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/* cairo - a vector graphics library with display and print output
*
* Copyright © 2004 Keith Packard
* This library is free software; you can redistribute it and/or
* modify it either under the terms of the GNU Lesser General Public
* License version 2.1 as published by the Free Software Foundation
* (the "LGPL") or, at your option, under the terms of the Mozilla
* Public License Version 1.1 (the "MPL"). If you do not alter this
* notice, a recipient may use your version of this file under either
* the MPL or the LGPL.
* You should have received a copy of the LGPL along with this library
* in the file COPYING-LGPL-2.1; if not, write to the Free Software
* Foundation, Inc., 51 Franklin Street, Suite 500, Boston, MA 02110-1335, USA
* You should have received a copy of the MPL along with this library
* in the file COPYING-MPL-1.1
* The contents of this file are subject to the Mozilla Public License
* Version 1.1 (the "License"); you may not use this file except in
* compliance with the License. You may obtain a copy of the License at
* http://www.mozilla.org/MPL/
* This software is distributed on an "AS IS" basis, WITHOUT WARRANTY
* OF ANY KIND, either express or implied. See the LGPL or the MPL for
* the specific language governing rights and limitations.
* The Original Code is the cairo graphics library.
* The Initial Developer of the Original Code is Keith Packard
* Contributor(s):
* Keith R. Packard <keithp@keithp.com>
*/
#include "cairoint.h"
#if HAVE_UINT64_T
#define uint64_lo32(i) ((i) & 0xffffffff)
#define uint64_hi32(i) ((i) >> 32)
#define uint64_lo(i) ((i) & 0xffffffff)
#define uint64_hi(i) ((i) >> 32)
#define uint64_shift32(i) ((i) << 32)
#define uint64_carry32 (((uint64_t) 1) << 32)
#define _cairo_uint32s_to_uint64(h,l) ((uint64_t) (h) << 32 | (l))
#else
#define uint64_lo32(i) ((i).lo)
#define uint64_hi32(i) ((i).hi)
static cairo_uint64_t
uint64_lo (cairo_uint64_t i)
{
cairo_uint64_t s;
s.lo = i.lo;
s.hi = 0;
return s;
}
uint64_hi (cairo_uint64_t i)
s.lo = i.hi;
uint64_shift32 (cairo_uint64_t i)
s.lo = 0;
s.hi = i.lo;
static const cairo_uint64_t uint64_carry32 = { 0, 1 };
cairo_uint64_t
_cairo_double_to_uint64 (double i)
cairo_uint64_t q;
q.hi = i * (1. / 4294967296.);
q.lo = i - q.hi * 4294967296.;
return q;
double
_cairo_uint64_to_double (cairo_uint64_t i)
return i.hi * 4294967296. + i.lo;
cairo_int64_t
_cairo_double_to_int64 (double i)
q.hi = i * (1. / INT32_MAX);
q.lo = i - q.hi * (double)INT32_MAX;
_cairo_int64_to_double (cairo_int64_t i)
return i.hi * INT32_MAX + i.lo;
_cairo_uint32_to_uint64 (uint32_t i)
