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Make numerous improvements

Browse source 
- Python static hello world now 1.8mb
- Python static fully loaded now 10mb
- Python HTTPS client now uses MbedTLS
- Python REPL now completes import stmts
- Increase stack size for Python for now
- Begin synthesizing posixpath and ntpath
- Restore Python \N{UNICODE NAME} support
- Restore Python NFKD symbol normalization
- Add optimized code path for Intel SHA-NI
- Get more Python unit tests passing faster
- Get Python help() pagination working on NT
- Python hashlib now supports MbedTLS PBKDF2
- Make memcpy/memmove/memcmp/bcmp/etc. faster
- Add Mersenne Twister and Vigna to LIBC_RAND
- Provide privileged __printf() for error code
- Fix zipos opendir() so that it reports ENOTDIR
- Add basic chmod() implementation for Windows NT
- Add Cosmo's best functions to Python cosmo module
- Pin function trace indent depth to that of caller
- Show memory diagram on invalid access in MODE=dbg
- Differentiate stack overflow on crash in MODE=dbg
- Add stb_truetype and tools for analyzing font files
- Upgrade to UNICODE 13 and reduce its binary footprint
- COMPILE.COM now logs resource usage of build commands
- Start implementing basic poll() support on bare metal
- Set getauxval(AT_EXECFN) to GetModuleFileName() on NT
- Add descriptions to strerror() in non-TINY build modes
- Add COUNTBRANCH() macro to help with micro-optimizations
- Make error / backtrace / asan / memory code more unbreakable
- Add fast perfect C implementation of μ-Law and a-Law audio codecs
- Make strtol() functions consistent with other libc implementations
- Improve Linenoise implementation (see also github.com/jart/bestline)
- COMPILE.COM now suppresses stdout/stderr of successful build commands
This commit is contained in:
Justine Tunney 2021-09-27 22:58:51 -07:00
parent fa7b4f5bd1
commit 39bf41f4eb
806 changed files with 77494 additions and 63859 deletions
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341
third_party/python/Modules/_decimal/libmpdec/fourstep.c vendored
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@ -40,12 +40,261 @@ libmpdec (BSD-2)\\n\
Copyright 2008-2016 Stefan Krah\"");
asm(".include \"libc/disclaimer.inc\"");
/* Bignum: Cache efficient Matrix Fourier Transform for arrays of the
form 3 * 2**n (See literature/matrix-transform.txt). */
/*
Cache Efficient Matrix Fourier Transform
for arrays of form 3×2ⁿ
#ifndef PPRO
The Matrix Fourier Transform
════════════════════════════
In libmpdec, the Matrix Fourier Transform [1] is called four-step
transform after a variant that appears in [2]. The algorithm requires
that the input array can be viewed as an R*C matrix.
All operations are done modulo p. For readability, the proofs drop all
instances of (mod p).
Algorithm four-step (forward transform)
───────────────────────────────────────
a := input array
d := len(a) = R * C
p := prime
w := primitive root of unity of the prime field
r := w**((p-1)/d)
A := output array
1) Apply a length R FNT to each column.
2) Multiply each matrix element (addressed by j*C+m) by r**(j*m).
3) Apply a length C FNT to each row.
4) Transpose the matrix.
Proof (forward transform)
─────────────────────────
The algorithm can be derived starting from the regular definition of
the finite-field transform of length d:
d-1
,────
\
A[k] = | a[l] × r**(k × l)
/
`────
l = 0
The sum can be rearranged into the sum of the sums of columns:
C-1 R-1
,──── ,────
\ \
= | | a[i × C + j] × r**(k × (i × C + j))
/ /
`──── `────
j = 0 i = 0
Extracting a constant from the inner sum:
C-1 R-1
,──── ,────
\ \
= | rᵏ×j × | a[i × C + j] × r**(k × i × C)
/ /
`──── `────
j = 0 i = 0
Without any loss of generality, let k = n × R + m,
where n < C and m < R:
C-1 R-1
,──── ,────
\ \
A[n×R+m] = | r**(R×n×j) × r**(m×j) × | a[i×C+j] × r**(R×C×n×i) × r**(C×m×i)
/ /
`──── `────
j = 0 i = 0
Since r = w ** ((p-1) / (R×C)):
a) r**(R×C×n×i) = w**((p-1)×n×i) = 1
b) r**(C×m×i) = w**((p-1) / R) ** (m×i) = r_R ** (m×i)
c) r**(R×n×j) = w**((p-1) / C) ** (n×j) = r_C ** (n×j)
r_R := root of the subfield of length R.
r_C := root of the subfield of length C.
C-1 R-1
,──── ,────
\ \
A[n×R+m] = | r_C**(n×j) × [ r**(m×j) × | a[i×C+j] × r_R**(m×i) ]
/ ^ /
`──── | `──── 1) transform the columns
j = 0 | i = 0
^ |
| `-- 2) multiply
|
`-- 3) transform the rows
Note that the entire RHS is a function of n and m and that the results
for each pair (n, m) are stored in Fortran order.
