-1 R-1
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,──── ,────
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\ \
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= | | a[i × C + j] × r**(k × (i × C + j))
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/ /
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`──── `────
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j = 0 i = 0
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Extracting a constant from the inner sum:
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C-1 R-1
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,──── ,────
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\ \
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= | rᵏ×j × | a[i × C + j] × r**(k × i × C)
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/ /
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`──── `────
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j = 0 i = 0
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Without any loss of generality, let k = n × R + m,
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where n < C and m < R:
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C-1 R-1
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,──── ,────
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\ \
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A[n×R+m] = | r**(R×n×j) × r**(m×j) × | a[i×C+j] × r**(R×C×n×i) × r**(C×m×i)
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/ /
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`──── `────
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j = 0 i = 0
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Since r = w ** ((p-1) / (R×C)):
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a) r**(R×C×n×i) = w**((p-1)×n×i) = 1
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b) r**(C×m×i) = w**((p-1) / R) ** (m×i) = r_R ** (m×i)
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c) r**(R×n×j) = w**((p-1) / C) ** (n×j) = r_C ** (n×j)
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r_R := root of the subfield of length R.
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r_C := root of the subfield of length C.
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C-1 R-1
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,──── ,────
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\ \
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A[n×R+m] = | r_C**(n×j) × [ r**(m×j) × | a[i×C+j] × r_R**(m×i) ]
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/ ^ /
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`──── | `──── 1) transform the columns
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j = 0 | i = 0
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^ |
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| `-- 2) multiply
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`-- 3) transform the rows
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Note that the entire RHS is a function of n and m and that the results
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for each pair (n, m) are stored in Fortran order.
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Let the term in square brackets be 𝑓(m, j). Step 1) and 2) precalculate
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the term for all (m, j). After that, the original matrix is now a lookup
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table with the mth element in the jth column at location m × C + j.
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Let the complete RHS be g(m, n). Step 3) does an in-place transform of
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length n on all rows. After that, the original matrix is now a lookup
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table with the mth element in the nth column at location m × C + n.
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But each (m, n) pair should be written to location n × R + m. Therefore,
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step 4) transposes the result of step 3).
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Algorithm four-step (inverse transform)
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───────────────────────────────────────
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A := input array
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d := len(A) = R × C
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p := prime
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d′ := d⁽ᵖ⁻²⁾ # inverse of d
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w := primitive root of unity of the prime field
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r := w**((p-1)/d) # root of the subfield
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r′ := w**((p-1) - (p-1)/d) # inverse of r
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a := output array
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0) View the matrix as a C×R matrix.
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1) Transpose the matrix, producing an R×C matrix.
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2) Apply a length C FNT to each row.
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3) Multiply each matrix element (addressed by i×C+n) by r**(i×n).
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4) Apply a length R FNT to each column.
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Proof (inverse transform)
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─────────────────────────
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The algorithm can be derived starting from the regular definition of
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the finite-field inverse transform of length d:
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d-1
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,────
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\
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a[k] = d′ × | A[l] × r′ ** (k × l)
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/
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`────
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l = 0
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The sum can be rearranged into the sum of the sums of columns. Note
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that at this stage we still have a C*R matrix, so C denotes the number
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of rows:
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R-1 C-1
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,──── ,────
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\ \
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= d′ × | | a[j × R + i] × r′ ** (k × (j × R + i))
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/ /
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`──── `────
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i = 0 j = 0
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Extracting a constant from the inner sum:
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R-1 C-1
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,──── ,────
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\ \
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= d′ × | r′ ** (k×i) × | a[j × R + i] × r′ ** (k × j × R)
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/ /
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`──── `────
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i = 0 j = 0
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Without any loss of generality, let k = m * C + n,
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where m < R and n < C:
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R-1 C-1
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,──── ,────
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\ \
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A[m×C+n] = d′ × | r′ ** (C×m×i) × r′ ** (n×i) × | a[j×R+i] × r′ ** (R×C×m×j) × r′ ** (R×n×j)
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/ /
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`──── `────
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i = 0 j = 0
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Since r′ = w**((p-1) - (p-1)/d) and d = R×C:
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a) r′ ** (R×C×m×j) = w**((p-1)×R×C×m×j - (p-1)×m×j) = 1
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b) r′ ** (C×m×i) = w**((p-1)×C - (p-1)/R) ** (m×i) = r_R′ ** (m×i)
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c) r′ ** (R×n×j) = r_C′ ** (n×j)
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d) d′ = d⁽ᵖ⁻²⁾ = (R×C)⁽ᵖ⁻²⁾ = R⁽ᵖ⁻²⁾ × C⁽ᵖ⁻²⁾ = R′ × C′
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r_R′ := inverse of the root of the subfield of length R.