q.lo = i;
q.hi = 0;
_cairo_int32_to_int64 (int32_t i)
q.hi = i < 0 ? -1 : 0;
_cairo_uint32s_to_uint64 (uint32_t h, uint32_t l)
q.lo = l;
q.hi = h;
_cairo_uint64_add (cairo_uint64_t a, cairo_uint64_t b)
s.hi = a.hi + b.hi;
s.lo = a.lo + b.lo;
if (s.lo < a.lo)
s.hi++;
_cairo_uint64_sub (cairo_uint64_t a, cairo_uint64_t b)
s.hi = a.hi - b.hi;
s.lo = a.lo - b.lo;
if (s.lo > a.lo)
s.hi--;
#define uint32_lo(i) ((i) & 0xffff)
#define uint32_hi(i) ((i) >> 16)
#define uint32_carry16 ((1) << 16)
_cairo_uint32x32_64_mul (uint32_t a, uint32_t b)
uint16_t ah, al, bh, bl;
uint32_t r0, r1, r2, r3;
al = uint32_lo (a);
ah = uint32_hi (a);
bl = uint32_lo (b);
bh = uint32_hi (b);
r0 = (uint32_t) al * bl;
r1 = (uint32_t) al * bh;
r2 = (uint32_t) ah * bl;
r3 = (uint32_t) ah * bh;
r1 += uint32_hi(r0); /* no carry possible */
r1 += r2; /* but this can carry */
if (r1 < r2) /* check */
r3 += uint32_carry16;
s.hi = r3 + uint32_hi(r1);
s.lo = (uint32_lo (r1) << 16) + uint32_lo (r0);
_cairo_int32x32_64_mul (int32_t a, int32_t b)
cairo_int64_t s;
s = _cairo_uint32x32_64_mul ((uint32_t) a, (uint32_t) b);
if (a < 0)
s.hi -= b;
if (b < 0)
s.hi -= a;
_cairo_uint64_mul (cairo_uint64_t a, cairo_uint64_t b)
s = _cairo_uint32x32_64_mul (a.lo, b.lo);
s.hi += a.lo * b.hi + a.hi * b.lo;
_cairo_uint64_lsl (cairo_uint64_t a, int shift)
if (shift >= 32)
a.hi = a.lo;
a.lo = 0;
shift -= 32;
if (shift)
a.hi = a.hi << shift | a.lo >> (32 - shift);
a.lo = a.lo << shift;
return a;
_cairo_uint64_rsl (cairo_uint64_t a, int shift)
a.lo = a.hi;
a.hi = 0;
a.lo = a.lo >> shift | a.hi << (32 - shift);
a.hi = a.hi >> shift;
#define _cairo_uint32_rsa(a,n) ((uint32_t) (((int32_t) (a)) >> (n)))
_cairo_uint64_rsa (cairo_int64_t a, int shift)
a.hi = _cairo_uint32_rsa (a.hi, 31);
a.hi = _cairo_uint32_rsa (a.hi, shift);
int
_cairo_uint64_lt (cairo_uint64_t a, cairo_uint64_t b)
return (a.hi < b.hi ||
(a.hi == b.hi && a.lo < b.lo));
_cairo_uint64_eq (cairo_uint64_t a, cairo_uint64_t b)
return a.hi == b.hi && a.lo == b.lo;
_cairo_int64_lt (cairo_int64_t a, cairo_int64_t b)
if (_cairo_int64_negative (a) && !_cairo_int64_negative (b))
return 1;
if (!_cairo_int64_negative (a) && _cairo_int64_negative (b))
return 0;
return _cairo_uint64_lt (a, b);
_cairo_uint64_cmp (cairo_uint64_t a, cairo_uint64_t b)
if (a.hi < b.hi)
return -1;
else if (a.hi > b.hi)
else if (a.lo < b.lo)
else if (a.lo > b.lo)
else
_cairo_int64_cmp (cairo_int64_t a, cairo_int64_t b)
return _cairo_uint64_cmp (a, b);
_cairo_uint64_not (cairo_uint64_t a)
a.lo = ~a.lo;
a.hi = ~a.hi;
_cairo_uint64_negate (cairo_uint64_t a)
if (++a.lo == 0)
++a.hi;
/*
* Simple bit-at-a-time divide.