Let the term in square brackets be 𝑓(m, j). Step 1) and 2) precalculate
the term for all (m, j). After that, the original matrix is now a lookup
table with the mth element in the jth column at location m × C + j.
Let the complete RHS be g(m, n). Step 3) does an in-place transform of
length n on all rows. After that, the original matrix is now a lookup
table with the mth element in the nth column at location m × C + n.
But each (m, n) pair should be written to location n × R + m. Therefore,
step 4) transposes the result of step 3).
Algorithm four-step (inverse transform)
───────────────────────────────────────
A := input array
d := len(A) = R × C
p := prime
d′ := d⁽ᵖ⁻²⁾ # inverse of d
w := primitive root of unity of the prime field
r := w**((p-1)/d) # root of the subfield
r′ := w**((p-1) - (p-1)/d) # inverse of r
a := output array
0) View the matrix as a C×R matrix.
1) Transpose the matrix, producing an R×C matrix.
2) Apply a length C FNT to each row.
3) Multiply each matrix element (addressed by i×C+n) by r**(i×n).
4) Apply a length R FNT to each column.
Proof (inverse transform)
─────────────────────────
The algorithm can be derived starting from the regular definition of
the finite-field inverse transform of length d:
d-1
,────
\
a[k] = d′ × | A[l] × r′ ** (k × l)
/
`────
l = 0
The sum can be rearranged into the sum of the sums of columns. Note
that at this stage we still have a C*R matrix, so C denotes the number
of rows:
R-1 C-1
,──── ,────
\ \
= d′ × | | a[j × R + i] × r′ ** (k × (j × R + i))
/ /
`──── `────
i = 0 j = 0
Extracting a constant from the inner sum:
R-1 C-1
,──── ,────
\ \
= d′ × | r′ ** (k×i) × | a[j × R + i] × r′ ** (k × j × R)
/ /
`──── `────
i = 0 j = 0
Without any loss of generality, let k = m * C + n,
where m < R and n < C:
R-1 C-1
,──── ,────
\ \
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)
/ /
`──── `────
i = 0 j = 0
Since r′ = w**((p-1) - (p-1)/d) and d = R×C:
a) r′ ** (R×C×m×j) = w**((p-1)×R×C×m×j - (p-1)×m×j) = 1
b) r′ ** (C×m×i) = w**((p-1)×C - (p-1)/R) ** (m×i) = r_R′ ** (m×i)
c) r′ ** (R×n×j) = r_C′ ** (n×j)
d) d′ = d⁽ᵖ⁻²⁾ = (R×C)⁽ᵖ⁻²⁾ = R⁽ᵖ⁻²⁾ × C⁽ᵖ⁻²⁾ = R′ × C′
r_R′ := inverse of the root of the subfield of length R.
r_C′ := inverse of the root of the subfield of length C.
R′ := inverse of R
C′ := inverse of C
R-1 C-1
,──── ,──── 2) transform the rows of a^T
\ \
A[m×C+n] = R′ × | r_R′ ** (m×i) × [ r′ ** (n×i) × C′ × | a[j×R+i] × r_C′ ** (n×j) ]
/ ^ / ^
`──── | `──── |
i = 0 | j = 0 |
^ | `── 1) Transpose input matrix
| `── 3) multiply to address elements by
| i × C + j
`── 3) transform the columns
Note that the entire RHS is a function of m and n and that the results
for each pair (m, n) are stored in C order.
Let the term in square brackets be 𝑓(n, i). Without step 1), the sum
would perform a length C transform on the columns of the input matrix.
This is a) inefficient and b) the results are needed in C order, so
step 1) exchanges rows and columns.
Step 2) and 3) precalculate 𝑓(n, i) for all (n, i). After that, the
original matrix is now a lookup table with the ith element in the nth
column at location i × C + n.
Let the complete RHS be g(m, n). Step 4) does an in-place transform of
length m on all columns. After that, the original matrix is now a lookup
table with the mth element in the nth column at location m × C + n,
which means that all A[k] = A[m × C + n] are in the correct order.