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r_C′ := inverse of the root of the subfield of length C.
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R′ := inverse of R
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C′ := inverse of C
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R-1 C-1
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,──── ,──── 2) transform the rows of a^T
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\ \
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A[m×C+n] = R′ × | r_R′ ** (m×i) × [ r′ ** (n×i) × C′ × | a[j×R+i] × r_C′ ** (n×j) ]
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/ ^ / ^
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`──── | `──── |
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i = 0 | j = 0 |
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^ | `── 1) Transpose input matrix
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| `── 3) multiply to address elements by
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| i × C + j
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`── 3) transform the columns
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Note that the entire RHS is a function of m and n and that the results
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for each pair (m, n) are stored in C order.
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Let the term in square brackets be 𝑓(n, i). Without step 1), the sum
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would perform a length C transform on the columns of the input matrix.
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This is a) inefficient and b) the results are needed in C order, so
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step 1) exchanges rows and columns.
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Step 2) and 3) precalculate 𝑓(n, i) for all (n, i). After that, the
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original matrix is now a lookup table with the ith element in the nth
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column at location i × C + n.
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Let the complete RHS be g(m, n). Step 4) does an in-place transform of
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length m on all columns. After that, the original matrix is now a lookup
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table with the mth element in the nth column at location m × C + n,
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which means that all A[k] = A[m × C + n] are in the correct order.
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──
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[1] Joerg Arndt: "Matters Computational"
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http://www.jjj.de/fxt/
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[2] David H. Bailey: FFTs in External or Hierarchical Memory
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http://crd.lbl.gov/~dhbailey/dhbpapers/
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*/
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static inline void
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std_size3_ntt(mpd_uint_t *x1, mpd_uint_t *x2, mpd_uint_t *x3,
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mpd_uint_t w3table[3], mpd_uint_t umod)
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{
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mpd_uint_t r1, r2;
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mpd_uint_t w;
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mpd_uint_t s, tmp;
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/* k = 0 -> w = 1 */
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s = *x1;
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s = addmod(s, *x2, umod);
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s = addmod(s, *x3, umod);
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r1 = s;
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/* k = 1 */
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s = *x1;
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w = w3table[1];
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tmp = MULMOD(*x2, w);
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s = addmod(s, tmp, umod);
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w = w3table[2];
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tmp = MULMOD(*x3, w);
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s = addmod(s, tmp, umod);
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r2 = s;
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/* k = 2 */
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s = *x1;
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w = w3table[2];
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tmp = MULMOD(*x2, w);
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s = addmod(s, tmp, umod);
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w = w3table[1];
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tmp = MULMOD(*x3, w);
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s = addmod(s, tmp, umod);
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*x3 = s;
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*x2 = r2;
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*x1 = r1;
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}
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/* forward transform, sign = -1; transform length = 3 * 2**n */
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int
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four_step_fnt(mpd_uint_t *a, mpd_size_t n, int modnum)
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{
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mpd_size_t R = 3; /* number of rows */
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mpd_size_t C = n / 3; /* number of columns */
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mpd_uint_t w3table[3];
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mpd_uint_t kernel, w0, w1, wstep;
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mpd_uint_t *s, *p0, *p1, *p2;
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mpd_uint_t umod;
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mpd_size_t i, k;
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assert(n >= 48);
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assert(n <= 3*MPD_MAXTRANSFORM_2N);