cairo_uquorem64_t
_cairo_uint64_divrem (cairo_uint64_t num, cairo_uint64_t den)
cairo_uquorem64_t qr;
cairo_uint64_t bit;
cairo_uint64_t quo;
bit = _cairo_uint32_to_uint64 (1);
/* normalize to make den >= num, but not overflow */
while (_cairo_uint64_lt (den, num) && (den.hi & 0x80000000) == 0)
bit = _cairo_uint64_lsl (bit, 1);
den = _cairo_uint64_lsl (den, 1);
quo = _cairo_uint32_to_uint64 (0);
/* generate quotient, one bit at a time */
while (bit.hi | bit.lo)
if (_cairo_uint64_le (den, num))
num = _cairo_uint64_sub (num, den);
quo = _cairo_uint64_add (quo, bit);
bit = _cairo_uint64_rsl (bit, 1);
den = _cairo_uint64_rsl (den, 1);
qr.quo = quo;
qr.rem = num;
return qr;
#endif /* !HAVE_UINT64_T */
#if HAVE_UINT128_T
cairo_uquorem128_t
_cairo_uint128_divrem (cairo_uint128_t num, cairo_uint128_t den)
cairo_uquorem128_t qr;
qr.quo = num / den;
qr.rem = num % den;
cairo_uint128_t
_cairo_uint32_to_uint128 (uint32_t i)
cairo_uint128_t q;
q.lo = _cairo_uint32_to_uint64 (i);
q.hi = _cairo_uint32_to_uint64 (0);
cairo_int128_t
_cairo_int32_to_int128 (int32_t i)
cairo_int128_t q;
q.lo = _cairo_int32_to_int64 (i);
q.hi = _cairo_int32_to_int64 (i < 0 ? -1 : 0);
_cairo_uint64_to_uint128 (cairo_uint64_t i)
_cairo_int64_to_int128 (cairo_int64_t i)
q.hi = _cairo_int32_to_int64 (_cairo_int64_negative(i) ? -1 : 0);
_cairo_uint128_add (cairo_uint128_t a, cairo_uint128_t b)
cairo_uint128_t s;
s.hi = _cairo_uint64_add (a.hi, b.hi);
s.lo = _cairo_uint64_add (a.lo, b.lo);
if (_cairo_uint64_lt (s.lo, a.lo))
s.hi = _cairo_uint64_add (s.hi, _cairo_uint32_to_uint64 (1));
_cairo_uint128_sub (cairo_uint128_t a, cairo_uint128_t b)
s.hi = _cairo_uint64_sub (a.hi, b.hi);
s.lo = _cairo_uint64_sub (a.lo, b.lo);
if (_cairo_uint64_gt (s.lo, a.lo))
s.hi = _cairo_uint64_sub (s.hi, _cairo_uint32_to_uint64(1));
_cairo_uint64x64_128_mul (cairo_uint64_t a, cairo_uint64_t b)
uint32_t ah, al, bh, bl;
cairo_uint64_t r0, r1, r2, r3;
al = uint64_lo32 (a);
ah = uint64_hi32 (a);
bl = uint64_lo32 (b);
bh = uint64_hi32 (b);
r0 = _cairo_uint32x32_64_mul (al, bl);
r1 = _cairo_uint32x32_64_mul (al, bh);
r2 = _cairo_uint32x32_64_mul (ah, bl);
r3 = _cairo_uint32x32_64_mul (ah, bh);
r1 = _cairo_uint64_add (r1, uint64_hi (r0)); /* no carry possible */
r1 = _cairo_uint64_add (r1, r2); /* but this can carry */
if (_cairo_uint64_lt (r1, r2)) /* check */
r3 = _cairo_uint64_add (r3, uint64_carry32);
s.hi = _cairo_uint64_add (r3, uint64_hi(r1));
s.lo = _cairo_uint64_add (uint64_shift32 (r1),
uint64_lo (r0));
_cairo_int64x64_128_mul (cairo_int64_t a, cairo_int64_t b)
cairo_int128_t s;
s = _cairo_uint64x64_128_mul (_cairo_int64_to_uint64(a),
_cairo_int64_to_uint64(b));
if (_cairo_int64_negative (a))
s.hi = _cairo_uint64_sub (s.hi,
_cairo_int64_to_uint64 (b));
if (_cairo_int64_negative (b))
_cairo_int64_to_uint64 (a));
_cairo_uint128_mul (cairo_uint128_t a, cairo_uint128_t b)
s = _cairo_uint64x64_128_mul (a.lo, b.lo);
s.hi = _cairo_uint64_add (s.hi,
_cairo_uint64_mul (a.lo, b.hi));
_cairo_uint64_mul (a.hi, b.lo));
_cairo_uint128_lsl (cairo_uint128_t a, int shift)