──
[1] Joerg Arndt: "Matters Computational"
http://www.jjj.de/fxt/
[2] David H. Bailey: FFTs in External or Hierarchical Memory
http://crd.lbl.gov/~dhbailey/dhbpapers/
*/
static inline void
std_size3_ntt(mpd_uint_t *x1, mpd_uint_t *x2, mpd_uint_t *x3,
mpd_uint_t w3table[3], mpd_uint_t umod)
@ -53,90 +302,32 @@ std_size3_ntt(mpd_uint_t *x1, mpd_uint_t *x2, mpd_uint_t *x3,
mpd_uint_t r1, r2;
mpd_uint_t w;
mpd_uint_t s, tmp;
/* k = 0 -> w = 1 */
s = *x1;
s = addmod(s, *x2, umod);
s = addmod(s, *x3, umod);
r1 = s;
/* k = 1 */
s = *x1;
w = w3table[1];
tmp = MULMOD(*x2, w);
s = addmod(s, tmp, umod);
w = w3table[2];
tmp = MULMOD(*x3, w);
s = addmod(s, tmp, umod);
r2 = s;
/* k = 2 */
s = *x1;
w = w3table[2];
tmp = MULMOD(*x2, w);
s = addmod(s, tmp, umod);
w = w3table[1];
tmp = MULMOD(*x3, w);
s = addmod(s, tmp, umod);
*x3 = s;
*x2 = r2;
*x1 = r1;
}
#else /* PPRO */
static inline void
ppro_size3_ntt(mpd_uint_t *x1, mpd_uint_t *x2, mpd_uint_t *x3, mpd_uint_t w3table[3],
mpd_uint_t umod, double *dmod, uint32_t dinvmod[3])
{
mpd_uint_t r1, r2;
mpd_uint_t w;
mpd_uint_t s, tmp;
/* k = 0 -> w = 1 */
s = *x1;
s = addmod(s, *x2, umod);
s = addmod(s, *x3, umod);
r1 = s;
/* k = 1 */
s = *x1;
w = w3table[1];
tmp = ppro_mulmod(*x2, w, dmod, dinvmod);
s = addmod(s, tmp, umod);
w = w3table[2];
tmp = ppro_mulmod(*x3, w, dmod, dinvmod);
s = addmod(s, tmp, umod);
r2 = s;
/* k = 2 */
s = *x1;
w = w3table[2];
tmp = ppro_mulmod(*x2, w, dmod, dinvmod);
s = addmod(s, tmp, umod);
w = w3table[1];
tmp = ppro_mulmod(*x3, w, dmod, dinvmod);
s = addmod(s, tmp, umod);
*x3 = s;
*x2 = r2;
*x1 = r1;
}
#endif
/* forward transform, sign = -1; transform length = 3 * 2**n */
int
@ -148,25 +339,15 @@ four_step_fnt(mpd_uint_t *a, mpd_size_t n, int modnum)
mpd_uint_t kernel, w0, w1, wstep;
mpd_uint_t *s, *p0, *p1, *p2;
mpd_uint_t umod;
#ifdef PPRO
double dmod;
uint32_t dinvmod[3];
#endif
mpd_size_t i, k;
assert(n >= 48);
assert(n <= 3*MPD_MAXTRANSFORM_2N);
/* Length R transform on the columns. */
SETMODULUS(modnum);
_mpd_init_w3table(w3table, -1, modnum);
for (p0=a, p1=p0+C, p2=p0+2*C; p0<a+C; p0++,p1++,p2++) {
SIZE3_NTT(p0, p1, p2, w3table);
}
/* Multiply each matrix element (addressed by i*C+k) by r**(i*k). */
kernel = _mpd_getkernel(n, -1, modnum);
for (i = 1; i < R; i++) {
@ -182,20 +363,17 @@ four_step_fnt(mpd_uint_t *a, mpd_size_t n, int modnum)
a[i*C+k+1] = x1;
}
}
/* Length C transform on the rows. */
for (s = a; s < a+n; s += C) {
if (!six_step_fnt(s, C, modnum)) {
return 0;
}
}
#if 0
/* An unordered transform is sufficient for convolution. */
/* Transpose the matrix. */
transpose_3xpow2(a, R, C);
#endif
return 1;
}
@ -209,30 +387,20 @@ inv_four_step_fnt(mpd_uint_t *a, mpd_size_t n, int modnum)
mpd_uint_t kernel, w0, w1, wstep;
mpd_uint_t *s, *p0, *p1, *p2;
mpd_uint_t umod;
#ifdef PPRO
double dmod;
uint32_t dinvmod[3];
#endif
mpd_size_t i, k;
assert(n >= 48);
assert(n <= 3*MPD_MAXTRANSFORM_2N);
#if 0
/* An unordered transform is sufficient for convolution. */
/* Transpose the matrix, producing an R*C matrix. */
transpose_3xpow2(a, C, R);
#endif
/* Length C transform on the rows. */
for (s = a; s < a+n; s += C) {
if (!inv_six_step_fnt(s, C, modnum)) {
return 0;
}
}
/* Multiply each matrix element (addressed by i*C+k) by r**(i*k). */
SETMODULUS(modnum);
kernel = _mpd_getkernel(n, 1, modnum);
@ -249,13 +417,10 @@ inv_four_step_fnt(mpd_uint_t *a, mpd_size_t n, int modnum)
a[i*C+k+1] = x1;
}
}
/* Length R transform on the columns. */
_mpd_init_w3table(w3table, 1, modnum);
for (p0=a, p1=p0+C, p2=p0+2*C; p0<a+C; p0++,p1++,p2++) {
SIZE3_NTT(p0, p1, p2, w3table);
}
return 1;
}