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/* Length R transform on the columns. */
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SETMODULUS(modnum);
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_mpd_init_w3table(w3table, -1, modnum);
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for (p0=a, p1=p0+C, p2=p0+2*C; p0<a+C; p0++,p1++,p2++) {
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SIZE3_NTT(p0, p1, p2, w3table);
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}
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/* Multiply each matrix element (addressed by i*C+k) by r**(i*k). */
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kernel = _mpd_getkernel(n, -1, modnum);
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for (i = 1; i < R; i++) {
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w0 = 1; /* r**(i*0): initial value for k=0 */
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w1 = POWMOD(kernel, i); /* r**(i*1): initial value for k=1 */
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wstep = MULMOD(w1, w1); /* r**(2*i) */
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for (k = 0; k < C-1; k += 2) {
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mpd_uint_t x0 = a[i*C+k];
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mpd_uint_t x1 = a[i*C+k+1];
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MULMOD2(&x0, w0, &x1, w1);
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MULMOD2C(&w0, &w1, wstep); /* r**(i*(k+2)) = r**(i*k) * r**(2*i) */
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a[i*C+k] = x0;
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a[i*C+k+1] = x1;
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}
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}
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/* Length C transform on the rows. */
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for (s = a; s < a+n; s += C) {
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if (!six_step_fnt(s, C, modnum)) {
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return 0;
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}
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}
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#if 0
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/* An unordered transform is sufficient for convolution. */
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/* Transpose the matrix. */
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transpose_3xpow2(a, R, C);
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#endif
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return 1;
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}
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/* backward transform, sign = 1; transform length = 3 * 2**n */
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int
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inv_four_step_fnt(mpd_uint_t *a, mpd_size_t n, int modnum)
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{
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mpd_size_t R = 3; /* number of rows */
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mpd_size_t C = n / 3; /* number of columns */
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mpd_uint_t w3table[3];
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mpd_uint_t kernel, w0, w1, wstep;
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mpd_uint_t *s, *p0, *p1, *p2;
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mpd_uint_t umod;
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mpd_size_t i, k;
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assert(n >= 48);
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assert(n <= 3*MPD_MAXTRANSFORM_2N);
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#if 0
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/* An unordered transform is sufficient for convolution. */
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/* Transpose the matrix, producing an R*C matrix. */
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transpose_3xpow2(a, C, R);
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#endif
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/* Length C transform on the rows. */
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for (s = a; s < a+n; s += C) {
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if (!inv_six_step_fnt(s, C, modnum)) {
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return 0;
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}
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}
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/* Multiply each matrix element (addressed by i*C+k) by r**(i*k). */
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SETMODULUS(modnum);
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kernel = _mpd_getkernel(n, 1, modnum);
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for (i = 1; i < R; i++) {
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w0 = 1;
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w1 = POWMOD(kernel, i);
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wstep = MULMOD(w1, w1);
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for (k = 0; k < C; k += 2) {
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mpd_uint_t x0 = a[i*C+k];
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mpd_uint_t x1 = a[i*C+k+1];
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MULMOD2(&x0, w0, &x1, w1);
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MULMOD2C(&w0, &w1, wstep);
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a[i*C+k] = x0;
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a[i*C+k+1] = x1;
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}
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}
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/* Length R transform on the columns. */
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_mpd_init_w3table(w3table, 1, modnum);
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for (p0=a, p1=p0+C, p2=p0+2*C; p0<a+C; p0++,p1++,p2++) {
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SIZE3_NTT(p0, p1, p2, w3table);
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}
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return 1;
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}
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