if (shift >= 64)
a.lo = _cairo_uint32_to_uint64 (0);
shift -= 64;
a.hi = _cairo_uint64_add (_cairo_uint64_lsl (a.hi, shift),
_cairo_uint64_rsl (a.lo, (64 - shift)));
a.lo = _cairo_uint64_lsl (a.lo, shift);
_cairo_uint128_rsl (cairo_uint128_t a, int shift)
a.hi = _cairo_uint32_to_uint64 (0);
a.lo = _cairo_uint64_add (_cairo_uint64_rsl (a.lo, shift),
_cairo_uint64_lsl (a.hi, (64 - shift)));
a.hi = _cairo_uint64_rsl (a.hi, shift);
_cairo_uint128_rsa (cairo_int128_t a, int shift)
a.hi = _cairo_uint64_rsa (a.hi, 64-1);
a.hi = _cairo_uint64_rsa (a.hi, shift);
_cairo_uint128_lt (cairo_uint128_t a, cairo_uint128_t b)
return (_cairo_uint64_lt (a.hi, b.hi) ||
(_cairo_uint64_eq (a.hi, b.hi) &&
_cairo_uint64_lt (a.lo, b.lo)));
_cairo_int128_lt (cairo_int128_t a, cairo_int128_t b)
if (_cairo_int128_negative (a) && !_cairo_int128_negative (b))
if (!_cairo_int128_negative (a) && _cairo_int128_negative (b))
return _cairo_uint128_lt (a, b);
_cairo_uint128_cmp (cairo_uint128_t a, cairo_uint128_t b)
int cmp;
cmp = _cairo_uint64_cmp (a.hi, b.hi);
if (cmp)
return cmp;
return _cairo_uint64_cmp (a.lo, b.lo);
_cairo_int128_cmp (cairo_int128_t a, cairo_int128_t b)
return _cairo_uint128_cmp (a, b);
_cairo_uint128_eq (cairo_uint128_t a, cairo_uint128_t b)
return (_cairo_uint64_eq (a.hi, b.hi) &&
_cairo_uint64_eq (a.lo, b.lo));
#define _cairo_msbset64(q) (q & ((uint64_t) 1 << 63))
#define _cairo_msbset64(q) (q.hi & ((uint32_t) 1 << 31))
#endif
cairo_uint128_t bit;
cairo_uint128_t quo;
bit = _cairo_uint32_to_uint128 (1);
while (_cairo_uint128_lt (den, num) && !_cairo_msbset64(den.hi))
bit = _cairo_uint128_lsl (bit, 1);
den = _cairo_uint128_lsl (den, 1);
quo = _cairo_uint32_to_uint128 (0);
while (_cairo_uint128_ne (bit, _cairo_uint32_to_uint128(0)))
if (_cairo_uint128_le (den, num))
num = _cairo_uint128_sub (num, den);
quo = _cairo_uint128_add (quo, bit);
bit = _cairo_uint128_rsl (bit, 1);
den = _cairo_uint128_rsl (den, 1);
_cairo_uint128_negate (cairo_uint128_t a)
a.lo = _cairo_uint64_not (a.lo);
a.hi = _cairo_uint64_not (a.hi);
return _cairo_uint128_add (a, _cairo_uint32_to_uint128 (1));
_cairo_uint128_not (cairo_uint128_t a)
#endif /* !HAVE_UINT128_T */
cairo_quorem128_t
_cairo_int128_divrem (cairo_int128_t num, cairo_int128_t den)
int num_neg = _cairo_int128_negative (num);
int den_neg = _cairo_int128_negative (den);
cairo_uquorem128_t uqr;
cairo_quorem128_t qr;
if (num_neg)
num = _cairo_int128_negate (num);
if (den_neg)
den = _cairo_int128_negate (den);
uqr = _cairo_uint128_divrem (num, den);
qr.rem = _cairo_int128_negate (uqr.rem);
qr.rem = uqr.rem;
if (num_neg != den_neg)
qr.quo = _cairo_int128_negate (uqr.quo);
qr.quo = uqr.quo;
/**
* _cairo_uint_96by64_32x64_divrem:
* Compute a 32 bit quotient and 64 bit remainder of a 96 bit unsigned
* dividend and 64 bit divisor. If the quotient doesn't fit into 32
* bits then the returned remainder is equal to the divisor, and the
* quotient is the largest representable 64 bit integer. It is an
* error to call this function with the high 32 bits of @num being
* non-zero.
**/
_cairo_uint_96by64_32x64_divrem (cairo_uint128_t num,
cairo_uint64_t den)
cairo_uquorem64_t result;
cairo_uint64_t B = _cairo_uint32s_to_uint64 (1, 0);
/* These are the high 64 bits of the *96* bit numerator. We're
* going to represent the numerator as xB + y, where x is a 64,
* and y is a 32 bit number. */
cairo_uint64_t x = _cairo_uint128_to_uint64 (_cairo_uint128_rsl(num, 32));
/* Initialise the result to indicate overflow. */
result.quo = _cairo_uint32s_to_uint64 (-1U, -1U);
result.rem = den;
/* Don't bother if the quotient is going to overflow. */
if (_cairo_uint64_ge (x, den)) {
return /* overflow */ result;
if (_cairo_uint64_lt (x, B)) {
/* When the final quotient is known to fit in 32 bits, then
* num < 2^64 if and only if den < 2^32. */
return _cairo_uint64_divrem (_cairo_uint128_to_uint64 (num), den);
else {
/* Denominator is >= 2^32. the numerator is >= 2^64, and the
* division won't overflow: need two divrems. Write the
* numerator and denominator as
* num = xB + y x : 64 bits, y : 32 bits
* den = uB + v u, v : 32 bits
uint32_t y = _cairo_uint128_to_uint32 (num);
uint32_t u = uint64_hi32 (den);
uint32_t v = _cairo_uint64_to_uint32 (den);
/* Compute a lower bound approximate quotient of num/den
* from x/(u+1). Then we have
* x = q(u+1) + r ; q : 32 bits, r <= u : 32 bits.
* xB + y = q(u+1)B + (rB+y)
* = q(uB + B + v - v) + (rB+y)
* = q(uB + v) + qB - qv + (rB+y)
* = q(uB + v) + q(B-v) + (rB+y)
* The true quotient of num/den then is q plus the
* contribution of q(B-v) + (rB+y). The main contribution
* comes from the term q(B-v), with the term (rB+y) only
* contributing at most one part.
* The term q(B-v) must fit into 64 bits, since q fits into 32
* bits on account of being a lower bound to the true
* quotient, and as B-v <= 2^32, we may safely use a single
* 64/64 bit division to find its contribution. */
cairo_uquorem64_t quorem;
cairo_uint64_t remainder; /* will contain final remainder */
uint32_t quotient; /* will contain final quotient. */
uint32_t q;
uint32_t r;
/* Approximate quotient by dividing the high 64 bits of num by
* u+1. Watch out for overflow of u+1. */
if (u+1) {
quorem = _cairo_uint64_divrem (x, _cairo_uint32_to_uint64 (u+1));
q = _cairo_uint64_to_uint32 (quorem.quo);
r = _cairo_uint64_to_uint32 (quorem.rem);
q = uint64_hi32 (x);
r = _cairo_uint64_to_uint32 (x);
quotient = q;
/* Add the main term's contribution to quotient. Note B-v =
* -v as an uint32 (unless v = 0) */
if (v)
quorem = _cairo_uint64_divrem (_cairo_uint32x32_64_mul (q, -v), den);
quorem = _cairo_uint64_divrem (_cairo_uint32s_to_uint64 (q, 0), den);
quotient += _cairo_uint64_to_uint32 (quorem.quo);
/* Add the contribution of the subterm and start computing the
* true remainder. */
remainder = _cairo_uint32s_to_uint64 (r, y);
if (_cairo_uint64_ge (remainder, den)) {
remainder = _cairo_uint64_sub (remainder, den);
quotient++;
/* Add the contribution of the main term's remainder. The
* funky test here checks that remainder + main_rem >= den,
* taking into account overflow of the addition. */
remainder = _cairo_uint64_add (remainder, quorem.rem);
if (_cairo_uint64_ge (remainder, den) ||
_cairo_uint64_lt (remainder, quorem.rem))
result.quo = _cairo_uint32_to_uint64 (quotient);
result.rem = remainder;
return result;
cairo_quorem64_t
_cairo_int_96by64_32x64_divrem (cairo_int128_t num, cairo_int64_t den)
int den_neg = _cairo_int64_negative (den);
cairo_uint64_t nonneg_den;
cairo_uquorem64_t uqr;
cairo_quorem64_t qr;
nonneg_den = _cairo_int64_negate (den);
nonneg_den = den;
uqr = _cairo_uint_96by64_32x64_divrem (num, nonneg_den);
if (_cairo_uint64_eq (uqr.rem, nonneg_den)) {
/* bail on overflow. */
qr.quo = _cairo_uint32s_to_uint64 (0x7FFFFFFF, -1U);
qr.rem = den;
qr.rem = _cairo_int64_negate (uqr.rem);
qr.quo = _cairo_int64_negate (uqr.quo);